Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

Yoshiyuki Suzuki and Ryoichiro Minai

Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611, Japan

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the It6 stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single time) statistics or probability distributions.

INTRODUCTION

Seismic design of building structures is based on the common design philosophy that for strong or violent earthquakes the building structures have to survive without collapse nor serious structural damage though inelastic behaviour is permitted. Hence the safety of structures should be guaranteed through dynamic response analyses of structures. Since there are, however, many uncertainties and randomness in supposing seismic excitations in the future and also in modelling the structural systems and their associated dynamic failure criteria, the seismic response analysis from the probabilistic point of view, in other words, the seismic reliability analysis of structures is needed. This seismic reliability analysis has the following significant features: The measures of seismic safety or serviceability are directly defined in terms of seismic responses, and the safety or serviceability is quantified in terms of probability by taking into account of the various kind of uncertainties and randomness.

The definite method of seismic reliability analysis and the reasonable seismic design procedure for hysteretic structures have not yet been obtained to date with consideration of the nonstationarities both in amplitude and spectral characteristics and various types of hysteretic characteristics and their associated dynamic failure criteria depending on the structural materials and structural types. Under these circumstances, it might be primarily necessary to obtain mathematical formulations of appropriate dynamical models for seismic excitations and dynamic failure criteria of structures or their structural components and to develop a method for seismic reliability analysis.

There has been an increasing interest in response analysis of hysteretic structures under random excitation,

since in early 1960's Caughey' proposed the statistical linearization technique of hystereses in terms of deformation and its time derivative. This approximate method and its modified ones have been successfully applied to a limited class of weakly nonlinear or stable hysteretic systems because no plastic drift of the centre of hysteresis was assumed. The differential formulations of hysteretic constitutive laws for a class of piecewise-linear and curved hysteresis were proposed 2-5. By using these differential formulations, the approximate methods of stochastic response analysis of hysteretic structures based on theory of Markov vector process, especially the Fokker-Planck equation approaches were developed. Since the hysteretic constitutive laws are mathematically expressed in the first order ordinary differential equations including single-valued nonlinear functions of the relevant state variables, the stochastic linearization method 6 which has been developed for nonhysteretic dynamical systems can be effectively applied to the seismic response analyses of hysteretic structures.

As regards the seismic reliability of hysteretic structures when considering the absolute maximum ductility ratio as the seismic safety measure, the reliability analysis of structures could be in the issue the problem of determining the probability distribution of the maximum ductility ratio or the first passage problem of the displacement response measure. A number of approximate methods have been explored because of difficulties of solution to the first passage problem. Especially, are widely known the level crossing methods 7 9 and the direct method solving the initial- and boundary- value problems of the Fokker-Planck-Kolmogorov equations '°-12. These methods have been applied effectively to the reliability analyses of linear dynamical

© 1988 Computational Mechanics Publication Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 1 43

Application of stochastic d([ferential equations: K Suzuki and R. Minai

systems and a special class of nonlinear dynamical systems. However, the application of these methods to the reliability analysis of hysteretic structures subjected to nonstationary seismic excitations gives rise to m~thematical difficulties, otherwise it needs troublesome analytical and numerical schemes. On the other hand, failure due to cumulative damages such as cumulative plastic deformation, dissipated hysteretic energy and low- cycle fatigue becomes important, as far as structures are subjected to repeated loadings like earthquakes. A large number of researchers have investigated these cumulative damage, in particular, fatigue damages. However, there remains the problem of determining the time-dependent probability density function as well as the statistics of cumulative damage when considering the reliability analysis of hysteretic structures under nonstationary seismic excitations. The application of level crossing methods to the above problem might lead to complex and troublesome analysis for determining multi-time statistics.

In this paper, an analytical method of seismic reliability analysis of hysteretic structures subjected to nonstationary seismic excitations is described mainly from the methodological aspect. This is the method based on stochastic differential equations through the theory of continuous Markov vector process. The aims of this method are in unifying the conventional stochastic response analysis and the reliability analysis by making use of differential representations of hysteretic constitutive laws and seismic safety measures, and in directly evaluating seismic reliability functions without recourse to the Fokker-Planck-Kolmogorov equation approach or the level crossing methods.

probability density function p(Z,t] K r) satisfies the Fokker-Planck equation or Kolmogorov's forward equation

~t = L*p = ~ [2\i?Z (Vp) - F(Z, t)p (3)

with initial condition

limp(Z,t] K~)=b(Z-Y) (4) f . T

where 6(- ) is the Dirac delta function. As will be shown in the following sections, the function F includes nondifferentiable nonlinear functions such as the unit step function and signam function. Hence, equation (3) is not interpreted in the strict sense as a parabolic partial differential equation, but it is only in the formal sense. Even if the Fokker-Planck equation is valid, the solution to equation (3) may not generally be simple to obtain analytically except for linear dynamical systems or special cases of nonlinear dynamical systems.

Now assume that ~, is a scalar-valued real function of Z and t which has continuous first and second partial derivatives with respect to Z and is continuously differentiable in t. Then ~ satisfies the stochastic differential equation

t5)

where the differential operator L is the formal adjoint of the forward diffusion operator L*, namely

STOCHASTIC DIFFERENTIAL EQUATION APPROACH

Basic concepts which are required for the application of stochastic differential equations to the seismic reliability analysis of hysteretic structures are presented. First, consider a whole nonlinear dynamical system which is composed of dynamical systems generating nonstationary seismic excitations, structural systems having hysteretic characteristics and output systems concerning seismic safety of structural components. It is supposed here that the whole system is written in the general form of the vector It6 stochastic differential equation

dZ( t ) = F(Z , t ) d t+V (Z , t ) dB ( t ) Zt, ,_o=Zo (1)

where t is time, Z is the state vector describing the whole system, F(Z, t) and V(Z, t) are, respectively, a vector and matrix of nonlinear functions of Z and t, and B(t) is Brownian motion or Wiener process with zero-mean and diffusion matrix Q, namely

E [ dB] = 0 and E[dBdB / ] = Q d t (2)

where E l ' ] denotes the expectation operator, and superscript T means the transpose of a vector or matrix.

It is generally known that the state vector Z(t) which is the solution to the It6 stochastic differential equation (1) is a Markov vector process and is completely determined by the transition probability density function p(Z, t I Y, z) of Z(t) at time t, given Zf f )= Y for t> ~. If there exist the continuous partial derivatives ?.p/i?t, ?,[pF]/~Z and ?2[ pF]/~Z 2 where F---FT= VQV 7, the transition

?Z (6)

This equation is known as It6's formula [e.g., Ref. 13]. Specifically, if ~, is an explicit function of only Z, then the differential equation for the expectation E[~p(Z)] is derived directly from It6's formula

d a-7 E[*] = E[L ] (7)

For the sake of simplicity, it is assumed that r = t o = 0 and Y=Z0=0 . The expectation operator E can be defined by the probability density function p(Z, t I O, O)= p(Z, t). By making use of equation (7), a set of the first order nonlinear differential equations for the moments of Z(t) can be derived. Unlike the cases of linear systems, the probability density function p(Z,t) for the nonlinear system to be called to account here is in general non- Gaussian, and may not be known exactly. The moment equations form generally an infinite hierarchy system of coupled equations. Hence appropriate closure schemes are required to reduce the infinite moment equations to a definite set of equations. By assuming an appropriate analytical form of joint probability density function in which coefficients are determined in terms of moments, the truncated moment equations will be numerically solved under nonstationary state. Then the statistics and the joint probability density function of Z can be determined.

44 Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 1

Application of stochastic differential equations: Y. Suzuki and R. Minai

In order to apply the above-mentioned stochastic differential equation approach to the seismic reliability analysis of hysteretic structures, it is necessary to formulate the state space equation of the whole dynamical system in the form of the stochastic differential equation (1). Since Gaussian white noise is the formal derivative of Brownian motion, equation (1) can be written by the first order vector differential equation

Z=F'(Z , t)+ V(Z, t)~(t) Z,=o = Z 0 (8)

where

1[ ( ~ \ t 7 T vT z ) Qv J (9)

the overdot denotes time derivative, and ~(t) is a Gaussian white noise vector process with zero-mean and intensity matrix Q. Equation (8) is in the form of a dynamical system driven only by Gaussian white noise. Hence in order to achieve the formulation of reliability analysis, it suffices to derive the same form of equation (8) instead of (1) as the governing equations of the whole system composed of sub-systems generating seismic excitations, hysteretic systems and sub-systems concerning seismic safety measures. For this purpose seismic excitations are modelled as the output of dynamical systems driven by Gaussian white noise. And, the hysteretic constitutive laws of structural systems and the seismic safety measures are described in the form of first order ordinary differential equations in terms of the relevant state variables.

DIFFERENTIAL FORMULATIONS OF HYSTERETIC CONSTITUTIVE LAWS

On the basis of experimental as well as theoretical studies of hysteretic behaviour of structural systems subjected to repeated loading processes during earthquakes, a large number of mathematical models of hysteretic characteristics have been presented without and with degrading or stiffening characteristics. The general constitutive laws of these hysteretic models cannot be, in nature, expressed only in terms of the relevant deformation and its time-derivative. In order to furnish a mathematically well-defined representation of hysteretic constitutive law, a complete set of state variables should be defined by introducing a new state vector in addition to the conventional state vector consisting of the relevant deformation and its time-derivative. Then, the hysteretic constitutive law is derived in the form of the first order nonlinear ordinary differential equations with only single-valued functions of the state variables. The hysteretic characteristics considered herein are mainly a general class of piecewise-linear hystereses such as the bilinear model, poly-linear model, origin-oriented model, peak-oriented model, slip model, Kato-Akiyama model and Clough model.

The nondimensional hysteretic characteristic ¢I) is expressed as a linear combination of the nondimensional linear component x and the nondimensional hysteretic component z as follows:

• = r x + ( 1 - r ) z (10)

where both • and z are so normalized that the nondimensional unit initial rigidity and yield

deformation are unity, and x is the normalized deformation with reference to yield deformation.

Bilinear hysteretic model The simplest hysteretic model would be the normalized

perfectly elasto-plastic hysteretic component as shown in Fig. 1. The constitutive law of this hysteretic component is given by the following differential form3:

= k[1 - U(k)U(z- l ) - U ( - k )U( - z - 1)] = gz(k, z)

(11)

where U(') denotes the unit step function, namely, a(x)-- 1 for x>~O, and =Ofor x<O. The constitutive law of bilinear hysteretic model having a rigidity ratio r of the second to first branches is completely described by equations (lO)and (11).

Poly-linear hysteretic model The normalized hysteretic component z of poly-linear

hysteretic model can be expressed by z

J Z= Z (rj--rj+l)ZJ (12)

j i

where rj is the rigidity of thej th branch, and 1 = rl > r 2 > •. . > rj > rj + 1 = 0. Here, zj is the perfectly elasto-plastic hysteresis with yield deformation fir, and is given from equation (11) as follows:

~j = ~[1 - u(~)u (z~- ~j ) - u ( - ~ ) u ( - z j - ,~j)] -~zj(~, 6)

(13)

where 6 i is the deformation at thej th turning point of the virgin curve, and 1 -- 6~ < 62 < - . • < 6j. For example, in the case where J = 2, equations (10), (12) and (13) give the hysteretic constitutive law of trilinear hysteresis.

Kato-Akiyama hysteretic model The hysteretic model presented by Kato and

Akiyama 14 has the stiffening or degrading characteristics of yield strength with the cumulative plastic deformation as shown in Fig. 2. The differential form of the normalized

Z

Fig. 1. Normalized perfectly elasto-plastic hysteretic component

Probabilistic Engineering Mechanics, 1988, Voh 3, No. 1 45

Application of stochastic di[lerential equations." Y. Suzuki and R. Minai

Z

s x

_. - - - - - -~-"~" 1

Fig. 2. Normalized Kato-Akiyama hysteretic component

Kato-Akiyama hysteretic component is expressed by 4

-- k[1 - (1 - s)U(k)U(z- z ~ ) - (1 - s)C(-.:OU(- z - z[)]

==_g=(Mz,u+,u -) (14)

- z L <~z<<.z[

where

S S

ZL+=I+ - u + I _ s Z L = I + I - - s u (15)

are the instantaneous yield strengths of the hysteretic component z. In the above equations, the superscripts + and - denote, respectively, the positive and negative directions of deformation-coordinate, and s is the rigidity ratio of the hysteretic component z. The yield strengths are governed by the new state variables u + and u- defined a s

g. (x ,z ,u +) ~ + = ( 1 - s ) k U ( k ) U ( z - z ~ ) = - +"

f i - = - ( l - s ) ~ c U ( - S c ) U ( - z - z [ ) = g ~ ( k , z , u )(16)

Here u + and u- are the one-directional cumulative plastic deformations in the hysteretic component z. Thus the description of Kato-Akiyama hysteretic model needs the state variables u + and u- controlling the stiffening and degrading of hysteresis in addition to the conventional state variables k and z. The one-directional cumulative plastic deformation ratios rl~- and t/~ in the hysteresis • given by equation (10) are related to u + and u- as follows:

q+ =(1 -r)u + t/~- =(1 - r ) u - (17)

The Kato-Akiyama model represents the stiffening or degrading characteristics according as s > 0 or s < 0. In a particular case where s = 0, the Kato-Akiyama hysteresis is reduced to the bilinear hysteresis.

Origin-oriented model The origin-oriented model shown in Fig. 3 has the zero-

memory and peak-oriented features. The constitutive law of the normalized origin-oriented hysteretic component is

given by

~= A+~[~- U ( k ) U ( z - HI U(~) + A - . ; c [ 1 - a ( - ~c)U(- z-1)]a(-z)=-gz(.;c,z,u +,u )

(18)

where

1 1 A + = A- = - - (19)

1 + u + 1 + u -

The state variables u + and u which govern the degrading of stiffnesses A + and A- are given by

+ = .;cu(~)U(z- 1 ) - g+. (k, z)

fi- = -kU(- . ;c )U( - z-1)=_g, (k,z) (20)

Here u + and u- are, respectively, the absolute values of the maximum and minimum displacements minus one, and they are utilized to memorize the current positive and negative peak deformations.

Peak-oriented hysteretic model As shown in Fig. 4 the peak-oriented hysteretic model

represents stiffness degrading characteristic which is

Fig. 3.

Z

t I

1 x

Normalized origin-oriented hysteretic component

Z

u--4 Fig. 4.

_ _ U ÷ _ ]

/ 1 X

U = U ÷ * U -

Normalized peak-oriented hysteretic component


Application o f stochastic differential equations: Y. Suzuki and R. Minai

Z where

/ / I u - - -

] ~ LI÷ I

1 ×

Fig. 5. Normalized slip hysteretic component

closely related to the total cumulative plastic deformation. This normalized hysteretic component is given by

2= ASc[1 - U(Sc)W(z- 1 ) - U ( - ~ ) U ( - z - 1)] = 9~()c, z, u)

(21)

f i= .~[U( .~)U(z-1)-U(- . ;c)U(-z-1) ] -g , ( .k ,z ) (22)

where

2 A - (23)

2 + u

and u is a sum of u + and u- defined by equations (20).

Slip model The normalized hysteretic component of slip model

with dead zone as shown in Fig. 5 is expressed by using the controlling state variables u + and u- defined by equations (20) as follows:

~= s~{ U ( x - u + )U (SO[1- U ( z - 1)]

+ U(x)U(- k)[1 - u ( - z ) ]

+ u ( - x - u - ) u ( - ~)[1 - v ( - z - 1)]

+ u ( - x ) v ( ~ ) [ 1 - v ( z ) ] }

- 9~(x, 5c, z, u +, u - ) (24)

The idealized slip model proposed by Tanabashi and Kaneta 15 is given by equations (10), (20) and (24). In the particular case that u + and u- in equation (24) are neglected, the idealized slip model reduces to the double bilinear hysteretic model proposed by Iwan 16

CIough hysteretic model The stiffness degrading hysteretic model presented by

Clough and Johnston 1 v for reinforced concrete structures is shown in Fig. 6. In this model, the differential representation of z is expressed by 4

~= 5cU(z){A + U(~)[1 - U ( z - 1)] + U ( - ~)}

+ s~u(- z){A- U(- ~)[1 - U ( - z - 1)] + U(~)}

-g~(x , Sc, z ,u +,u - ) (25)

1 - z l + z A + A- - (26)

l + u + - x l + u - + x

and u + and u- are given by equations (20)• At this point it is important to note that the state

variables u + and u- introduced in the origin-oriented model, slip model and Clough model are related to the maximum and minimum ductility ratios ~/+ and ~/~ beyond the elastic limits as follows:

~+ = l + u + ~ > 1

q,~ = - 1 - u ~1~, ~< - 1 (27)

General formulations o f hysteretic constitutive laws The differential formulations of the hysteretic

constitutive laws of some typical hysteretic models confined to piecewise-linear hysteretic characteristics are presented• As stated in the Kato-Akiyama model, origin- oriented model, peak-oriented model, slip model and Clough model, the state variables which control the stiffening or degrading of hystereses are contained in the differential forms in addition to the conventional state variables x, ~ and z. Hence they are called state- dependent hysteretic characteristics 4. On the other hand, the hysteretic characteristics such as the bilinear, double bilinear and poly-linear models are said to be state- independent because the hysteretic constitutive laws of them do not explicitly depend on the controlling state variables•

For a general class of parallel composite hysteretic characteristics, the general hysteretic constitutive law of the normalized hysteretic component is given by

J J

z = ~ cjzj ~ cj = 1 1 >~ cj >~ 0 (28) j = l j = t

where cj are appropriately chosen coefficients and zj is the j th normalized hysteretic component given by

kj = 9~j(x, 5c, z j, u + , u f ) (29)

• + X uf = g,j ( , 5c, z j, u + ) (30)

fi; =9~ (x, Sc, zj, u - ) (31)

Z

1 U÷. [

/ / / . , . . - " -x

t - u - q

Fig. 6. Normalized Clough hysteretic component

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 1 47

Application of stochastic differential equations: Y. Suzuki and R. Minai

The general constitutive laws of the normalized symmetric hysteretic components including the piecewise-linear hysteretic characteristics stated herein and a class of curved hysteretic characteristics presented by Wen s are written as

= +)

+U(- - z )Fz ( -X , - z , u ,u +)]

+ U(-x)[U(z)F2(x ,z ,u+,u-)

+ U ( - z ) F ~ ( - x , - z , u )]} (32)

fi + = 2U(2)G(z, u +)

{ ~ - = - 2 U ( - 2 ) G ( - z , u ) (33)

where

function. The quantity is defined as the maximum absolute value qm(t) of the nondimensional displacement response referred to the corresponding yield deformation within time interval [0, t], namely

r/,.(t) = max Ix( )f (36)

As shown in Fig. 7, this is clearly a continuous and nondecreasing function of time. The differential form of the maximum ductility ratio is given by 18

(37)

The relevant state variables of the maximum ductility ratio are x, 2 and r/m.

F1 <~F2 G>~O

and the values of F~ and F z at the origin are unity. The slope of hysteresis is given in each quadrant of the 2z- plane as follows:

Oz ?mx=Fl(x,z,u + )

= F z ( - X , - Z , U ,u +)

=Fz(x , z ,u +,u )

= F l ( - x , - z , u - )

for 2 > 0 and z > 0

for 2 > 0 and z<O

for 2 < 0 and z>O

for 2 < 0 and z<O (34)

The derivatives of u + and u- with respect to x are given by

~,U + - G ( z , u +) for 2 > 0

8x

~U- - G ( - z , u ) for 2 < 0 (35)

8x

It is noted here that the hysteretic characteristics are symmetric with respect to x, 2 and z, and u + and u- are nonnegative.

Cumulative plastic deformation The total cumulative plastic deformation ratio is

defined as a sum of absolute values of nondimensional plastic deformations normalized by the yield deformation as follows 19:

08) where q~ and ~/p are, respectively, one-directional cumulative plastic deformation ratios in positive and negative directions. For the piecewise-linear hysteretic characteristics stated in the preceding section, the differential forms of these cumulative plastic deformation ratios are given by

0v = (1 - r) sgn(2)(2- g~) -= g, (39)

42 = U(2)2p = (1 - r )U(2)(2- gz) = g ; (40)

r j p = - U ( - 2 ) 2 p = - ( 1 - r ) U ( - 2 ) ( 2 - g z ) - g ~ (41)

where g~ denotes the nonlinear functions which give the time derivative of the normalized hysteretic component z, and the symbol sgn(') is the signum function.

Dissipated hysteretic energy The dissipated hysteretic energy ratio t/h(t ) is defined as

the total amount of hysteretic energy dissipated within the time interval [0, t], and can be expressed as z°

DIFFERENTIAL FORMULATIONS OF SAFETY MEASURES

In doing the seismic reliability analysis, it is necessary to define appropriate safety measures as seismic responses and corresponding failure criteria. However, these measures and failure criteria might depend on the function of structures and the types of structures. No definite method to identify the measures and failure criteria seem to exist. The maximum ductility ratio, cumulative plastic deformation ratio, dissipated hysteretic energy ratio and low cycle fatigue damage are well known as safety measures. In general, these safety measures can be defined as continuous and nondecreasing functions of time. And, the differential forms of the safety measures can be written in terms of single-valued nonlinear functions of the state variables including safety measures.

Maximum ductility ratio The maximum ductility ratio may be considered as the

simplest measure of structural safety as well as structural

~lh(t)= O(r)2(z) dz -Ep( t ) (42)

where r/h is normalized by twice the potential energy at the yield point, and Ep is the potential energy determined along unloading curve. For the hysteretic characteristics in which the unloading rigidity is equal to the initial

x

"am # . ,

Fig. 7. Maximum ductility ratio


A

Application of stochastic differential equations." Y. Suzuki and R. Minai

~ : : : : ~

W.-

/ x Ep

Fig. 8. Dissipated hysteretic energy

rigidity, Ep is given by

Ep = 1 [rx2 -t- (1 - r ) 2 2 ] (43)

From equations (42) and (43), the dissipated hysteretic energy ratio is given by in the differential form

h = (1 - r ) z ( 2 - 9~) =- 9h (44)

Specifically, for the bilinear hysteresis the dissipated hysteretic energy ratio defined by equation (44) is identical to the toal cumulative plastic deformation ratio given by equation (39), since in plastic range sgn(2)z = 1.

Low-cycle fatigue damage When considering the so-called low-cycle fatigue

phenomenon of structures subjected to severe earthquakes, it is rather difficult to determine the universally available damage rate function g: depending on the structural material and the loading processes. In this study, since under the recognition that failure due to low-cycle fatigue damage is substantially a brittle fracture phenomenon, it is assumed that the macroscopic relationship between deformation amplitude and number of cycles to failure is given by the following simple power law 2 ~ :

cF/ 4N (45)

where ~ is the total deformation amplitude, N is the number of cycles to failure, cr is the static fracture ductility ratio and a is the parameter which governs the degree of fatigue accumulation. Similarly, the relationship between plastic deformation amplitude and number of cycles to failure is assumed to be

cp / 4N (46)

where cp denotes the static fracture plastic deformation ratio.

By transforming to continuous processes through sinusoidal time variation based on the linear damage

accumulation hypothesis, the differential forms of fatigue damage factors corresponding to equations (45) and (46) are, respectively, obtained as follows22:

a (47)

(48)

The fatigue damage factors ,r/f, and .q:p are normalized so as to take the value 1 when failure occurs. Especially, in the case where a = 1, equation (48) is identical to the total cumulative plastic deformation ratio r/p defined by equation(39). On the other hand, when the parameter a becomes sufficiently large, equations (47) and (48) approach the corresponding maximum ductility ratio, and maximum plastic deformation ratio, respectively.

In the above, the differential forms of typical seismic safety measures are given in terms of single-valued nonlinear functions of state variables. From the above definitions and differential formulations of safety measures, a composite safety measure is defined by normalizing the safety measures dividing by the ultimate value of each safety measure like the low-cycle fatigue damage factor and assembling them as follows:

K

,7 = ~ dk,~l~? c~ k >~ 0 1 >~ d k > 0 (49) k = l

where d k a r e coefficients, and nqk are the normalized safety measures. The other composite safety measure may be also written as

K

.7 = 1 - E (1-. .k) #' / k>0 (50) k = l

The safe domain for the normalized safety measure ,7 is in the range [0, 1).

STOCHASTIC MODEL FOR SEISMIC EXCITATIONS

One of the most simple dynamical systems which generate seismic excitations having nonstationarities both in amplitude and spectral characteristics will be a class of time-dependent linear filter subjected to amplitude modulated white noise processes 23'z4.

The seismic excitation f ( t ) may be given by 4

f ( t ) = ~lv(t) + Y2~2(t) (51)

where v(t) is the output process of which the time- dependent linear dynamical system driven by Gaussian white noise is given by

v=Lz~p Llq~=73~l(t) (52)

with the zero initial conditions di49/dti[, = o = 0 for i = 0, I, . . . . m - 1. Here the differential operators L 1 and L 2 are given by

m d i l d ~ LI = L2 =jS o (53)



where a,, = 1, m > 1. In the above equations, ~1 (t) and ~ 2 ( t )

are zero-mean Gaussian white noise processes, 71(0, ~)2 (t) and 73(0 are deterministic amplitude-modulation functions, and ai, i = 0, 1 . . . . . m - 1 and bj, j = 0, 1 . . . . . l are generally time-variant coefficients.

By defining the m-dimensional state vector ~Z of the time-dependent linear filter as follows:

E Z = [ ~ I ~)2 " ' " ~)m] T (54)

where

d i - 1(~

49i- dt i_ 1 i= 1, 2 . . . . . m (55)

the state space equations of the time-dependent linear filter are given by

EZ =EAEZ +EN EZt: o = 0 (56)

where EA is an m-by-m deterministic matrix, and EN is the m-dimensional input vector

0 - 1 0 0

0 0 - 1 0

E A = -- • . .

0 0 0 .. --1

ao a l a2 • " am 1

0

0

EN = -- i (57)

0

~'3~1

The seismic excitation f ( t ) defined by equation (51) is expressed by making use of EZ as follows:

f ( t ) = y abr~Z(t) + ?'2 ~2(t) (58)

where

b r = [bo bl . . . b, 0 . . . 0] (59)

Equation (56) is the form of a linear dynamical system driven only by Gaussian white noise. It is clear that equation (56) is equivalent to the corresponding linear stochastic differential equations which belong to a special class of equation (1), that is,

dEZ(t)=EA(t)EZ(t)dt+EV(t)dB(t) Zt_o=O

(60)

The shaping filter considered in the present discussion is only the linear type, but the extension of the concept to nonlinear shaping filter systems is straightforward, and equation (60) is then written in the more general form

dEZ(t)=EF(eZ, t )d t+eV( t )dB( t ) Z t : 0 = 0 (61)

SEISMIC RELIABILITY ANALYSIS

The stochastic vector equations of motion of a class of hysteretic structural systems subjected to a nonstationary seismic excitation f ( t ) given by equation (58) may be

written as

s ~ = sAs Z + sG(s Z) + sEAEZ + s N

where

sZt= o = 0

(62)

s Z = x] 2

Z

It

sG=

g . d

sEA =

o 1 [o -71 ibT - i ~2 i

k 0

(63)

In the above equations, x is the displacement vector associated with all hysteretic characteristics of the structural system considered, z is the normalized hysteretic component vector, u is the state vector which controls the state-dependent hystereses, 9z and 9. are the single-valued nonlinear vector functions concerning the differential forms of z and u, respectively, and i is the coefficient vector which represents locations related to the seismic excitation, sZ is the state vector of the hysteretic structural system, sA is the coefficient matrix of the linear part of the hysteretic structural system, sG is the nonlinear vector-valued function of the nonlinear part of the hysteretic structural system, and sEA and sN are, respectively, the coefficient matrix and the input vector concerning the first and second terms of equation (58). Also, the differential representation of the safety measure vector ~/Z can be expressed in the general form

~2 = MG(sZ, ,~Z) (64)

By defining the state vector Z of the whole system as an assemblage of the state vectors EZ, sZ and MZ, the state equation of the augmented system is determined as follows:

= F(Z) + N F(Z) = AZ + G(Z)

where

Z = ~Z A =

Z

~A 0 0

sEA sA 0

0 0 0

Z t - o = O

(65)

G= sG N = s (66)

~t G

In the above equation, the inhomogeneous vector N is expressed as the product of the amplitude modulation matrix V and zero-mean white noise vector ~. Equation (65) is in the form of nonlinear dynamic system driven only by white noises. There fore, equation (65) is rewritten by the stochastic differential equation in the form

dZ(t) = F(Z, t) d t+ V(t) dB(t) Zt - o = 0 (67)



where the vector F(Z, t) is a function of time and the state vector Z(t), and the matrix V(t) is a function of time. This is a special class of the It5 stochastic differential equation.

By defining the scalar function ~k(Z) as

0 = (-I Z~' ki>~O i= 1, 2 . . . . . n (68) i = 1

then the differential equation for the moment E[~b(Z)] can be derived from equation (7)

1 t~ T

i= 1 i,j= 1 LOZi t~Zj Fij (69)

where Fi and Z i are the ith components of F and Z, respectively, and F o is the ij element of F. In particular, the differential equation for the first order moments EEZ] is given by

/~[Z] = EEF(Z, t)] EEZ],= 0 = 0 (70)

Defining Z d as

Z d = Z - E[Z] (71)

the stochastic differential equation (67) is written in terms of Z e

dZ e = Fe(Z d, t) dt+ V(t) dB(t) (72)

where

Fd(Zd, t) = F(Z d + E[Z], t ) - EEF(Z a + EEZ], t)] (73)

Then, the differential equation for E[O(Za) ] is

r ~3 E[~k(Zn)]=EE{~(F~Y+Fd(Zd't)}~d~(Zd']\ ~Zd/ (74,

Especially, the second order moment equations in terms of the covariance matrix K is given by

Ii2 = EEFe(Zd, t)Z~] + E[ZnFr(Ze, t)] + r K, = o = 0

(75)

where K = E [ Z d Z~]. By denoting the moments M(k~,k2 . . . . . k.; t) of the

state vector Z

M(k~, k2, . . ., k,) = E[O(Z)] (76)

then, a set of the first order nonlinear differential equations for the higher order moments is derived as

M(kl, k2, . . . , k.)

= ki E FiZ k, - t zkj i = 1 j =

j # i

1 L ~-1 [ k,-1 kj-lpO Zkpp ] +~ k~kyqE Z~ Zj i = l j = =

+~ ~ k,(k,-1)F,,E Z~ '-2 1-I Z~, i = L j = l

?1

= ~ k,E[F,~k(Z)/Z,] i = l

n - 1

+ ~ L FqkikjM(kl . . . . . k i - 1 . . . . . k j - 1 . . . . . k.) i = 1 j = i + l

1 +~ L Fuki(ki-1)M(ki . . . . . k i - 2 . . . . . k.) (77)

i = 1

with zero initial conditions and the normalization condition

M(kl, k 2 ..... kn)t= 0 = 0 M(0, 0 . . . . . 0)-- 1 (78)

The expectations of the first summation in the right- hand side of equation (77) cannot directly expressed in terms of moments, since the nonlinear functions Fi cannot be expanded in a power series of the state variables. Even if the power series expansion is possible, the solution to equation (77) cannot be obtained unless the moment equations are truncated to a finite set of equations. In order to definitely determine the expectation operator E appearing in equation (77) and to approximately truncate the infinite moment equations to a finite set of simultaneous first order nonlinear differential equations, an analytical form of the joint probability density function p(Z; t) is assumed. An effective way of achieving this approximation of the probability density function is by expressing as a finite orthogonal series expansion in terms of different polynomials depending on the properties of state variables, especially on the regions of the state variables concerning the hysteretic characteristics and safety measures 2°'25. The coefficients and parameters in the series expansion are determined in terms of moments. By solving the moment equations, which are modified by the assumed probability density function, under nonstationary state, the time-dependent probability density function p(Z; t) can be determined.

Since the probability density function p(MZ; t) of the safety measures is obtained as the marginal probability density function, the reliability functions of the structural system and structural components are given by

R(t; S") = ;m p~uZ; t)dMZ (79)

(. R(t; S ~ ) = | P(MZu; t)duZ u

js',, #=1 , 2, . . . , m

(8o)

where S m is the m-dimensional safe domain of the whole structural system and S~ is the one-dimensional safe domain of the/~th structural component. The reliability functions are obtained by simply integrating the probability density functions of the safety measures over the relevant safe domains, since the state variables concerning seismic safety measures are defined as nondecreasing functions of time. Also, one of merits of the proposed method is that the reliability function of the whole structural system can be evaluated through taking account of statistical dependence of seismic safety measures of structural components.


Application o f stochastic differential equations: Y. Suzuki and R. Mina i

CONCLUSIONS

The basic idea and concepts of the reliability analysis of hysteretic structures subjected to nons ta t ionary seismic excitations have been described on the basis of the theory of cont inuous Markov vector processes. This method is materialized through the differential formulat ions of the seismic safety measures as well as those of hysteretic consti tutive laws. The safety measures considered herein are the max imum ductility ratio, cumulat ive plastic deformation ratio, dissipated hysteretic energy ratio, low- cycle fatigue damage factor and their possible composite measures, all of which are defined as cont inuous and nondecreasing functions of time. As the hysteretic characteristics, the bilinear and polylinear hysteretic models and more generally the state-dependent hysteretic models such as peak-oriented model, slip model, Kato- Akiyama model and Clough model are considered. The seismic reliability analysis of hysteretic structures is formulated in the form of the It6 stochastic differential equat ions which govern the augmented nonl inear dynamical system consisting of a shaping filter system generat ing nons ta t ionary seismic excitations, hysteretic structural system and the output system composed of the pert inent seismic safety measures of structural components .

The proposed method has the advantage that the convent ional stochastic seismic response analysis and seismic reliability analysis of hysteretic structures are simply and systematically unified into a single-time parameter analysis based on the stochastic differential equat ions by in t roducing t ime-dependent multiple series expansion of different or thogonal polynomials. Also, the t ime-dependent statistics and joint probabil i ty density function of the state variables including the seismic safety measures can be determined with due regard to the statistical dependence of the state variables. And thus the seismic reliability function of the structural system is obtained by simply integrat ing the marginal jo int probabil i ty density function concerning the seismic safety measures over the safe domain. The proposed method of seismic reliability analysis is versatile to apply to nonl inear dynamical systems without recourse to the Fokke r -P lanck -Ko lmogorov equat ion approaches or the level crossing methods.

REFERENCES

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52 Probabilistic Engineering Mechanics , 1988, Vol. 3, No. 1

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Application of stochastic differential equations to seismic reliability analysis of hysteretic structures