等価履歴システムを用いた非線形履歴構造物モデルの不規則応答解析(概要)

(1)

Architectural Institute of Japan

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ArchitecturalInstitute of Japan

[iag.,,!,,).,,.,,,,,.,

{8T.".a,L.Oli.S.`;"8f`"A'rl'r"Na.lis,",s,`'N:tg':,",,Ef,gggeering

s.,pt,,ee\et?im:x,ts,Mfivag.i:

APPROACH

TO

STOCHASTIC

RESPONSE

ANALYSIS

'

OF

PIECE-WISE-LINEAR

HYSTERETIC

STRUCTURAL

SYSTEMS

'

-For

Bilinear

Model

by

MASANORI

IZUMI*,

LI

ZAIMING'"

and

HIROSHI

KATUKURA"",

Members

ef

A.

I.

J.

1. Introduction

Structural

systems under

dynamic

loading

usually exhibit nonlinear

hysteretic

behaviour,

especially under

strong

earthquake excitation.

For

reasons of safety and economy,

it

is

essential

that

such

behaviour

of

the

structures,

particularly

of

the

modern

base-isolated

structures,

be

taken

into

account

in

design

and analysis.

Indeed,

during

seismic response analysis

rnost

of

practical

structures are expressed

inte

such nonlinear

hysteretic

models as

a

bilinear

model,

double

bilinear

model,

_poly-linear

model, origin-oriented model,

peak-oriented

model, slip model,

Clough's

medel,

and

Takeda's

model, etc,

,

Such

models

have

a

common

_property

that

the

hysteretic

characteristics censist of

_{piece-wise-linear}

behaviour.

Therefore,

in

this

paper

we call such a model as a

piece-wise-linear

hysteretic

model or

p.w.L

hysteretic

model

for

short.

'

On

the

other

hand,

the

stochastic response analysis of

the

nonlinear

structtiral

models

has

been

widely studied

during

the

_past

decades.

In

general,

the

up-to-date approaches

to

the

analysis can

be

categorized as

follows:

1. the

Monte-Carlo

simulation

approach

6. the

perturbation

approachi3)

2. the

Fokker-Planck

equation

approach')")

7. the

Wiener-Hermite

expansien approa ¢

hi`)

3. the

stochastic

linearization

approachS}-9)

'

8. the

Markov

chain

theory

approachi5)

4. the

stochastic

differential

equation appioach'O)･i')

9,

the

other approximate

approachesiE)

5. the

cumulant-neglect

closure

approachi2)

Among

these

existing

approaches,

the

$tochastic

linearization

approach seems

to

have

the

greatest

potential

in

terms

of

_practical

applications.

In

_particular,

Utku,

et al,5)

have

_proposed

a

more

direct

and simplified version of

linearization

technique.

Due

to

its

direct

and simple

formulation

and

good

accuracy,

the

Utkuis

linearization

technique

has

found

wide

applications

in

the

stochastic response anaiysis o'fnonlinear

dynamic

systems,

However,

it

has

been

showni?)

that

this

rinearization

technique

is

essentially

inapplicable

to

a

_p.

w.

1. hysteretic

system

which

has

large

nonlinearity

or

is

excited

at

high

iesponse

level.

This

limitation

can

be

explained

as

follows.

Since

in

ap. w.

1,

hysteretic

system

the

transition

of

the

response

from

the

elastic

domain

to

the

plastic

domain

is

not smooth,

the

differential

expression

of

the

characteristics and

hence

the

differentiation

of

the

governing

equation

of

motion with respect

to

its

state variabies

exhibit

singularity or

discontinuity.

As

a consequence,

this

paper

is

especially

intended

to

_present

a

both

convenient and reliable approach

to

the

stochaStic response

analysis

of a

_general

_p.

w.

1. hysteretic

structural system under earthquake

excitation.

In

this

paper

we

wi!1

develop

a

distinguishing

approach

to

smooth

the

_p.w,1.

hysteretic

systern

in

an equivalent

probabilistic

sense

so

that

the

singutarity may

be

eliminated and

the

application of

the

Utku's

linearizatiQn

technique

may

become

_possible,

The

smoothing

technique

will

enable

us

both

to

conveniently obtain rather satisfactory results of

the

covariance response

matrix

and

to

directly

evaluate

the

maximum

displacement

response.

Moreever,

the

*

_Professor

_of

_Architecture

_Department,

_Engineering

Faculty.

Tohoku

University,

Dr.

_of

Engineering

*'

Graduate

Student

_of

Architecture

Department,

Engineering

Facutty,

Tohoku

University

**S

_Ohsaki

_Research

Institute,

Shimizu

Censtruction

Co.,

Ltd.,

Dr.

_of

Engineering

(ManuscTipt

[eceived

Febi"ary

8,

198S)

(2)

-59-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan

present

approach will

be

applied concretely

to

the

bilinear

hysteretic

model and

further

be

examined with numerical simulation.

2. Formulation

of

P.W,

L. Hysteretic

Structural

Model

The

diffeTential

formulations

of

the

hysteretic

characteristics

for

_p.

w.

L

hysteretic

structural systems

haye

been

developed

by

a

lot

of researchers.

For

a

general

review, refer

to

Referenceii).

In

Referencei7),

the

authois

have

developed

asomewhat

different

differential

formutation

for

_p,

w.

1. hysteretic

structural moclels and expressed

it

in

a

general

form

as

follows.

'

The

nondimensional

nonlinear

restoring

force

for

a

_p,w.1.

hysteretic

system can

be

described

by

di==ax+(1-a)z-'"''-''･･---･-･･-･･--･･-･･-･-･-･--･･-･･･-･･･---･-･-･-･･----･---(1)

Here

a

is

the

_{post-to-preyield}

stiffness ratio,

x

is

the

displacement

_;.and

z

is

the

so-called

hysteresis

and related

to

x

through

a

first

order

differential

equation

in

the

following

_general

form

_:

2=Aic[1-U(th)U(z-1)-U(-t)U(-z-1)]=f(ic,z)･････-･･･････--･･････････････-････････-･･--････････････････････(2}

where

U

is

what we call

here

the

_U-step

function

and

defined

by

u(.)-(g

:t:---･---･-･----･---""""""-"."H"H".","-".HH"H"H-."""."<,)

and

A

is

what we call

herein

the

hysteresis

coefficient which

depends

on

the

characteristics of

the

hysteieticsystem,

For

bilinear

model

A=1･-･･-･･････････-････････-････-･-･･･-･･････････-･･････-･-･-････-･･-･･-････-･-･-･-･

(4a)

For

double

bilinear

moclei

A=1-U(x)U{-dr)U(-2)-U(-x)U(t)U(2)'''''''･･--･''H''"''H''"-'-''-(4b)

For

origin-oriented model

_{A=-1+v.u{z)1+}

v-u(-z)

'''''''''''''''''''''''''''''''-'''-''''''''''''H''H''''(4C

),

A='E-+

_v?+

_v-

""hH"H-H-H-""h'"'H''"'H'"'-H"H'"'"'M"'k"'"H'

(4d

)

For

peak-oriented

model

For

slip mode]

A=

U(x-

V')U{dr)+

U{-x-

V-)U(-dr)+

U(x)U(-th)U(z)

+U(-x)U(dr)U(-2)･----･-･･-･･-･･--･---･･-･----･----(4e)

For

Clough's

model

A=A'

U(th)U(z)+A'

U(-th)U(-2)+

U{-th)U(z}+

U(X)U(-z)'''''''''

(4

f

)

For

Takeda's

model

_{A=A'U(th)U(z)+A'U(-th)U(-z)+B'}

U(-dr)U(2)+B'U{th)U(-z)

(4g)

Here

V',

A',

B'

and

B'

are state-dependent variables and

given

by

i

U'=thU(dr)U(z-1)･-･--･･--･---(4h)

U---abU{-th)U(-z-1)-･--･･-･---(4i)

.-

1-2

1+z

A

-1+

v+-.

-'""'"H"H"H'""H"H"

(4

j

)

A'=1+

v.+.

'v'-"-H"""'""""HH"

(4k)

'

B'=Q+lv+)e"""'-'-''"'"""'""H"H"(41)

B'=a+lv.)fi--"H:H'-'"''""--H"H''"(4m)

The

hysteresis

expressionfi

for

poly-Iinear

model and

Kato-Akiyama's

modet can

be

formed

similarly

on

the

basis

of

the

_general

form

(2)

although

they

take

somewhat

different

forms

from

expression

(2>,

'

3. Smoothing

of

P.W.L

Hysteretic

Structural

Model

As

vve can see clearly,

the

nondimensional

differentiaL

expression

(

2 )

of

the

hysteresi's

2

for

the

_p.

w.

1. hysteretic

structural model

invo!ves

the

_U-step

functions

and

therefore

exhibits singularity at

z==

±

1. As

mentioned

above,

this

singularity of

the

hysteresis

expression results

in

rather

_poor

accuracy

to

the

direct

application of

the

Utku's

linearization

technique

to

the

system.

Consequently,

in

order

to

apply

the

linearization

techniqde

it

may

be

_quite

reasonable

to

search

for

a

smooth

hysteretic

system

which

is

essentially eq'uivalent

to

the

p.

w.

1. hystevetic

system.

By

term

"equivalent"

here

we mean

that

the

smooth

hysteretic

system

has

similar

hysteretic

characteristics with

the

original

_p.w.1.

hysteretic

system and meanwhile almost･thg same

_pTobability

information

about

the

variables

that

are concerned can

be

ddrived

after smoothing as well,

This

aim can

be

achieved as

follows.

Approximate

the

U-step

functions

U<th)

and

U(z-1)

througH

'

u(dr)-Sa+sgn

th)==

(

o.so

i

/:'

=>

<oO

o

---･･･-･-

I･--･･-･-･---･-･･-･-･--･-･---･---

_{sa

₎

u(2-1)=elzl"(l+sgn

z}

<l21$1)

for

a

_positive

n---･--･--･---･--･･---(5b)

(3)

-60-Architectural Institute of Japan

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Therefore

the

U-step

functions

U(th}

and

U(2-1)

become

continuous and

have

no

singularity

any

longer.

Expression

(5b)

holds

since

there

exists

the

_property

lzlsl,

as may

be

derived

from

expression

(2).

The

smoothness

of

U(z-1)

is

controlled

by

n,-

as shown

in

Fig.1.

Similarly,

rewrite

U(-th)

and

U{-z-1)

approximately

into

:

1 tl

th<o

U(-ab)==2(1-sgn

th)=

t

O.5 dr=O-,",H-.･-"--."-"-""-,"---,･-,-･"-.---･---(5C)

NO

dr>o

1.e

Z,5

z.z

U(z-1)

z2

1 t

n=ls'--V

:::;--.7i--i,"i4,:

n.m-.r)(H-"7Cl

1

lr'

1

IJ

/x

2x

o

-1

tFig.1

Z

l

smoethness ofU{z-1>

Fig.2

z2

1

0 -1

-2

-3Fig.5a

z2

smoothness ofhysteresis

-2

-1

_O

_t

₂

origin-oriented model

x

-2--3

-2

-1

Fig.3a

z2

CM

012 3x

bilinear

model'

CM3x

'z2

1 o

-1

z2

1 o

-1 n=3

-2

CM

-3

-2

-1

O

1

2 3x

Fig.4a

Takeda's

model

1

n=5

o

-l

1 o

-!-2

-3Fig.5b-2

-1smoeth

O

1

2

origin-oriented

CM3xmodel

-2-3

-2Fig.3b

-:

_O123X

CM

smoeth

bilinear

model

CM

-1

_O12

_3x

peak-ofiented

model

Z2

r

o

-1

-2

-3

-2

Fig,6a

z2

1 o

-t

-2

n=5

CM

O12

3X

peak-oTiented model

-2

n=3

-3

--2

-1

Fig.4b

smooth

z2

-3Fig.6b-2

-1smooth

!

o

-1

CM

Dt2

3x

Takeda's

model

CM

-2

-3

-2

-1

O

1

2 3X

Fig.7a

Clough's

model

z2

:

o

-1

-2

n=3

-3

-2

-1

_O

Fig,7b

smooth

12Clough's

3

model

-61-CMx

(4)

Architectural Institute of Japan

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u(-2-O=Slzl"(1-sgn2}

(kl'$O

f6r

a

_positive

n-･･･-･･-･･･-･･-･･･-･･････････････-･･･-･･--･--･･･････-(sd)

As

a result,

by

inserting

the

approximations

(5

a-d)

into

(2

),

we

can

sinooth

the

differential

expression

(2)

into':

'

2;A[th-O.slzl"th-O.slthlz121"'i]

for

a

_positive

n･-･･-･･･----･-･-･---･･･-･･-･･--i-･-･･-･-･{6)

Clearly,

expression

(6>

is

continuous and, at

the

same

time,

smooth,

i.e.

mathematically

having

continuous

differentiations

with respect

to'th

and

z,

The

smoothed

hysteresis

coefficient

A

can

be

obtained

from

A

by

making

use

of

the

above smoothing

technique.

It

is

very

interesting

to

notice

that

the

smeothed

differentia}

expression

(

6 )

has

almost

the

same

form

with

the

Y.K.

Wen

hysteretic

expression which

is

described

by

2=

icth-fllz["th-71thlzlzl"-i

for

an

integer

n････-･･-･･-･･･-･･･--･･･-･････-････-･･･････････････-･--･･-･･-･<7)

where

k,

fi,

_r

and n

are

the

parameters.

A$

in

the

Y.

K. Wen

model{7),

the

effect of

n

in

expression

(6)

is

to

control

the

transition

smoothness

between

the

elastic

response

and

the

plastic

response and can

be

illustrated

by

the

t

virgin

loading

curves

for

the

bilinear

model

(AttA=1),

as shown

in

Fig.2.

Theoretically

when

_n.co,

the

smoothed

hysteretic

model

becomes

the

original

p.

w.

1. hy$teretic

model.

Some

illustrations

of

the

smoothing are

given

in

Figs,

3-7.

.

We

should notice

that

parameter

n

be

so cljosen

that

it

is

large

enough

to

_give

the

same

_{probabilistic}

information

about what

is

cohcerned after smoothing and meanwhile

it

is

kepl

small enough

to

apply

the

Utku"s

linearization

technique.

In

this

paper,

parameter

n

is

selected

by

the

criterion

which

describes

that

the

hysteretic

system smoothed

with

enough

small

n

and

the

original

_p,

w.

1,

system

dissipate

almost

the

same

amount

of

hysteretic

energy

during

one

cycle

of a stationary excitation

f<t}i[psintot

nearly at

the

response

level

D=

3,

i.e.

max

displ.

_x..,

D=

yield

displ,

=

1

=Xrmax=i

3''"''"H-'-H-"''''"-"''"-H-"'""''"'""""''"'""-"-'-"'<s)

4. Evaluation

ef

Covariance

Response

Matrix

t

Without

loss

of

_generality,

consider

the

response analysis of a single-degree-of-freedom

_p.w.1.

hysteretic

structural system, with

the

following

nondimensional equations of motion

:

X+2ht+ax'+(1-a>z==f{t)･-L････････-･･････････････r･--･･-･･-･''････''''''"'''''''''''-':''''"'''''''-''"''"(9a)

2==Adi[1-U{dr)U{z-1}-U(-th)U(-z-1)]･･･-････-･-･･･-････-･･･････-････････････････････････････････････--･--･･･(9b)

'

where

h

is

the

viscous

damping

ratio,

As

described

above, expression{2) assumes

the

following

smooth

form

:

2==A[dr-o.slzl"th-o.sldrI21zl"'i]---･---･--･---･(6)

=Ag(th,2)--･-I---･---･---･-･-･･･---(6')

Consequently,

we can apply

the

Utku's

linearization

technique5}

which

describes

as

fo!lows,

For

a vibration system

/

'

of

the

form

･'

'

g(Zi,22,23)=f(t)---･--t･････--･････----･-･･--･･･-･-･-･----･-･---･･---･----･･---･(10a)

'

one

has

the

equivalent

linear

system of

(loa)

as

E

[

_aazg,

]zi

+E

[

_aazg,

]z!+E

[

_aazg,

]zs=f{t)-･･-･---･---･--･-･･----･----.-

---･---ao

b

)

where

E=expected

value.

Using

the

above

techriique,

expression

(

6 )

can

be

rewritten

into

:

2=a(bth+c2)--･･･--･･-････-･･････-･･････--･-･･･-･･････-･'･-･'･･･'-''-H''H''H'''''''''H''''HH'''''''''''''''''''''"{10

The

linear

coefficients a,

b

and c are

_given,

respectively,

by

a=E[A]---･---･----･---･-･･--･---･:･---･---･--･--(12a)

b=E

[

g\.

]=i-

2"'.2'i

:zll(

:

)r(

icii

)r(

"-2h+i

)

(1

-p:)ki:pn-h.2-2X/

-i

r(

";i

)an.

---t--o2

b

)

c

..

E

[

gg

]=

-

n

2"i.'-'

:\,L

(

nIi

)r(

kii

)r(

n-2k+i

)

a

-pz)kftpn-

ic-i aia!-i

Tn

2Ytili'i

_F(

n;1

)p.i.n.-i

H""M,-,"-","-"---･--･---'-'"""H-'"'""""'-"H"H(12C

₎

where

h=even--･---･---･---･---H"""'-HH"H"""H"'"H""""""'"'"'-""-H(12d)

NII-Electronic

(5)

Architectural Institute of Japan

NII-Electronic Library Service

'

E[thz]

'

･

P=

opg.

'''"H'''H'''''''''H'''''''''-'''"'''""'''''''-''-''H'-m'-:''"''"'''''''''"''H''"''-"''"''"''"'''''(12e)

and

r

is

the

_gamma

function

defined

by

'

_{r(x)=X'coe'ttx'idt-･--･-･-･･-･････････-･--･-･-･････--･･l-････L･･-･･･--･--･････--･･-････････････-･･･--･･--･･-･･-･･(12f)}

Through

introducing

the

state

variabLe vector

Y

with

_y,=x,

_y,==z

and

_y,=th,

the

equations

of

motion

(9a

)

and

{11)

can

be

rewritten

into:

･

'

dY

･

dt=GY+F-"-""-""""-"-''"HHH"H-H--'"-H""'"'"'"H"H"H"v"""H"'"-'---･･--(13)

where

'

G=[-i

-(iC.)

i2bh]"'L"-'-""H"H"H'(14a)

'.F=[;t

From

the

linear

random vibration

theory'S),

the

covariance response matrix

S

following

first

differe,ntial

equation

:

d,S,

response vector

V. B,,=O

(

i,

j=1,

3)

except

Bs3=l{t),

time-dependent

covariance response matrix

S

can

be

obtained

by

solving

eqttation

Runge-Kutta

method.

5. Evaluation

of

Maximum

Displacement

Response

The

maximum value of

the

displacement

response

x(t)

within

the

time

defined

by

'

v(t)=maxx(r)･････････････-･-････････-････-･･-･･････････--･-(16)

Xs

osrst

as

$hown

in

Fig.8.

_u(t)

satisfies

the

following

differential

equa-

4 tioni])

o

ij==drU(th)U(x-o)･･･････-･･･-･･-･･････-･･･-･･-･･-･･･-･･････-(17)

--4

Since

_lx)sn,

it

can

be

derived

that

for

_nio

-8

u(x-n)=u(:l;-i)==ilil"u(x}･---････-･･----{isa)

D

2 Fig.8

1 lxln

=2

rpn

(1+Sgnx)

(by

(5b))

foralarge

lnteger

Subsituting

(18)

into

(･17)

yields

nnij=dru(th)lxl"u(x)

(by

(17a))

=o.2s<irlxl"+lxllxl"+drxlxl""+1thIxitln-i)

(by

(sa)

_and

It

is

worth

to

mention

that

expression

(18)

meets

the

initial

condition,

i.e.

expectations

for

the

two

sides

leads

to

Mn+i=(n+1)X't

aeaZ2["-3ii2

(･X.

_r

(

n;2

)+t;

where

E[thx]

P=

(Itaic

"'･-･･....-..-..

u=(i-p2}{i+nf!)r(n;i)+2va

_7pir(F(n;3

.ni5

p-2

Here

M. is

the

moment

functions

of

the

maximum

displacement

response,

i,e.

Mh=

assumed

to

have

joint

Gaussian

density

distribution.

In

solutions

if

the

integer

n

is

large

enough.

with

'

GS+SGt+B････････-･･-･･････-･･-･･-･･-･-･･･-･･････-･･･-･･--･-･･-･･-t-ny･-･･･････････J･J･J･･･-･･-･--･---･-･{15)

where

t

means

the

transpose

and

B

is

amatrix of

the

expected values of

the

_products

of

the

forcing

vector

F

and

the

intensity

function

of

the

excitation

f(t),

Therefore,

the

(15)

numeri cally, e,g, with

'

interval

[O,

t]

for

a vibration system

is

n-･･･-････-････-･･-･･-･･-･･-･･-･-･-,.-,...

.,""---･---･(14b)

Sw=E[yiys]

satisfies

the

T

468

10

maximum

displacement

.H"H"H.."",-""".""---'--"-H(18b)

(17b))･---(19)

'x=n=O

as welL

Taking

the

)

dt･･-･･･-･･-･･･-･･-･･--･･･-･･-･･･-････-･-･･-･･･-･･-･･-･･-･･(20)

'

".".""H"H"",,"""""kHHH"HM"-"---･-･･---･--t---(21a)

)mr(

)3a

p,))""･-･-･---･･-･-(21b)

E[o"]

;

and

di

and

x

haye

been

theory

the

approximate

expression

(20>

_yields

almost exact

(6)

Architectural Institute of Japan

NII-Electronic Library Service

'

6. Application

and

Accuracy

In

the

_preceding

sections,

the

direct

construction of

the

covaTiance response matrix

and

explicit

expression

of

the

moment

function

of

the

maximum

displacement

respense

for

a

_general

_p.

w.

1. hysteretic

structural system

have

been

'

presented

in

detail.

In

this

section,

they

will

be

applied

to

the

bilinear

model; and

through

nurnerical simulation

the

'

accuracy of

the

approach wili

be

examined.

6.1 Monte-Carlo

Simulation

The

excitation

f(t)

imposed

here

is

a

Gaussian

white noise with

E[f(t)]i=O･･･-･･-･･-･･･････････-･:-(22a)

E[f(t)f(t+T}]=S,a(T)･････････････-･-･･-･･-･･-(22b)

Such

a white noise

is

_generated

in

this

study on

the

basis

of

the

Fast-Fourier

Transform

_(FFT).

The

flowchart

shown

in

Tabie

1 describes

roughly

_the

_generation

of

the

white

noise.

It

is

of

interest

to

note

that

zero

mean

on

a

very

short

limited

time

domain

should

not

be

taken.

Otherwise

the

power

of

the

white noise

at

the

infinitesimal

frequency

interval

wc[-E, e]

for

any small e will

be

lost

and,

therefore,

the

drift

response of

the

hysteretic

system

tends

to

be

cut

down

greatly.

The

generatednoises

last

10 times

longer

than

the

natural

_period

(2

rr)of

the

hysteretic

system.

The

equations

ef

motion

(9a

)

and

<9b)

is

solved

numerically

by

Runge-Kutta

method

in

this

_paper.

By

introducing

the

state vector

Y

as

in

expression

(13),

equations

(9a)

and

{9b)'

can

be

rewritten

into

dd't

==

[

AY3[i-Y[

y,;I

U,(

y,2,-,

I

Y

I,:-,,U.(

I,

!3LU,;;

yt-i)]

]

-e(

y･

t)･･--･････････････-･･-･･-･･-･･･････-･･････-･･･(23)

Obviously

equation

{23)

is

a

3-dimensional

first-order

differential

equation.

From

the

viewpoint of

_physical

Table1

Flowchart

of

the

_generation

of white nOise

_Table2

_Related

_pararneters

Generatedi

uniform randon variable -ithin

(O,2z)

,

Fam

F(o}

F<de

_)-exp(i

_di

_>

_1toIS12

iv

F<to)=O

icoll12f

FFTt

At=O.06

x<t)=R(t)+il(t}

REAL

f{t>=real{x(t}}

lstandarddeviation

na2r-£ t=1f2(t,} 765a32to n

tsetaf(t)

f<t)=olsetpo

-f{t)=VI2g6f(t}

o2

'6

2 power=So caseSoahT case1O.6o.oO.Olt!2-case2O.6O.10.01t12" case3O.4O.5O,Olt!2" case4O.2O.6O,Olt12za

3

2

1 o

axL--os a: 246810

g(c}

ceEe 2

t

Tao246810

9(e)

[ase 3 1.2D o.eo O,40 T

_D.

a e laD 9Ca) case 1 1 ] s246S10

9(g)

case 4 T 1 o

-1

PXI Pgt/ e 1 o

t-i

PXE. p"s o T.1 PHt pn P"y o T

-1

246e lo a2

9(b}

case1 9(d}

Fig.9

prepriety

of smoothing

466 10 a24ss 10 e

case2 9Cf･) cnse3

(solid

line

:original

billnear

uuu L,t,)ken

line

:

smooth. 2i6

9(h)

case 4

bilinear)

B tDT

-64-NII-Electronic Mbrary

(7)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan 765t321o 1 o

-1

-exvitheutsmeethtns'i--i-r.Su.itr-l'.'ii''''t''-.'Ux-.-'JJ''-t-JP-tJJre-"

Og

'

.o2d68

10

.]O(a)

case 1 o

7654321o

3 2

l

TD 1 o ax'tt'

F

os

--02a6s lo 10(c) case 2 2

1

o T-t 2

1

o246B ID 10(e) case

3

2 T TO OH

l

2 1

o

o2a6e to

10(g)

cese 4 T-l To T-1

246B to D2#6e

]O

o2ass ID O246e le

10{b) case1 _le(d) _ca,.2 le(f) case3 10(h} case4

Fig.10

response

history

for

bilineaT

rnodel

(solid

line:by

M.C.S.

broken

line:by

_theory)

le

8

6

4

2 6

4

2

Psl-"---. Ptx T T

o246s ieT

o246B

ioT

Oo _246e

_]oT

11 (a)

Z"-rtMg

_ll{b)

'I'ML

11 (c)

EI-M,･

Fig.11

maximurn

displacement

response

(solid

line:by

M.C.S.

broken

Iine:by

theory)

interpretatien,

the

solutions

of

equation

(23)

are

the

state variables of

the

vibration system,

i,

e. _,

the

displacement,

hysteTesis

and velocity response, and

therefore

they

are continueus although

e(Y,

t)

is

not smooth.

As

a consequence, equation

(23)

can

be

solved

directly

by

Runge-Kutta

method with any

desirable'accuracy.

The

for

the

case studies are

tabulated

in

Table

2

and

the

covariance

matrix

of

the

respense variables

is

evaluated

by

taking

the

sample

average

over

100cycles

of oscillation.

6,2

Analytical

Solutien

In

referencei7J,

the

authors

have

set

parametei

n

foT

the

bilinear

model

to

be

equal

to

1. With

n[1,

the

original

bilinear

_p.w.L,

hysteretic

model

has

become

much smoother and almost

the

same

_probability

information

still remained except

for

asystem of very

small

nonlinearity.

In

this

_paper,

_parameter

_n

has

been

chosen

to

be

₃

with

_the

criterion

_presented

in

section

3

and

the

hysteretic

characteristics shown

in

Fig.

_3a

becorne

much

smoother,

as shown

in

Fig.

3b,

The

_propriety

of

the

smoothing approximation

is

investigated

by

numerical simulation

in

Fig.

_9.

From

Fig.

9,

it

is

obvious

that

the

smoothing

is

rather

satisfactory

for

small nonlinearity and

large

nonlinearity as well.

It

is

clear

from

expression

(4

a

}

that

for

the

bilinear

model we

have

A=1,

hence

Z=:1

and

therefore

_a=1

as well.

The

equivalent

linear

coefficients

b

and c can

be

obtained

easily

from

expressions

(12

b

)

and

(12

_b

)

by

substituting

n

with

3

and are

found

_,te

be

b==1-th

a.3p(3-p:)-ff aZ･･････････-･･････････････--･･-･･････-･･･-･･･････････････-･･-･-･--･-･･････････ny･･･････(24a)

c=-th

q.i

ae(1+p2}-3epq.!q}

_{-･--･---･---･･---･-･---･-･----･-･-(24b)}

(8)

Architectural Institute of Japan

NII-Electronic Library Service

where

p=

E:lllliX.]

--･･-･･---･･---･--･-･-･-･･--･-･-･-･･-･---･･･----･･--･-･---･-･--･-･--(24

c

)

'

Substitutmg

(24

a

)

and

<24

b

)

into

(14

a

)

together

with a==l,

the

covariance response matnx

S

can

be

obtained numerically

by

solving equation

(15).

After

solving

the

covariance

matrix

S,

the

moment

functions

of

the

maximum

displacement

response

xma.

can

be

obtained

through'expression

(2o).

Some

case studies are

_performed

and

the

results are compared against

the

Monte

Carlo

simulation,

as

illustrated

in

Fig.10

and

Fig,11.

7. Conclusions

,

In

this

_paper,

we

have

studied

the

stochastic response analysis of

_{piece-wise-linear}

hysteretic

structural models

to

which

the

direct

application

of

the

existing

linearization

techniqtte

is

basically

impossible.

A

new

differential

fofmulation

for

the

_p,

w,

1. hysteretic

systems

have

been

_presented

in

a

_general

fgrm.

First

we

have

deve16ped

a

pewerful

approach

to

smooth

the

general

differential

expression of

the

p.

w.

1. hysteretic

systgms so

that

it

has

become

possible

to

apply

the

Utku's

linearization

technique

to

the

system.

Meanwhile,

the

propriety

of

the

smoothing

has

been

verified

by

the

Monte

Carlo

siinulation.

Based

on

the

tsmoothing

technique,

we

have

_presented

the

direct

and reliable constructions

of

the

differential

equations

for

the

covariance

response matrix and of

the

explicit

integral

expression

for

the

moment

functions

of

the

maximum

displacement

response.

'

Furthermore,

the

_present

approach

has

been

appl.ied

tg

the

bilinear

structural model concretely.

The

accuracy

has

been

investigated

through

numerical simulation,

In

summary,

the

following

results

have

been

concluded.

1)

the

_present

approach

has

6een

proven

to

produce

Tather accurate and ieliable approximate solutions of

the

covariance response rpatrix

both

for

small

nonlinearity and

for

large

nonlinearity,

2)

the

evaluatien expression

(20)

for

the

maximum

displacement

response seems

to

be

more

depehdable

to

estimate

the

_4th

moment

functions

of

the

maximum

displacement

resiponse and

inStead･tends

to

underestimate

the

higher

ones

but

overestimate

the

lower

ones

for

large

nonlinearity.

However,

thig

evaluation can

_give

very

satisfactory

expectation values

of

the

maximum

displacement

response

for

small

nonlinearity.

Finally,

as

the

continuing work

it

is

planned

to

apply

the

_present

approach

to

other

_p.

iy.

1,

hysteretic

structural

'

models concretely.

''

'

t

tt

Reterences

1)

Caughey,

T,K.,

"Derivation _and

_Application

_of

_the

_{Fokker-Planck.,Eqgation}

_to

Disc[ete

Nonlinear

Dynamic

Systems

Subjected

to

White

Noise

Excitation",

JournFI

of the

4goustic,al

Society

_.of

America,

VoLfi5,

No.ll,

November

pp.]638-1692,

l963.

2)

Asano,

K.,

"Stochastic

_Earthquake

Response

Analysis

_ef

_the

Lumped

Mass

Structural

Systern

_with

Elasto-Plastic

Characteristics",

Trans.

of

A.I.J.,

No.247;

September,

pp.75-82,

1976

(in

Japanese).

'

3)

Matsushima,

Y.

,

"NonlineaT

_Random

_Response

of

Single-Degree-of-Fieedom

$ystem

Subjected

to

White

Neise

Excitation",

Trans.

of

A.I.J,

No.Z55,

May,

pp.17-23,

1977

{in

Japanese).

'

4)

Ishimaru,

S,

_,

"Stochastic

Seismic

Response

of

Hysteretic

Structures",

Trans,

of

A,

I,J.

_,

No,

265,

March.

pp.

71-80,

1978,

5)

AtaLik,

T,S.

&

Utku,

S.

,

"Stochastic

_{LineaTization}

ef

Multi-Dggree-of-Freeq,em

Nonlinear

System",

Earthquake

Engin.

&

Struc,

Dyn.,

Vol.4,

pp.411-420,

1976,

6)

Wen,

Y.K.,

"Equivalent

_{Linearizatiori}

_fof

_Hysteretic

Systems

Under

Random

Excitation",

J.

_of

Applied

Mechanics,

ASCE,

VeL47,

Marclt,

_pp.150-154,

1980.'･

7)

Asano,

K. &

Iwan,

W,

D.

,

"An

_Alternative

_Approach

_to_the

_Random

Response

_of

Bilinear

Hysteretic

Systems",

Earthquake

Engineering

&

Structural

Dynamics,

VoL12,

No.2,

March-April,

pp.229-236,

1984,

.

s)

Shibata,

A.

_, et al. _,

'`Inelastic

Randorn

Respense

Anaiysis

of

Reinforced

Concrete

Frames",

Prec.

of

Annual

Meeting

ef

A.LJ.,

Oct.,

No.

B,

pp.649-650,

1987'

_(in

Japanese},

_,

9)

Kobori,

T.,

Minai,

R,

&

Suzuki.

Yl,

"Stochastic

_{Linearization}

_Techniques

of

Hysteretic

Structures

to

Earthquake

Excitations",

Bulletin

of

the

Disaster

Preyention

Research

Instiute,

Kyoto

University,

'Vol.23,

Parts

3-4,

Ne.215,

Decernber,

pp.111-135,

1973,

'

10)

Suzuki,

Y. &

Minai,

R.

,

"A

_Method

of

Seismic

Response

Analysi

of

Hysteretic

'Structures

Based

on

Stochastic

DifferentiaL

Equations",

Proc.

of

8th

WCEE,

San

Fiancisco,

VoL.IV,

_pp.177-186,

July,

19S4.

10 Suiuki,

Y.

_,

Seismic

Reliability

ARalysis

of

Hysterbtic

Strubtures

Based

on

Stochastic

DiffetentiaL

Equatio]s,

Ph.

D

thesis,

(9)

Architectural Institute of Japan

NII-Electronic Library Service

12)

l3)

14)

15)

16)

17>

18)

Disaster

Prevention

Research

Institute,

Kyoto

University,

December

1985.

Wu,

W.

F. &

Lin,

Y.

K.

_, "Cumulant-Negleet

_Closure

_for

_Non-Linear

_Oscillators

_UndeT

_RandoTn

_Parametric

and

External

Excitations",

Int.

J. Non-linear

Mechanics,

Vol.19,

No.4,

_pp.349-362,

1984.

Burton,

T,D.

,

"A

_Perturbation

_Method

_for

_Certain

_Non-Linear

_{Oscillators",}

_Int.

J. Non-linear

Mechanics,

Vol.

19,

No.5,

pp.397-407.

1984.

Jahedi,

A. &

Ahmadi,

G.

, "Application

of

Wiener-Hermite

Expansion

to

Nonstationary

Randorn

Vibration

ef

A

Duffing

Oscillator",

Journal

of

Applied

Mechani'cs,

ASCE,

Vol.50,

pp.436-442,

June,

1983.

Izumi,

M. &

Kimuia,

M.

_, et al. _,

"Response

Analysis

of

Hysteretic

Structures

Under

Random

Excitation

By

Markov

Chain

Model",

Trans.

of

A.I.J.,

No.324,

_pp.18-Z7,

Feb,,

1983

(in

Japanese}.

Matsushima.

Y.

"Random

_Response

ef

Hysteretic

Single-Degiee-of-Freedom

Systems

Subjected

to

Eaiquake-Like

Disturbances",

Trans.

of

A.I.J.,

No.382,

_pp.50-55,

Dec.,

1987

(in

Japanese).

Izumi,

M. &

Li

Zaimng,

et al. ,

"A

_Modified

_Stochafitic

_{Linearization}

_Technique

_to

_Randorn

_Response

_Analysis

of

Nonlinear

Structural

Model",

J.

ef

Structural

Engineering,

A.I.J.,

Voi.34b,

pp.57-92,

Mareh,

1988.

Lin,

Y.K.,

Probabilistic

Theory

of

St[uctu[al

Dynamlcs,

MeGraw-Hill

Beok

Co.,

New

York,

1967.

--

67

(10)

Architectural Institute of Japan

NII-Electronic Library Service

Arohiteotural エnstitute 　of 　Japan

【

論

　

文

1

UDC ：

624 ．

e42

．

7 ；620

．

1 日本建築学会構造系論文報告集第 390 号

・

昭和

63

年

8

月

等価

履

歴シ

ステム

を用

い

た非線形

履歴

構

造

物

モ

デ

ルの

不

規

則応

答解析（

概

要

）

一

_バ

_{イリ}

_ニ

_{アモ}

_デ

_ル

_に

_つ _{いて}

一

正

会

員正会員正会員

和

李

勝

泉

倉

正

再

哲

＊

明

＊＊

裕

＊＊＊

　

1．

序

　

構

造

物

の

_{非線形不規則応答解析}

に

関

しては_，

_種

々の

方

法

が

提案

されている

（

文献リ

ス

ト参照）

が_，その

中

で，

実用面

から

見

て

有力

な

手法

の

一

つとして

，Utku

らによる

等価線形化手法

が

挙げ

られる

。

しかしながら

，

この

_等

価線形化手法を

，

非線形性

の

大

きな

piece

・

wise

・

linear

履歴

システム

（

piece

・

wise

・

linear

履歴

システムとは

，

区

分的

に

線形特性を持

ち，

弾性領域

から

塑

性

領

域

への応

答

遷移

が

滑

らかで

な

い

履歴特性を持

つモ

デ

ル

を指

し，

以後

p

．

w

．

1．

_{履歴}

システムと

呼

ぶこ

と

にする

）

の

問題

に

適

用する

時

には_，

応答評価

の

精度

が

悪

くなるという

問

題が

指

摘

されている

。

_p

．

w

．

1 ．

_{履歴}

_システムでは，

一

_般

的

に

，

支配運動方程式

の

状態量

に

関

する

偏微分

が

不連

続

であり，これが，

非線形性

が

大

きい

場

合

，Utku

らの

等価線

形手法

の

適

用

を大

いに

妨

げるからである。

　

本論文

は

，

非

線

形

性

の

大

きな場

合

にも

十分

な

精

度

が

得

られることを

前

提

に

_，

地

震外乱

を

受

ける

一

般的

な

p

．

w

．

1．

_履

歴

構造物

モデルにおける

不

規

則

応答

解析

のための

簡単

かつ

有

力

なスム

ー

ジン

グ

手

法

を

提

案

したものである

。

したがって，

本

論

文

で

展開

する

等価線形化手法

は

，

“

p

．

w

．

1 ，

履歴

システムのスム

ー

ジン

グ

” を

基礎

としている

。

すなわち，スム

ー

ジン

グ

により

応答

の

確

率情報

の

等

価

な

滑

らかな

_履

歴システムを

導

入し，これに

等価線形手

法を適用

することにより

不規則

応

答

を

求

め

，

これを

p

．

w

．

L

履歴

システムの

不規則応答

と

当

てはめるものである

。

　

本論文

では

，

はじめに

，

多

くの

_p

．

w

．

1．

履歴

システムを

表

現

する

一

般式

を

誘導

し

，

応答

の

確率情報

の

等価

な

滑

らかな履歴システムを

見

つ _け

_出

_{すため}のスム

ー

ジングの

考

え

方

を

述

べ，この

滑

らかな

履

歴システムに

Utku

らの　＊東北大学　教授

・

工博＊＊ _{東北大学　大学院生}

・

_{工修} i

＊

清水建設大崎研究室　工博　　（昭和 63 年 2月8日原禍受理｝