局部座屈により耐力低下を生じる鋼柱の軸力比制限

【論

　

文

】

UDC 　624

．

014

．

2

；

624

．

075

．

2　691

．

714　62

−

462

　　　

R

本 建 築 学 会構 造 系 論 文輾 告集 第

4Z5

号

・

1991

年

7

月

Journal

　of　

Struct

．

　

Constr

、

　

Engng

、

　

AIJ

，

　

No

、

425

，

　

Juty

，

1991

LIMIT

　

OF

　

AXIAL

　

FORCE

　

RATIO

　

FOR

　

DEGRADING

　

　

　

STEEL

　

BEAM

−

COLUMNS

　

INVOLVING

　

LOCAL

　

．

　

　　

　　

　　

　　

　　

　　

BUCKLING

局部

座

屈

に よ り

_{耐力低 下}

を 生

じ る

鋼柱

の

軸 力 比

制

限

Yasuhire

　

UCHIDA

＊



_and



Shosufle



MORIIVO

＊ ＊

　　　　　　　　

内

田

保 博 ，森 野 捷 輔

　

Amethod

　

for

　evaluating 　

the

　seismic 　safety 　of 　

degrading

　

steel

　

beam−

columns 　

falling

　

in

　

local

−

buckling

　under 　repeated 

bending

_{moment 　was}

_prQposed



in



_this

_study

_，



based



_on



_the

_convergence

condition 

for



the

_centroidalstrain 　and 　

indicators

　representing 　

the

　rate 　of 　strain 　accumulation 　and

the

_{rate 　of 　moment 　capacity}

deterioration

_．



The

　

_proposed

　method 　was 　used 　

to

　

investigate

　

li

_皿

its

　

_of

the

　axial 　

force

　ratio 　

for

　

the

　square 　

hollow

　steel 　

beam−

columns

　

in

　which 　

the

　width

−

thickness

　ratio of　

plate

　

elements

，

　curvature 　amplitude 　and 　

yield

　strain 　vary 　as　

_parameters

．

　

KegWOizlS

：steei 　

beam

−

column

，

　

iocal

　

buckling

，

α蕩α

1

　

force

　ratio

，

　 strain 　accumutation

，　 strength 讒

一

　 　 　 　

t

θr：oratl（m

　　　　　　　　　

鋼 柱

，局 部 座 屈

，軸 力比

，

ひ

ず

み の

累積

，

耐 力

低 下

1

．

　lntroduction

　　

When

　

the

　steel 　

structure

　

is

　

subjected

　

to

　

the

　

fiuctuating

　

horizontal

　

force

　

brought

　

by

　strong earthquakes

，　

the

　

beam

−

column

　often 　

degrades

　

due

　

to

　

the

　

local

　

buckling

_，　 and 　

the

　

strain

　

accumulation

occurs

．



The



strength

　

deterioration

　 of 　

the

　

beam

−

column

　

significantly

　 correlates 　 with 　

the

　

_strain

accumulation 　

of

　

the

　

cross

　

section

_．

　

The

　

accumulation

　

of

　centroidal 　

_strain

　

_or

　

_axial

　

deformation

　

_of

compact 

steel

　

beam

−

columns

　under 　repeated 　

bending

　

has

　

been

　

investigated

　

by

　

many

　

researchers

， 　

_such

as　

Takanashii

］ ，　

Suzuki2

）

，

　

Sakamoto3

］_，　

Mukudai

⇔ ，　

Makinos

］

，

　

Matsui6・

’

，

　

Yamada7

）

・

s ）

，

　

Imai9

】

，

　

SaishoiD

）

and

Igarashi1

】〕

_．

　

The

　

effect

　

of

　

axiaHorce

　ratio 

and

　

width

−

thickness

　ratio 　

on

　

_plastic

　

deformation

　capacity 　

for

steel 　

beam

−

columns

　was 　 studied 　

by

　

Kato

職 13 】

_and



Mitani14

）

_．

　

When

　a　

thin

　

walled

　

beam

−

column

　

is

　subjected 

to



alternately

　repeated 　

uniaxial

　

or

　

biaxial

　

bending

moment



in



which

　

Iocal

　

buckling

　

occurs

_，　

itmay

　

reach

　a 　steady 　state 　

and

　

the

　

hysteretic

　moment

．

curvature

relation 

converges

　

after

　

a

　

certain

．

number 　

of

　

the

　

load

　

cycles

_．

　

On

　

the

　

other

　

hand

_，

　

_the

　centroidal 　

_strain

may 

keep



increasing

_{with 　 a}

_gradual



deteriQration

_of

_the



bending



_moment

_capaCity

_in



_another

beam −

column

_．



The



boundary

of　

these

　

two

　

behaviors

　may 　

be

　related 　

to

　

_parameters

　such 　as　

axial

　

force

ratio

，　

width

−

thickness

　ratio 　of 　

the

　

plate

　

element

，

　and 　curvature 　

amplitude

，

　

in

　

a

　

complex

　manner

．

　

This

paper

　

presents

　

a

　

method

　

to

　evaluate 　

the

　

seismic

　

resistance

　capacity 　of 　

a

　

_given

　member 　

_at

　

the

　

_ultimate

stage 

under



the



repeated



loading



condition

_，

　

inclllding

　a　member 　

subjected

　

to

　

biaxial

　

bellding

_，

　and

proposes

　

expressions

　of 　

the

　

limits

　

of

　

the

　

axial

　

force

　

in

　

terms

　of 　

the

　

width

−

thickness

　ratiQ 　

of

　

the

　

_plate

elements 　

based

　

on

　

the

　

convergence

　condition 　

of

　

the

　

axial

　strain 　and 　

the

　rate 　of　

the

　strain 　

_accumulation

本 論 文は

，

文 献 17 ｝

−

2D を ま と め た も のて

．

あ るu

　

＊

Assoc

．

　

Prof

．

，

　

Dep

亡

．

　 of 　

Architecture

，

　

Faculty

　of　

Engineering

，

　　

Kagoshima

　

Univ

．

，

Dr

．

　

Eng

．

＊＊

Prof

．

_，

　

Dept

．

　of　

Architecture

，

　

Faculty

　of　

Enginee

血

g

，

　

Mie

　　

Univ

．

，

Dr

．

　

Eng

鹿 児 島大 学工学部建築 学 科

　

助 教 授

・

工博

三重 大 学工学 部建 築 学 科

　

教 授

・

工博

一

₅₇

− ．

一

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

2.

Cyclic

Behavior

of

Beam-Columns

with

Square

Hollow

Section

2.1

Mathematical

Model

Figure

1

shows

the

model of a square

hollow

section,

which

is

divided

into

a number of elements

for

the

numerical

treatment,

The

following

dimensions

are

_given

to

the

model

:

b=d==48

mm, and

t=

3.

2

mm.

The

number

of

elements

n

is

taken

equal

to

11.

The

model

is

subjected

to

a constant axial

force

whose ratio

_p

to

the

yield

axial

force

P.

is

equal

to

O.

1

or

O.

5,

and

alternately･repeated

biaxial

bending

with

nondimensional curvature

e5.

=

di.d!e.

cos

e+

_¢

.blE.sin

e,

as

shown

in

Fig.

2,

where

the

yield

stiain e.=O.

14

%.

The

value

of

curvature

amplitude

is

taken

equal

to

3

and

direction

of

the

curvature

vector

g==O

or if4.

The

hysteretic

relation

between

stress

a

and

strain

e

considered

in

this

study

is

shown

in

Fig,

₃

where

s=ala,,

e=EIE,,

with

the

tension

being

taken

positive.

a,

denotes

theyield

stress. u-e relation

here

includes

the

effect

of

local

buckling

in

a

macroscopic

way.

Therefore,

it

is'not

the

stress-strain Telation

for

an

infinitesimal

element.

o and e

here

are

defined

as

the

axial

force

divided

by

area

and

the

change

of

Length

divided

by

the

original

length

for

a

bar

element,

The

stress-strain

reiation

is

composed of

three

relations

:

linear

relation

for

elastic

loading

and

unloading

_;

strain

hardening

type

linear

_plastic

relation

for

tensile

loading

;

and

d'egrading

type

non-linear

piastic

relatien

for

compressive

loading,

which

is

derived

by

replacing

a

locally-buckled

plate

by

a

number

of

buckled

bars

in

compregsioni5'.

Nondimensional

stress-strain

relations

_.f},{e)

and

g.(e)

represen'ting

the

stress-strain

,relations

in

the

plastic

range

are

specified

in

terms

of

the

strain ef

the

i-th

element as

follows:

-f}'(e')==lf::l)--e,+i)l:li'l:l''''''''''''''''''

''

'

''''''H'

''

'H-'''''''''`i)

g.(ei)=#(e,-1)+1･･-････････-･･････････････-･･････････････････････････････････-･-･･･--･･-･･････････---･-････(2)

f(e,>=-

4

A'E2(e,)+1

+2

AE<ei)-'----'''-''''''''''''''''''''''''''''''''''''''""'''''''''''''-'''''r

(

3

)

E(ei)=

-2(e,-eH)Ey-(ei-eB}2ei

A-h(d!t)

where

?,=maximum

strain

that

the

i-th

element

has

experienced

in

the

past;u#strain

hardening

coefficient;

eb=E,le.;

eR=critical

strain

at

which

the

local

buckling

occurs

;

dlt=width-thickness

ratio

of

the

_plate

element;and

h=a

_parameter

introduced

to

compensate

the

error

involved

in

the

formula

derived

by

treating

the

plate

buckling

as

the

bar

buckling.

T･he

yalue

of

e.

is

determined

from

the

test

results

of

the

local

buckling

of

tubular

beam-columnsi6),;

eB=-2.78(t12

d)21E..

The

present

analysis

omits

the

case

that

en<-1,

i.e.,

the

plate

element

buckles

elastically.

The

value

of

h

is

assumed

to

vary

linearly

along

the

side of

the

model

section

from

O

at

the

corner

(

i--1)

to

O.

15

at

the

center

(i--n+11Z).

The

stress-strain

relation

of

the

_plate

element

plays

a

key

rol

in

determining

whether

the

accumulation

of strain occurs

or

not,

which

is

one

of

the

criteria

for

the

evaluation

of, earthquake

resistance

cdpacity,

Therefore,

a

realistic

model of stress-$train

relatien

must

be

employed,

but

it

is

difficult

at

present,

because

of

the

lack

of

experimental

data.

Although

the

stress

deterioration

with

the

Ioad

repetition

and

2d

g2yJfi+1)/2212E"

itt

E.1)12

x

Fig.1

Model

section

58

-¢

_y--.'''''',`

`3'xNM

-31-dih-str-

_i3

_¢ x''-i]it-3

Fig.2

Curvature

history

se･

9p(e)

tan"iv

1-..za.g

'

::L'

eB1//1

-111

11 e

/dF.tlllp't

f'(e)'li

_P

-1

the

stiffness

degradation

due

to

the

resisual

deformation

are

not

taken

inte

account

in

the

_present

model,

this

model may

by

reasonabie

in

view

of

the

analogy

with

the

behavior

of

a

bar

subjected

to

the

repeated

axial

loading.

2.2

Results

of

Numerical

Analysis

The

model

_given

in

_2.

₁

are

treated

numerically on

the

assumption

that

plane

sections

before

bencling

remains

_plane

after

bending,

and

the

shear

deformation

can

be

neglected.

Results

of

the

numerical

analysis

of

the

model

section

in

Fig.

₁

subjected

to

alternately repeated

bending

are

shown

in

Figs,

4(a)

through

4(d).

The

nondimensional

bending

moment

m

and centroidal strain

e,

are

defined

as

follows

:

1O

1.oM

H-'r

-4

-･

-4

'

O.6

O.2

-O,2-O.6

-1

{a)

p

=

-O.

r

(b)

p

=

-O

O.8m

-4

₄

¢ r

---O.8'

O.8m

-4

₄

_¢

-.4-o.

.o1.,

e.

ee

1.oM

O.6

O.2･-O.2

-O.6

g

16

11

'211'

-1.0.1,

1.0

O.6O.2

-O.2-O.6

-1

Cc)

p=

-o.

1

.5'

o

8

1

C=

450

m ,C=

oe

Om

O.6

O.2

4

¢ r-O.2

(d)

p

=

Fig.4

Cycl

-O.6

-1.0-O.5,

C

=

ic

behavior

20/r

r

14eeo

4seof

moclel

in

Fig.1

eo eo

59

-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

m==MleIM.cos

e+M.IM.sine;

Ml,,

M,=bending

moments

about

x-

and

_y-axis,

･respectively;M.==

full

_plastic

moment

about

x-axis;e,=e,/E.;and

E,=strain at

the

centroid.

'

,･

It

is

noted

from

Fig.

4

that

the

_plastic

strain

at

the

centroid

converges

to

a

certain

value

when

the

axial

force

ratio

is

small,

but

otherwise

it

seems

to

diverge.

The

question

whether

or

not

the

deterioration

of

bending

moment

capacity

with

the

increase

in

the

load

cycles stops

is

strongly

related

to

convergence

or

divergence

of

the

centroidal

strain.

The

accumulation

of

the

centroidal

strain

and

the

reduction.of

the

rnoment

capacity

of

the

model

with

e=･450

are

both

smaller

than

those

for

the

model with

eFOe,

when

_p

=-o.

1,

and

the

strain

accumulation

is

slower

in

the

former

case.

On

the

other

hand,

when

_p=

-O.

5,

the

centroidal strain

does

not

converge

in

either

case, and

the

computatio.n

is

terminated

when

the

stress

at

the

extreme

fiber

in

the

tension

side

becomes

zero.

3.

Convergence

Condition

for

the

Centroidal

Strain

of

a

Beam-Column

3.1

Analytical

Model

.

In

order

to

investigate

_,the

problem

of

strain

_qccumulation

in

a

beam-coluinn'

subjectecl

to

uniaxial

bending,

consider

a rectangular

cross

section

2

b

×

2

d

as

shown

in

Fig,

5,

in

which

the

curvature

¢

.=

ZF

occurs

as

a

result

of

initial

bending.

Then,

alternately

repeated

bending

Ml,

is

subsequently

applied

with

the

curvature

amplitude

di..

Figure

6

shows

the

curvature

history,

in

which

the

numerals

denote

the

turning

_points

of

the

repeated

loading.

The

nondimensional

curvature

ip"

and

the

axial

strain

e"

at

the

turning

_point

n

are

_given

as

follows:

ipn=6+

¢

.

en==e:+vipn･-･････････････---･-････････････････-･----･-･････-･-･････-･--･･････-･--･-･･-････(4,

5)

where

_rp=yld

_;

_¢

=

di.1

O.o

;

O.o=e,ld

;

and

the

supersc'ript

n

indicates

the

values at

the

turning

point

n,

The

_present

study

treats

the

following

case

only

:

The

strain

of

extreme

compression

fiber

goes

into

the

degrading

stress

region

of

the

compression.

'

3.2

Convergence

Condition

Suppose

now

the

strain

distribution

in

the

cross

section

changes

from

the

one at

the

turning

point

n

to

the

other at

the

turning

_point

(n+112),

as shown

by

solid

and

dashed

lines

in

Fig.

7(a),

respectively,

with

the

assumption

that

the

centroidal strain

at

the

_point

n

is

in

the

inelastic

range

in

compression,

i,

e.

,

e7<-1,

The

curvature

change

is

equal

to

2

ip.

and

the

strain

increment

at

the

centroid

is

Ae,,

If

the

nondimensional

stress-strain

relation

is

given

a

_priori

as shown

in

Fig,

3,

the

stress

distribution

in

2d

y

y

mx

-2b

Fig.5

Model

section

%-"-1---2.-.n----¢

T-"L.J

5

¢

d---1232

n+12cyce

Fig.6

Loading

condition

e

n

n

eo _n

-.tf2hOrsN.ll"egn+.,

1r,

.4'N-s'

tt-sl

X/n+i

sTen.

y

(a)

Fig.7

Stress

, 1

.tl

n

compl

(b)

and strain

distributions

n

-60-the

section

becomes

as

shown

in

Fig.

7(b).

The

strain

increment

Ae,

is

determined

as

a

function

of

e:,

¢

.

and

the

axial

force

ratio

p

from

the

equiiibrium

condition

of

the

axial

force

on

the

section.

It

is

attempted

here

to

obtain

an

approximate

solution

for

_Ae,.

If

it

is

assumed

that

_there

is

no

change

in

the

strain at

the

centroid

in

the

_process

of

loading

from

the

turning

_point

n

to

(n+lf2),

i,

e.

_,

Aeo=O,

the

strain

and

stress

distributions

in

the

section

become

as shown

in

Figs.

8(a)

and

(b).

In

_general,

the

stress

at

the

_point

(n+112),

dashed

line

in

Fig.

_8(b),

_is

_not

in

equilibrium

with

the

external

axial

force

_p.

The

unbalanced

axial

force

Ap

is

_given

as

follows

:

Ap=(f7f(e)do+J[i'g(e)do)12-p･･---･-･-･-･･-･---･-･･････-･････････････････････---･-･･････-･･(6)

where

_{a=nondimensienal}

distance

between

_the

_{centroid and}

_the

point

of

zero

stress,

and

f(e)

and

_g(e)

are

functions

of nondimensional

strain

expressing

the

nondimensional

stress-strain

Telations

in

_the

compression

side

and

the

tension

side,

respectively.

If

the

centroidal

strain

has

already

converged

to

a certain

value

in

the

inelastic

range,

the

condition

that

Ap

=

O

is

satisfied.

Noting

that

dn=(ip-¢

.)de,

e"'if2=e:+(iS-

¢

.)rp,

a=f(e?)121ip.,

andextreme

fiber

strains

at

the

load

_point

(n+112)

are

_given

by

e:

±

(5-

ip.),

the

condition

for

_the

convergence

of

the

centroidal

strain

i.e.

,

Ap=O,

is

given

as

Ap=2

_(6!

¢

.)

[.Cl

O-",t,Ce,O.IMe!e'-i]!2f(e)de+J[1:'.";[.-l:,07.')

..

.,v,

g(e)de]-p=o''''''''''''''-'''-

<

7

)

3.3

Accumulated

Strain

If

the

solution

for

_e,

of

Eq.

(

7

)

is

not

found,

the

centroidal

strain will not

conyerge.

Then

the

strain

accumulation

Ae,

_must

occur

at

the

centroid

during

the

loading

_process

from

the

load

_point

_n

_to

(n+112),

in

order

to

compensate

the

unbalanced

axial

force

Ap.

The

actual

stress

distribution

becomes

the

one

shown

by

solid

line

in

Fig.9(a),

instead

of

the

distribution

shown

by

dashed

line,

The

difference

in

the

axial

force

AP

between

the

two

sets

of

stress

distribution

is

_given

by

_the

shaded

area

in

Fig,

9(a).

Foi

simplicity

of

the

manipulation,

the

shaded

area

is

appreximated

by

the

dotted

area

in

Fig.9(b).

Then,

the

expression

for

Ap

is

obtained

as

AP=

2

(ziL

ip.)

[y[1:O.",):,-..,,,

f(

e)de-J[1:O.",i...,

f(e)de

e

'

n seR n sNs2ersN n+tN..VNs r-･--2(.1n+ia sTen, n

iiANssNsN--p'--.-tJn

Comp.

(a)

(b}

Fig.8

Stress

and strain

distributions

on

the

assumption

that

AeC"i:=O

C12AiLder)s

/Ae,\O

n Aeo=e s

...zt.1,",.FSI･L:･s,-I'/'[.111,t･.lil,,./-"."i,1,x...1.ttA.tttt:./s//:;';"'/"'li';.L-･

AeefO

,;,),!(<t..--,iiL･jAeo=O

"

(a)

tb)

Fig.9

Stress

distributions

on

the

assumption

that

_Ae:'i/Z"=O

-61-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan

+(e#"!2-

eV)

lg(

e:+

g)-f(

e:)l]

･--････"--･･･････-･---･-･-･･;････････････-･-･･････････--

(

s

)

g=11+"(eg-1)-f(e?)l!(2-rd

[Phe

value

of

g

divided

by

2

(ll-

_¢

.)

indicates

the

distance

between

the

centroid

and

the

point

of

tension

yielding,

as

shown

in

Fig.9(a).

The

equilibrium

condition

for

the

axial

force

requires

Ap+Aff=O-H''''''-'''''''''''''''''''''''''''''''r''''''''''''''''''"'''"'''H''''''--''''''･t･･'HH･･･-･-(9)

Substituting

Eqs.(6)

and(8)into

Eq.(9)leads

to

the

expression

for

the

increment

of

the

centroidal

straln

as

e?+ii2r

e:=

-

[Le.O"'

+aM"r)

gc

e)de

+

zae.O.1'J;-',aJ

..,

f(

e)de

-2

_p(th

-

¢

r)]Ag(

e?+

g)-f(

e:)I''

'

(

10

)

The

nondimensional

bending

moment

at

the

load

_point

(n+112)

is

derived

as

follows,

m"'ii2=

2

(Jl

_o.),

[J[1

O"'feL"r),g(

e)

(e-

e:)de

+y[1:O.";.i;.Z,-,-..,

f(

e}

(e

-

e:"!z)de

+(e:'ii2--eC)lg(e:+O-f(e:)He:-e?'i!2+gl12]･････'･'''･････････････････-･･-'･'･･････(11)

3.4

Evaluation

of

Earthquake

Resistance

Capacity

Earthquake

resistance

capacity

of,a

thin-walied

steel

beam-column

may

be

evaluated

from

three

_points

:

the

centroidal

strain

converges

within

a

limiting

value

e,.

and

the

bending

moment

capacity

at

the

converged

state

is

large

enough

;

the

bending

moment

at

the

load

point

n

for

an

assumed

strain

level

e?=

e,.

exceeds

a

iimit

7n,.,

and

the

deterioration

factors

₇

_℃

'if' and

7".'

tf2

are

less

than

corresponding

limits

x,..

and

_7s,..,

where

7Z"/2

and

_7".'if2

are

factors

indicating

the

accumulation

rate

of

the

centroidal

strain

and

the

deteribrati6n

rate

of

the

bending

moment'carTiying

capacity

during

the

loading

process

from

the

'

load

point

n

to

(n+112),

defined

as

,

7n.+i!!=

e:+i!21e:-1

;

7n.+i12=1-mn+i!:lmn

---･--････････････････i････--･-･･････････

(12)

where

m=

mk+m.

for

the

biaxial

bending,

'

Earthquake

resistance

capacity

may

be

evaluated

in

two

steps

:

1.

If

the

solution

of

Eq.(7)

for

e?

is

found,

say

e,',

check

for

leo"L<e..

;

m(e:}>m..････---･'･--･---･･･--･････--･･･････-･･･････････････････----･･-･-･(13-a,b))

2,

If

e:

is

not

found

within

e,.,

assume

e::=

e,.

for

Eqs.

(14)

and

C15-a),

and

e:'ift==

e,.

for

Eq.

(15-b),

and

check

for

m"'i12>mc.････-'-'''''"H'''"'''-'"'''''H"'''''''-'"'"'''''''"''''-'''''''''''H"''''''''''''''''''"''(14)

'

rZ"f2<?t,,.,

r".'i12<h,.･･･''･･･''''･-'''･･･''''-･'''････''''''''''''''-'''""''''''･''･'････----･･(15-a,b)

The

values

of

e..,

m,.,

)t,,.

and

_7in..

are

specified

a

priori

as

design

criteria.

In

this

study,

the

earthquake

resistance

capacity

of

the

beam-column

is

evaluated

under

the

alternately

repeated

bending

applied

in

a

regular

manner,

which

differs

from

the

real,

random

situation

brought

by

the

earthquake.

However,

the

results

obtained

from

this

study

must

be

useful

as

a

approximate

measure

for

evaluating

the

real earthquake resistance

capacity.

4.

Analysis

of

Beam-Columns

Using

Discrete

Element

Model

4.1

Formulation

for

Discrete

Element

Model

For

the

case

that

the

cross

section

is

divided

into

a

certain

number

of-discrete

elernents,

Eqs.

C

7

)

and

'

(10)

can

be

modified

as

follows

:

Si

f(er･

)at+Sli

g(e7)ai-p=o-･･'･･････････''''''･''''''-'''''""''-HH'"""""'''''''''''''''''"'-"-'

(i6)

i=1 i`1

.

e:'if2-

e:

=='-I

X..

,

fC

e?"f2)at+

tT.li,

_g(e?･

)at-p)laZ

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

(17)

-where

a,=A,IA

;

A,=::

area

of

i-th

element

_;

A

;=

total

area

;

aZ=

Clg(e?+

g)-f(e:)l12/(

¢

-

ip.)

;

C

== ratio

62

--NII-ElectronicMbrary

pf

web

area

to

the

total

area

;

and

m.,

7nt==number

of

_yielded

elements

in

_compression,

_er

ih

_tension,

respectively.

Expression

for

the

nondimensienal

bending

moment

carried

by

the

discrete

element

model at

the

load

point

(n+112)

is

simply

derived

as

m""!2=-tl;.i,

f(e?"!i)a,

_nyi+te.t,

_{g(e?)a,oi+(e:'i!!-}

_{e?)a:r:-･･･-･''-''･･''-･-''''--･''''''''"''--''}

(18)

where

_{n,=nondimensienal}

_coordinate

_of

_i-th

_{element,r.=(g-e:'i!2+e:)!2/(ii-ip.);and}

m=

M/(Aayd),

4,2

Biaxial

Bending

The

approach

to

the

_problem

of

accumulated

strain

explained

_above

can

be

extended

te

the

case

of

a

beam-column

subjected

to

alternately repeated

biaxial

bending

with

the

nondimensional

curvature

history

shown

in

Fig.

_10.

The

strain

occurring

in

the

i-th

element

shown

in

Fig.

11

when

the

curvature

reaches

its

maximum

value

is

_given

as

follows:

ei=

eo-

ri

cos

e

(ip

_siR

_X?

₊

_¢

_rsin

e)+

r;

sin

e;L

(di

cos

_ie+

ipr

_cos

g)

=eo+r,Iipsin(a-fi)+ip.sin(en-e)1･････-････t-･--･･･････-･i･････････-･････--･--･･-･･････--･--･･-ag)

where

r,=R,ld=nondimensional

distance

between

the

centroid

and

the

i-th

element.

The

expressions

of

the

convergence

condition

and

the

accumulated strain

for

_the

case

of

biaxial

bending

_are

_thus

obtained

by

substituting

Eq.

(19)

into

Eqs.

<16)

_and

_(17),

_{respectiveLy.}

_The

_{nondimensional}

_bending

moments

about

x-

and

_y-axes

can

be

derived

in

a

similar manipuiation

_to

_the

derivation

_of

Eq.

(18).

4.3

Expressions

for

Limits

of

Axial

Force

Ratio

In

this

section,

simple

expressions

for

the

limits

of

the

axial

force

ratio according

to

the

_evaluation

criteria

for

_the

earthqttake iesistance

capacity

are

derived

for

a

beam-colurnn

with

square

hollow

section

idealized

to

a

3-element

model

shown

in

Fig.

12,

whieh

has

multi-linear

type

of

stress-strain

relation

shewn

in

Fig.

13.

First,

consider

a

beam-column

_with

_a

_square

holiow

_section

_subjected

_to

_biaxial

bending

about

_m-mLaxis

(axis

for

zero

stress)

shown

in

Fig.

_12(a)

with

the

curvature

amplitude

di.

and

the

initial

curvature

e=O.

This

corss

section

is

iclealized

to

a

3-element

model in

Fig.

12(b)

subjected

to

bending

about

m'-m'

axis

which

is

_parallel

to

m-7n

axis.

Nondimensienal

area

a,

and

angle

e

are

determined'

on

the

assumption

that

the

distance

from

the

centroid

is

d

for

the

elements

₁

and

3,

and

zero

for

the

element

₂

i.e.

,

r,=

rs=1

and

r,=O;

and

the

full

diy

i--tltoN}

',-' ¢

_-'''

B

¢

Fig.10

Curvature

history

for

biaxial

bending

y

A

a

R,1'x

e

e

i

Fig.

11

Discrete

element model

qtyNNxd

1

dxexNsNN

3

Nm1

(a}

_(b)

Fig.12

Three-element

model

1s

7,ttan'iu

11

eD

_eB-1:

:D,

_iec::le

,C::F,1t

-1uan

''-c2

-1

BA

tan-1pcl

Fig,13

Stress-strain

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

plastic

moment

of

the

square

hollow

section

bent

about

m-7n

axis

with

the

axial

force

ratio

_p

is

equai

to

that

of

the

3-element

model

bent

about

7n'-m'

axis.

Then,

_a,=

a3==

a,

a2=1-2

a,

and

a

and

e

are

given

as

follows:

(i)

For1+2p-tane>O

a=

4tan2e+(3-4p2-tan2e)!18･････････-････････J････････-･･････-･--･････-･････--･---･･････-･･･-･･(20)

e=tanJ'

l2

tan

e/(3-4

p2-tan2g)}+

n12

(iD

For1+2p-taneSO

a=(1+p)

l(1+tan6)2-2(1+p)tanigl2+l(1+tane)t-2(1+p)l2/2･-･-････---･･･-･-･･･--･･-･-(21)

e=tan"[i(1+tane)2-2(1+p)tan2elll(1+tang)'-2(1+p)l]+n12

Each

segment

of

the

compressive

multilinear

stress-strain

relation shown

in

Fig.

13

can

be

expressed

iri

_general

as

follows:

s=",(e-e)+g･･-････････-･･-･････-･--･--･･･-･･--･----･-･-･･-･･････--･･･-･･---"''-････････'''"･--･-(22>

where

_p,=pt.i,

?=eB

and

g=-1

for

the

segment

BC,

for

example.

The

expression

for

the

limiting

value

of

the

axial

force

ratio controlled

by

the

convergence

condition

for

the

centroidal

strain

within a

limiting

value

e..,

Eq.

(13-a),

is

derived

by

substituting

Eq.

(22)

into

the

convergence

condition,

Eq.(16),

with

ee=e..,

as

follows:

p)dwucw+d,xtef+5-H""-'"'''''""""'H"'''''"-""'""'''''""H'''''''''''''''''''''"'-'''''''(23)

where

a.=(1-2

a)(ecT-e)

af=alec.-

¢

.sin(e-e)-el

5=

att

ig5rsin

(e-6)+ee.I+Q-2

a)

gw+a(1-st+Sf)

The

expression

for

the

bending

moment

at

the

converged

state

with

e,=e,.

is

obtainecl,

as

follows

:

m=al"(ec.+

¢

.sin(e-e)-1)+1-pcf(e..-ip.sin(e-e)-?)-gfl---････････････-････---(24)

In

the

expressions

above,

the

subscnpt

w

indicates

the

quantities

related

to

the

center

element

of

the

3-element

model,

i,e.

,

the

web

plate

element,

and

f

the

flange

plate

element

in

compression.

The

values

of

_",.,

_u.x,

e,

g.

and

S.

are

determined

from

Eq.

(22)

based

on

the

condition

that

e:=e..,

Expressions

for

the

limits

of

the

axial

force

ratio according

to

other

criteria,

i,

e.,

Eqs.(13-b),

(14)

and

(15),

can

be

derived

with

a

similar

manipulation

shown

above.

All

expressions

can

be

written

in

the

ciosed-form

except

for

the

one

based

on

Eq.

C15-b),

The

expressions

for

these

limits

are

_quite

Iengthy

and

thus

omitted

here

[see

Refs,18)

through

21)].

4.4

Numerical

Examples

For

the

square

hollow

section

approximated

by

a

3-element

model,

the

relations

between

the

limits

of

the

axial

force

ratio

and

the

width-thickness

ratio

are

numerically

analysed,

based

on

the

criteria

given

in

3.

4.

The

curvature

history

assigned

is

as

shown

in

Fig,

10,

and

it

is

assumed

that

ip=O,

ip.=3,

fi=O,

e=oe

or

4sO.

The

parameters

of

stress-strain

relation

in

Fig.

13

is

specified

that

_g=O.

02,

ec-

eB=

-5,

ep-eB=-100,

e.=O.11

%

(a.=2.4t!cm2)

or

O.

20

%

(a,=4.2tlcm2).

The

values

xt,,

and

pt,2

are

taken

so

that

the

assumed

stress-strain

felation

in

the

compression range might approximate

the

one

given

in

Eqs.(1)

and

(3

).

h

is

constantly

O.

15

for

all

elements,

The

values of

nendimensional

areaa

of

elernents

1

and

3

in

Fig.12

is

taken

as

O.

38

for

e=OO

and

O.

35

for

e=450.

The

results of numerical analysis

are

shown

in

Figs.

14

through

18

in

the

form

of

the

relations

between

the

axial

force

ratio

_p

and

the

width-thickness

ratio

2

df

t.

which

are

determined

from

the'criteria

for

the

evaluation

of

the

earthquqke

resistance

capacity

given

by

Eqs,

(13)

through

(15>,

The

solid

line

ih

Fig.

14

indicates

the

maximum

value

of

p

determined

by

Eq.

(13-a),

when

the

centroidal

strain

just

converges

with

e#=e.

(=e./e.;e.=-11u+1).

The

condition

e:=e.

is

selected

since

the

bending

moment

becomes

nearly zero

when

e:

becomes

eqal

to

e.

in

the

process

of

stiain

accumulation.

The

dashed

and

dotted

lines

in

Fig.

14

are

determined

by

Eq.

(15-a),

where

?t,..

is

taken

equal

to

O.3

or

O.5

with

e?=eB-5

or

eB-10,

The

solid

and

dashed

lines

in

Fig.

15

indicate

the

maximum

value

of

p

detrmined

by

Eq.

(14)

with

m..=o

or

o.

3,

while

the

dotted

and

dash-dotted

lines

are

for

p

determined

by

Eq.

(15-b)

with

7h,.=O.

1

-64-p-e,s

-O.6

-O,4

-O.2

rscr=o.3

=etrB-s)B-10) e

20

40

2d/t

Fig.14

Limit

of

_p

based

on

the

convergence

condition and

the

value of

_x,

p-O.8

-O,6

-O.4

-O.2

213

o･

2e

40

_2d/t

Fig.15

Limit

of

_p

based

on

_the

values of

);,

and m

p-o.e

nO.6

･-O.4

-O.2

o

)

20

40

Fig.16

Effect

of curvature2dltamplitude

p-O.8

-O.6

-O.4

-O.2

o

20

)

402dlt

Fig.17

Effect

of

_yield

stress

p-O.B

-o.

-O.4

-O.2

)

O

₂₀

_{40 2d/t}

Fig.18

Effect

of

direction

of

curvhture vector

or

o,2,

both

calculated

with

e:=eB-5

or

eB-10.

Figure16

shows

the

p-2dlt

relations

for

the

curvature

amplitude

ip.=3

or

_6.

The

solid

line

is

determined

by

Eq.

_(13-a)

with

e:=e.,

The

dashed

line

indicates

the

maximum value of

_p

determined

by

Eq.

_(15-a)

with

_?c,..=O,

5

and

e?=eB-5.

The

value

of

_p

determined

by

Eq.

(15-a>

decreases

_with

the

increase

in

_ip.,

while

_the

value

of

_¢

does

not

affect

much

the

value

of

_p

determined

by

Eq.

(13-a).

Figure

17

compares

the

effect

of

_the

_yield

stress

level,

i.

e.,

2.4

tlcm2(E.=0,

11

%)

and

4.

1

tlcm2(E.

=o.

20

%

),

_on

the

_value

_of

p.

The

rimit

of

the

axial

force

ratio

need

to

be

set

smaller

with

_the

increase

of

the

_yield

stress

level.

The

effect

of

the

bending

direction

is

shown

in

Fig.

Is

with

g=O"

_,

uniaxial

bending

and

4se

_;

biaxial

bending.

The

direction

_angle

_e

has

_a

little

_{effect on}

_the

_value

_of

p

for

e,.=

e.,

but

otherwise

the

uniaxial

bending

iequires

the

smaller

limit

of

_p

than

the

biaxial

bending.

5.

Conclusions

In

order

to

ascertain

the

safety

in

the

ultimate

state

of

thin-walled

beam-columns

which

show

degrading

behavior

due

to

local

buckling

under

strong

earthquakes, a

_quantitative

evaluation

rnethod

and

_proper

criteria

for

the

earthquake

resistance

capacity

are

_proposed,

based

_on

_the

_convergence

condition

for

the

centroidal strain

ancl

the

indicators

representing

the

rate

of

strain

accumulation

_and

･the

rate

of

moment

capacity

deterioration,

_which

_are

derived

from

_an

_approximate

stress

distribution.

The

proposed

method

and

criteria

involve

the

axial

force

ratio,

width-thickness ratio

of

plate

element,

direction

of

the

curvature vector, curvature

amplitude,

_yield

strain

and strain-hardening coefficient

_as

parameters,

and

the

effects

of

_parameters

are

investigated,

The

_results

_obtained

from

_this

_stttdy

are

-65