鋼構造K形筋かい付骨組の終局水平耐力

(1)

NII-Electronic Library Service

[it

g]

Journal

of

StructuTal

and

Construction

Engmeenng

RgNees#ft-msvaXftscra=fi

uDc:624.

o14.2:624.

o7zg

CT[ansactiofls

of

AJJ)

No.

398,

Ap[it,

1989

ng3gse

･1gsgff

4

A

ULTIMATE

LATERAL

SHEAR

CAPACITY

OF

STEEL

FRAMES

WITH

INVERTED

V

BRACES

by

Toshibumi

FUKUTA*

and

Hiroyuki

YAMANOUCHI",

Members

of

A.

I.

J. 1,

lntroduction

In

high

seismic zones,

it

is

important

for

structural

designers

to

_pTedict

appropriately

the

seismic

_performance

of structures

designed

or underdesigning, especially

to

estimate

the

tiltimate

lateral

shear

capacity

of

them.

Refs.

1 and

2 discussed

simple evaluation methods on

the

ultimate

lateral

shear capacity of one-bay

and

one-story

frames

with

inverted

V

braces

_(chevron

shaped

braces).

However,

the

applicability of

these

methods

to

the

multi-bay and

multi-story

frames

has

not

yet

been

verified.

On

the

other

hand,

an evaluation method on

the

restoring

force

characteristics

of

the

frames

with

inve[ted

V

braces

was

_proposed

recently

by

Inoue

and

ShiTnizu

{Ref.

_3).

The

method,

however,

is

so complicated

that

it

is

not convenient even

in

designing

of

the

low-

and medium-rise

buildings.

Therefore,

structural

designers

stiLl need

1)

sufficient

knowledge

on

the

manner of

failure

of

the

braced

bay

beam

and

on

the

contribution of

this

failure

to

the

ultimate

lateral

shear capacity of

the

multi-bay and multi-story

frames,

and

2)

a simpLe evaluation method on

the

uLtimate

lateral

shear capacity

of

the

frames.

Thus,

this

_paper

discusses

the

low-

and medium-rise steet

buildings

with multi-bay, where

the

shear

deflection

becomes

a major

_part

of

the

inter-story

lateral

displacement

and

the

intermediate

braces

with

the

slenderness ratio of

70 to

130

are mainly

designed.

The

following

significant

iterns

are

investigated

in

identifying

lateral

shear

force

characteristics of

these

frames;O

evaluation

of

post-buckling

strength

of

a

pair

of

braces

in

a

frame,

and

2)

evaluation of contribution of

braced

bay

beam

to

lateral

strength of

the

total

frame.

Then,

these

items

are

incorpoiated

in

the

evaluation method

for

the

uitimate

lateral

shear capacity of

the

frames.

The

method

is

applied

to

the

scaled model

frames

which were

tested

as

_part

of

the

U. S,

-Japan

Cooperative

Research

Program

Utitizing

Large

Scale

Testing

Facilities

_(Ref.

4),

and

its

applicability

to

the

frames

with

inverted

V

braces

are

_generally

discussed.

2. Evaluation

Method

on

Ultimate

Lateral

Shear

Capacity

2-1

Lateral

Load

Carrying

Mechanism

Frames

with

inverted

_V

braces

have

a

lateral

load

carrying

mechanism

different

from

diagonal

braced

Irames

or

moment-resisting

frames.

The

evaluation method on

the

uLtimate

lateral

shear capacity of

the

frames

should

be

developecl

on

the

basis

of

their

characteristics of

the

lateral

load

carrying mechanism,

in

_particular

the

inteTaction

between

braces

and

braced

bay

beams.

Then,

in

order

to

make clear

the

above

_peculiarity,

the

experimental results

of

the

seismic

tests

on

the

scaled model

frames

with

inverted

V

hraces

(Ref.4)

afe

introduced

hore:

The

elevation of

the

typicai

frame

of

these

tests,

Test

Frame

No.

1,

is

shown

in

Fig.

1. Just

at an early

stage

of

the

lateral

story

drift,

such as

₁₁₃₉₆

radian of

the

Test

Frame

No.

1,

the

compression-side

braces

with

the

effective

slenderness ratio of

54

were

buckled.

Then,

with an

increase

of

the

story

drift,

tlte

compression-side

brace

reduced

its

axial

force.

On

the

other

hand,

the

tension-side

brace

was stretched, and

its

axial

force

Tose within a

limit.

The

absolute values of

the

forces

in

both

the

bTaces

were not

identical

in

this

range.

The

concenttated

force

that

balanced

with

the

difference

between

the

axial

forces

in

a

pair

ef

braces,

applied

vertically

to

the

mid-span of

the

braeed

bay

beam.

The

ultimate strength of

the

braced

bay

beam

was not so

large

that

it

_yielded

and was

_pulled

downwafd

by

the

concentrated

force.

The

downward

deformation

induced

the

additional shortening of

the

compression-side

brace

and

A

_part of this _paper was reperted in Ref.]z.

*

_Dr.

_of

_Eng.,

_Sefiioi

_ReseaTch

_Engineer,

_Structural

_EngineeTing

_DepaTtment,

_Building

_Researeh

_Institute,

_Ministry

of

Construction

(BRI)

*'

_Dr,

_et

_Eng.,

_Head

_of

_Structural

_Dynamics

_Division,

_Structural

_Enginee[ing

_Dcpartment,

_BRI

CManuscript

received

Ailgust

30, lgS8/Paper

Accepted

February

7,

1989)

(2)

-99-NII-Electronic Library Service

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jeck

v"---."

l

--"""""d

-i

-s

-t-r-ib-ution

beam

S-columttS7SOM-colu"n

..."..m--llll?ll

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tQeti

3150

m:

::

o o pa N N-celumn

･･-vr･II?tt

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Xkv dCtUdt reacti vnll unitimm

Fig.1

Test

Frame

No.1

accordingly resulted

in

almost no

increase

in

shovtening of

the

brace

exceeded

ten

times

the

_yield

axial

d'

story

drift

range.

Fig.3

shows

the

_peak

axial

force

in

the

tensien-side

single

difference

between

these

two

frames

is

the

detail

of

beams,

and

the

steel

beams

of

the

Test

Frame

No,

are

strengthened

by

the

concrete

floor

slabs.

2.

42 times

as

large

as

that

of

beams

in

the

Test

Frame

No.

the

test

frames

experienced

the

buckling

of

the

braces.

frames

had

almost same

axial

forces.

After

this,

the

diff

The

strength

_propoition

of

the

beam

to

the

brace

in

the

design.

Therefore,

as afesuLt of

the

experiments,

inverted

V

braces

_;one

is

to

evaluate

the

compressive stress

shear

capacity, and

the

other

to

estimate

the

strength of

2-2

Basic

Equation

of

Ultimate

Lateral

Shear

A

simple

method

is

_generally

useful

for

earthquake example,

the

load-displacement

relation

of

a

steel

frame

following

simple method.

The

braced

frame

is

portion.

The

poTtions.

This

method

is

applied

to

the

prediction

of

the

u

braces

As

illustrated

in

Fig.4,

the

frame

with

inverted

V

Table1Out-Line

of

Test

Frames

testframeNo.

-ltem

₁₂₃₄₅₆

compositeslabyesyesnononono

braces

in'lninout-lnno

mechanism

bbbbcb

loadingpassgmgggg

cf.in:

in-plane

buckling

expected

to

occur

out: out-of-plane

buckling

expected

to

occur

b:

beam

collapse

type

c: colunn collapse type

g:

gradually

increasing

cyclic

leading

m: more complicated cyclic

loading

the

elongation of

the

tension-side

brace.

As

shown

in

Fig.

2,

isplacement and

the

axial

force

dropt

down

inthe

axial

the

large

brace

of

the

Test

Frames

No.

1

and

No.3

in

each cycle.

A

beams;namely,

the

Test

Frame

No,3

has

bare

steei

1 have

the

same

dimensions

as

those

of

the

Test

Frame

No.3

but

The

calculated moment capacity of

beams

in

the

Test

Frame

No,

1 is

3 for

pesitiye

bending.

In

the

second cyclic

loading,

both

Up

to

this

loading

sequence,

the

tension-side

braces

in

these

erence

between

the

axial

forces

in

these

braces

became

large,

experiments

pertains

to

that

usually used

in

_practical

two

items

are

pointed

out as

general

problems

of

the

frames

with

level

of

the

brace

in

the

range of

the

ultimate

lateral

the

braced

bay

beam.

Capacity

response analyses as welt as

the

design

of structures.

For

with

X-type

braces

can

be

reasonably

_predicted

by

the

disintegrated

into

the

moment-resisting

fTame

_portion

and

the

brace

load-displacement

curve

of

the

total

frame

is

then

composed

of

the

load-displacement

curves of

the

two

Itimate

lateral

shear

capacity

of

the

frames

with

inverted

V

tens

1.0P/PyCOTTIP.

-t---Pu/Py

-s

ns.

o

-1.0

10

20 AIAy

Fig,2

Axial

Force-Axial

Displacement

Relation

of

S-Brace

in

2nd

Story

ef

Test

Frame

No.1

--100-braces

isdisintegrated

1.0 into

the

two

_portions.

The

PtlPy

o.o

No.1

t

Ittl SN'o'":3""li.. .t

brace

buckling

t

IZI]

Fig3EveryFirst

1234

loading

cycle

Peak

Tensile

Axial

Force

Story

of

Test

Frames

Ne.Iof

Braces

in

and

No.3

(3)

NII-Electronic Library Service

=

₊

Frame

with

Moment-Resisting

Inverted

V

Frame

Portjon

BracesFig.4

Disintegration

of

Frame

with

Inverted

V

Two

Sub-Structures

BracePortion

Braces

into

800Q(kN)

400 O

2

4

6 sstory

Frame

No.ln-a

drift(cm)

Fig,5

Representative

Comparison

between

Ultimate

Lateral

Shear

Capacity

given

by

Eq.1

and

Analytical

Load-Displacement

Ctirve

by

tic

Hinge

Method

of

Ref.IO

moment-resisting

frame

_portion

consists of

beams

and

columns

which

have

the

same sectional

_properties

and member

length

as

those

of

the

members

in

the

original

frame.

The

braces

and

the

braced

bay

beams

are

the

members of

the

brace

_portion,

The

beams

are

simply

supported

with

the

same span

length

and

full

_plastic

moment as

those

of

the

original

beams

in

the

braced

bays.

Further,

the

braces

have

the

same

buckling

load,

_tensile

_yield

strength and ultimate _cornpressive strength as

those

_of

the

_original

braces,

Now,

the

_ultimate

lateral

_shear _capacity

Q.

_of

_the

original

frame

with

inverted

V

braces

is

_given

as

follows;

Qu=Qbu+QTy--H--'"""'"H'"'"-'"H"'""'"'H"HHH'HH"'"'"H"'H'"HH"''"'-'-''"'"''h''''-'''"'(1)

where,

Qb.

is

the

_ultimate

lateral

_shear _strength _of

the

brace

_portion,

_and

Q..

is

_that

_of

_the

_{moment-resisting}

frame

portion.

Qbu

and

Q.,

are estimated

independently.

Stresses

in

an actual

frame

are redistributed

in

post-yielding

of

the

frame.

The

above

equation.

however,

does

not

incLude

the

effect of

the

stress redistribution

on

the

ultimate

lateral

shear capacity.

Now,

_this

effect will

be

studied

by

using

the

analytical results of many

cases

of

frames

in

the

section

2-3.

Fig.5

shows a representative comparison

between

the

ultimate

lateral

shear capacity

_given

by

Eq.

1

and

the

analyzed

load-displacement

curve of

the

frame

No.

1-1-a

in

Tabie

2,

Here,

the

a-factor

in

evaluating

Q..

of

Eq.

1

was assumed

to

_be

_1,

_O.

_This

comparison

implies

that

the

_{stress redistribution}

in

the

post-yielding

can

be

neglected

in

evaluating

the

ultimate

lateral

shear

capacity

in

the

light

of

the

structural

clesign

of

braced

frames.

2-3

Evaluatien

of

Qb.

The

ultimate

lateral

shear strength ef

the

brace

_portion

is

_given

by

the

following

equation;

Qbu=cQbu+tQbu'""'-'-'--'''"'"'"'"''-'"''"'"'''''''H-H'H'--'-'""H"HH'H-"'"-H-'"-'H'"HHHH(2)

where, _.Q,.

is

the

ultimate

lateral

shear

strength

of

the

brace

in

compression, and _,Qb.

is

_that

of

the

brace

in

tension.

As

mentioned

in

the

section

_2-1,

_the

axial

force-axial

displacement

relation of

the

compression-side

brace

has

a

trend

that

the

magnitude of

the

compressive

deformation

becomes

remarkably

large

and

its

stress

level

ultimately comes

to

_about

₂₀

_%

_or

₃₀

_%

of

the

yield

axial

force

in

the

range of

the

ultimate

lateral

s.hear capacity of

the

total

frame.

This

stress

level

is

defined

to

be

the

ultimate compressive strength

P.

of

the

brace.

Then,

.Qb.

is

representecl

by

the

cosine

component

of

P..

Estimation

_ojr

P. Braces

in

a

frame

are surrounded

by

beams

and

columns

through

_junction

elements

such

as

_gusset

plates

so

that

bending

moments _{as well as axial}

forces

are

transmitted

to

the

braces

corTesponding

_to

end-rotations and

deflections.

According

to

the

conclusions of

Ref.

5,

the

axial

force

in

the

_{post-buckling}

of

the

braces

restrained

by

beams

and

columns

is

equal

to

that

of a

pin-ended

brace

with

the

identical

sectional size and a

half

mernber

length

of

the

original

brace.

Therefore,

in

evaluating

P.,

the

brace

in

the

brace

_portion

is

idealized

to

the

pin-ended

brace

with

the

identical

section and a

half

member

length

of

the

original

brace.

There

have

been

many analytical and experimental researches on

the

_{post-buckling}

behavior

of

the

_pin-ended

brace.

Fig.

6

shows

the

load-displacement

curves of

the

_pin-ended

brace

in

Refs.

6 to

g.

In

_these

_methods,

_the

Paris's

one

gives

the

lowest

_estimation _on

the

resisting

forces.

Since

the

Paris's

method

_generates

the

displacement

without considering

the

axial

_plastic

deformation

at

the

_plastic

hinge,

it

_gives

an underestimation on

the

resisting axial

force.

(4)

-101-NII-Electronic Library Service

In

the

laige

deformation

range,

the

Kato's

method

_gets

the

highest

axial

force

in

these

methods

because

his

method

is

able

to

take

account of

the

expansion of

the

plastic

zone

that

eccuTs at

the

mid-span of

the

brace.

Here,

we will

_present

a

formula

that

woulcl enable us

to

evaluate

the

ultimate compressive strength

P.

of

the

brace,

The

axial

force-axial

displacement

curves of

the

pin-ended

braces

with slendemess ratio ef

35 to

65

are obtained

by

the

Kato's

rnethod.

According

to

the

foregoing

discussion,

in

tlte

_{post-buckling}

these

braces

correspond

to

the

braces

welded

to

the

frames

whose slenderness ratios are

70 to

130,

which a[e often used

in

low-

and medium-rise

braced

building

stTuctures,

aLmost

constant

in

the

range of around

10･A.,

where

assumed

that

the

ultirnate compressive strength of

the

following

equation

is

obtained as evaluating

P.

of

Pu==(O.5-O.065Ae)Ps''''''''''''''''''''''-'''''''''''''''

where, using

the

_yield

strain

E.,

X. is

defined

as

k,=

the

radius of

_gyration

of

the

section,

and

P,

the

experimental result of

the

Test

Frame

No.

].

the

experimental axial

force

in

the

large

dis

Estimation

of

,Q,.

As

discussed

in

the

section

2-1,

in

the

compressioll-side

brace

and

the

total

shear

force

(Pt.-P.)sine=Pgu････････････････････････････････････-･･

wheretension-side

brace

is

the

cosine component of

Pt.

as

tQbi

Ptu

cose==

Pg.

cote+

P. cose･･----･---Here,

P..

is

represented

by

a.P....

The

factor

iateral

shear strength of

the

tension-side

brace,

i.

e.,

shear

strength of

the

braced

bay

beam

simply supported

the

beam

and

L. is

a

half

span

length

of

the

braced

bay.

Then,

substituting

these

notations

into

Eq.5,

tQb.=min.(a.2Mgp!h+,Qb.,

where

his

the

story

height.

Here,

Eq.

Estimation

of

the

foctor

a

Now,

the

value

of

the

analytical results,

Fig.drift

of

the

cycLic

loading,

of

the

braces.

Here,

Lg

in

Pgye

times

as

large

as

Pgve

reached

to

two

tirnes

as

large

as

P,,.

in

the

_post-buc

span of

this

beam

should

have

a reverse sign of

the

distribution

should

be

like

the

one

drawn

in

Fig.

Next,

we analyzed some

frames

to

braees,

and

the

framing

on

the

a-factor

of

Eq.

6.

with a

_pair

of

inverted

V

braces

or with only

one-story-three-bay symmetric

frames

with

braces

in

The

analyticat method used

here

is

based

on

-102-Fig.6

1.0 PIPy

O2468

10 AIAy

'

Comparison

among

Analytical

Methods

as

to

Displacement

Curves

of

Pin-Ended

Braces

(slendernessratio==70)

From

Fig.

6,

it

is

found

that

the

axial

force

of

the

_braces

become

A,

is

yield

axial

displacement

of

the

brace,

Therefore,

it

is

brace

shouid

be

equal

to

the

axial

force

at

lo･A..

The

the

brace,

---･--･-･-････････-･-･-･---･----･--･-･---･･--(3)

VE;

Lli,

L

is

the

member

length

of

the

brace

in

the

frame,

_i

is

the

_yield

axial

force

of

the

original

brace.

This

equation

is

applied

to

The

dotted

and

dashed

line

in

Fig.

2,

given

by

Eq.

3,

weH explains

placement

range.

the

axial

force

P,.

in

the

tension-side

bTace,

the

axial

force

P. P..

in

the

braced

bay

beam

at

its

mid span satisfies

Eq.4.

".H"""h"".-H"-,H-H."..H-"."H-,,k",-"".---(4}

e

is

the

angle

between

axes of

the

brace

and

the

braced

bay

beam.

The

ultimate

iateral

shear strength

of

the

described

by

the

following

equation;

""".--.".,･-･-･･-･--･---･･-･---･-･････････--･(5)

a

implies

the

participation

of

the

braced

bay

beam

to

the

ultimate

the

ultimate

lateral

shear capacity of

the

frame.

P...

is

the

_yielti

{=2

M..IL.),

and where,

M..

is

the

full

plastic

mement

of

The

axial

force

in

the

tension-side

brace

does

not exceed

P..

P.cose)--･-･･--･･-･---･---･-･･"-･･--'･-'H"H""'H-HH-'""''C6)

6

means

that

tQb.

should

be

smaller value

in

the

two

terms

in

the

parentheses.

a-factor

in

Eq.6

wiLl

be

discussed

by

using

the

experirnental and

7

shows

the

total

shear

force

P,.

in

the

second

floor

braced

bay

beam

of

the

Test

Frame

No.

3

at

the

peak

story

P..

in

terms

of

P,..

were

back

calculated

by

the

measured

forces

{strain

_gauge

Teadings)

was estimated as a

half

length

of

the

clear

span

of

the

braced

bay,

_P..

reached

to

two

in

the

_positive

direction

loading

and reached

to

P...

in

the

reverse

direction.

The

fact

that

P..

kling

range

implies

that

the

shear

force

in

the

right-hand side

half

shear

force

in

the

left-hand

side

half

span,

i.

e,,

the

moment

8-b)

oT

Fig,8-c).

evaluate

the

effect of

the

strength

_{proportioning}

among

beams,

col"rnns and

Table

2

shows

the

analyzed

frames,

i.

e. , one-story-one-bay

frames

tension

brace,

ene-story-two-bay asymmetric

braced

frames,

the

interior

bay,

aRd

three-story

braced

frames.

(5)

NII-Electronic Library Service

2 ---

-F,

s'-==..r.'----'-:a

it

-`'-...s Xa tl

.:

i

;;;=fa

-i---,---p,gu---.o

`li

Pgu=PgyoM

Q'ngp

a)

a=1.0

MgpPgutl.5.Pgyong

Q'Mgp

b)

a=1

kpQ

Pgu=2.0.Pgyo

_ngp

Mgp

.5 c) a=2.0

Pgyo=2ngp/Lg

-O.02

0

O.02

O.06 R

Fjg.s

_Moment

_Distribution

_Pattern

_of

_Braced

_Bay

story

drift

angle(rad.)

Fig.7

Total

Shear

Force

P..

at

Mid-Span

of

M

Beam

vs.

Story

Drift

of

First

Story

of

Test

Frame

No.3

"tinax

Mpc

moment-curvature relationships of

beams

and columns weie assumed

to

be

a

bi-linear

model

(see

Fig.9).

Here,

in

Table

2,

the

deformability

_ip

ma./ip. of

beams

and columns was

assumed

to

be

2.0in

the

frames

witha

¢

_y

_¢

_max

sign of

_prime,

and

to

be

4. 4in

the

frames

_withoutasign

_Mpc:ful1

plastjc

moFvbent

of

_prime.

The

hysteresis

rule of

the

brace

adopted

in

_the

Fig.9

Moment-Curvature

Relation

of

Beams

and

analysis

was

theone

_proposecl

by

Jain.

A.K･

_{(Ref･11)･}

_columns

_used

_in

_Analysis

The

frames

were

defined

to

become

ultimate

when

the

bending

mornent

ln

beams

or columns reached

to

M...,

Only

the

lateral

_shear

force

_was _applied

_to

_the

frames.

The

lateral

shear

force

distribution

along

the

height

in

the

three-story

frames

was

_proportional

to

the

first

mode

of

the

natuial _vibration _of

the

frame.

The

a-factor

in

Table

2 is

the

ratio of

the

total

shear

force

ef

the

braced

bay

beam

at

its

mid span

to

the

sheai

force

P...

when

the

frame

becomes

ultimate.

Framing

and configuration of

the

structures

and

deformability

and strength

of

the

members are

impoTtant

factors

Table2

Contribution

of

Braced

Bay

Beam

to

Ultimate

Lateral

Shear

Capacity

_of

Frarnes

foi

Analytical

Case

Study

fral[be1)MblMc2)eember3)aframingframestoryMblMcmemberaf[aming

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3)c:stressexceedsfiLaximummoment

3..3t321O.8SO.93O.64

ibb1.681.6S1.52 atcolum

(ex:exterior

celumn,

in:interior

column)

b

:stress exceeds maximun momellt at

beam

* _:stresses

_did

_not _exceed

_hnx

_of _member

(6)

NII-Electronic Library Service

affecting

the

contribution of

the

braced

bay

beam

to

the

ultimate

lateral

shear capacity of

the

frame.

Frarnes

with many

bays

around

the

braced

bay

concerned

have

a

larger

a-factor

than

a one-bay

frame.

The

multi-story

fiame

has

also a

larger

a-factor

than

a

one-story

frame.

The

a-factor

of

ductile

frames

also

is

laTger

than

that

of

less

ductile

ones.

The

minimum value of

the

a-factor

is

about

1,

which

appears

in

a one-story

frame

with a single

bay

and

brittle

members.

According

to

the

above

discussion,

the

a-factors of

the

case

stndy

frames

are

between

1.

0

and

2.

0

exclusive of

the

brittle

frames,

and

depend

on

the

framing

and

ductility

of

the

structure.

As

for

design,

simplicity

of

the

evaluation

equation

is

one

of

the

important

factors.

Theiefore,

the

value of

the

a-factor

can

be

1.0 in

Eq.6

as a safety side estimation,

2-4

Evaluation

of

Q..

The

lateral

shear

strength

_Q..

is

defined

as

the

lateral

shear at

the

limit

state of

the

moment-resisting

frame

portion,

and

is

calculated

by

using

the

virtual work methocl.

In

_general,

this

method

for

calculating

Q.,

can

be

applied

to

the

frames

where

beams

and colurnns

have

excellent

deformability,

the

ultirnate strength of

the

joints

exceeds

the

full

plastic

state of each member, and

the

iateral

buckling

of

beams

are restrained

by

the

appropriate supports.

In

the

section

2-3,

the

a-factor

in

Eq.

6 is

assumed

to

be

1,

O. Therefore,

the

mornent

distribution

in

the

beams

and columns

becomes

the

one

illustrated

in

Fig.8-a)

for

evaluatio" of

Q.,.

In

the

above

evaluation

method

for

Q..,

the

full

_plastic

moment of

beams

is

obtained under

the

assumption

that

the

beams

are not subjected

to

the

axial

ferce.

The

full

_plastic

moment of

the

column

is

modified

by

the

axial

force

derived

from

the

sheai

force

of

the

beam,

the

additional

force

from

the

braces,

and

the

dead

and

live

load

of

the

upper

stories.

The

additional

force

from

the

braces

is

assumed

to

be

the

sine component of

the

buckling

force

of

the

brace.

After

the

buckling

ofthe compression-side

brace,

its

stress

i$

actually

reduced, and

the

axial

force

in

the

tension-side

brace

is

a

little

different

from

the

buckling

force.

Therefore,

this

assumption

_gives

a

little

overestimation

to

the

axial

force

in

the

column at

the

mechanism

state

of

the

moment-resisting

frame

_portion,

but

this

discrepancy

has

little

effect on

the

ultimate

lateral

shear strength of

the

mornent-resisting

frame

_portion.

According

to

the

analytical results of

the

frames

in

Table

2 except

for

the

frame

No.

1-1-c',

the

a-factor

exceeded

1.0

and

increased

with

the

increase

of

the

story

drift

<Ref.

12).

In

Fig.5,

the

lateral

shear

force

rises up after

the

story

drift

of

4

cm,

and

the

moment

distributions

in

the

beams

and columns

in

this

state are

like

the

one

illustrated

in

Fig.8-b)

or

Fig.8-c)

instead

of

the

one

in

Fig.8-a),

This

is

because

the

full-plastic

rnoment of

the

columns

increases

due

to

the

strain

hardening

and

because

the

braced

bay

beam

shares

the

increase

of

the

lateral

shear

force,

compensating

for

the

discrepancy

of

the

column

shear

force

from

the

lateral

shear

force

evaluated

for

Fig.8-a),

3. Experimental

Verification

The

method

for

evaluating

Q.

of

the

frames

with

inverted

V

braces

was applied

to

the

half-scale

three-story

steel

fTames.

These

frames

weie subjected not

_pnly

to

the

lateral

force

but

also

to

the

vertical

forces

at

the

top

of

the

thiid

stoiy columns.

In

modifying

the

full

_plastic

moment of columns.

the

axial

forces

in

the

columns were assumed

te

be

the

maximum values

measured

in

the

experiment.

Fig.10

shows

the

comparison

between

the

ultimate

lateral

shear eapacity analyzed

Cdashecl

lines)

and

the

experimental

load-displacernent

curves of

the

frames,

It

is

found

that

the

analytical

values

give

good

approximation

to

the

maximum

lateral

shear

force

of

the

test

frarnes.

In

these

test

frames,

width-to-thickness raties

(b!t

ratios) of

the

column section are

less

than

8. 2s

in

the

flange

plate

and

39.

4 in

the

web

pLate.

The

blt

ratios of

the

beam

section

do

not exceed

9.

61 in

the

flange

and

44.

0 in

the

web

_plate

(details

are

discussed

in

Ref.

4).

These

ratios are within

the

Blt

limitation

for

the

most

ductile

member

grade

in

the

Japanese

Building

Standard

Law

and

Enforcements.

The

slenderness ratio

Lli.

of

the

beam

about a weak axis

is

less

than

170,

where

L

is

the

span

length

and

i,

is

the

radius of

gyration.

of

the

beam

about a weak axis,

Thus,

the

moment-resisting

frame

portion

is

stable against

local

buckling

and

lateral

buckling

in

the

_{post-buckling}

of

the

braces

in

state such as

O,

02

radian of

the

story

drift.

Now,

the

brace

portion

shareF about only

_sO

%

of

the

lateral

shear

force

in

this

story

drift,

while

it

carries over

80 %

of

the

lateral

shear

foTce

in

elastic range.

The

behavior

of

the

total

frame

becomes

stable

because

of

the

contribution of

the

ductile

moment-resisting

frame

portion,

even

after

the

brittle

buckling

of

the

braces.

The

ultimate

lateral

shear capacity of

this

frame

can

be

well and sirnply evaluated

by

(7)

NII-Electronic Library Service

Q

kN400

No.3lststory

---".04

O.04storyraddrift

4oo

"--

_ultimate

_lateral

shear

capacity

Fig.10

Comparison

between

Ultimate

Lateral

Shear

Capacity

and

Experimenta]

Results

'Eq.

I.

4. Conclusions

The

main

purpose

of

this

paper

is

to

propose

the

reasonable and simple method

for

evaluating

the

ultimate

lateral

shear capacity of steel

frames

with

inverted

V

braces.

The

frames

_covered

by

_the

_proposed

_method

have

braces

_with slenderness ratio of

70 to

130,

and

ductile

beams

and

columns.

The

ultimate

lateral

shear capacity of

the

frames

is

_given

by

Eq,6.

The

a-factor

in

this

equation

_presents

the

contribution of

the

braced

bay

beam

to

the

_ultimate

lateral

_shear _capacity

_of

_the

frame,

_and

_has

_{a value}

_between

_1.

₀

and

_2.

₀

except

that

in

the

case of

brittle

frames.

As

a safety side estimation

in

_the

design,

the

value of

the

_a-factor

can

be

1.0 in

Eq.6.

This

evaluation method _was _verified

to

_give

reasonable ultimate

lateral

shear capacity of

the

frames

with

inverted

V

braces

in

comparison with

the

experiments.

5. AcknowIedgment

The

authors wish

to

express

the

deepest

appreciation

to

Prof,

Ben

KATO,

the

University

of

Tokyo,

for

his

valuable

advice

and encouragement,

The

authors are also

thankful

to

Dr.

Isao

NISHIYAMA

of

Building

Research

Institute,

Mr.

Naokuni

ENDOH

of

(8)

-105-NII-Electronic Library Service

Toda

General

Contractors

discussions.

6. References

1)

2)

3)

4)

5)

6)7)

8)

9)

10)

11)

12)

Co.

and

Mr,

Tomoyuki

WATANABE

of

Maeda

General

Contractors

Co.

for

their

useful

.

Shibata,

M.

and

Wakabayashi,

M..

"Ultimate

Strength

of

K-type

Braced

Frarne",

Tfansactions

of

A.

I. J,

,

No.

326,

April

l983,

pp.1-9

{in

Japanese}.

Muto,

K. ,

Tsugawa,

T.

and

Gote,

Y.

_, "A

DernonstTative

Study

of

Aseismic

Design

on

A

Large

Tufbine

Building

with

K-Type

Braced

Frame

Part2",

Transactions

of

A.I.J.,

No.360,

Feb.

1986,

pp,44-53

(in

Japanese).

Inoue,

K.

and

Shimizu,

N.

_,

"Plastic

Coliapse

Load

of

Steel

Braced

Frames

Subjected

to

HorlzontaL

Force",

Transactions

of

A.I,J.,

No.388,

June

1988,

pp.59-69

(in

Japanese}.

Fukuta,

T. ,

Nishiyama,

I,

andYamanouchi,

H.

_, "Elastic ancl

Plastic

Behavior

ofSteel

Frames

with

Concentric

K-Braces",

Transactions

of

A.I.J.,

No.392,

Oct.

1988,

pp,56'67.

Fukuta,

T.

and

Yamanouchi,

H. ,

"Post-Buckling

Behavior

of

Steel

Braces

with

Elasticaily

Restrainecl

Ends",

Transactions

of

A.LJ..

No.364,

June

1986,

pp.10-21,

Paul

C,

Paris,

"Limit

Oesign

ef

Co}umns",

Journal

of

the

Aeronautical

Scienees,

January

1954,

pp,43-49.

Igarashi,

S.,

Inoue,

K.,

Kibayashi,

M.

and

Asano,

M.,

"Hysteretic

_{Characteristics}

_of

_Steel

_Braced

Frames,

Part

₁

Behavior

of

Biacing

Members

uncler

Cyclic

Axial

Forces",

Transactions

of

A.I.J.,

No.196,

June

1972,

_pp.47-54

(in

japanese).

Kato,

B.,

Akiyama,

H.

and

Ineue,

K.,

"Post-Buckling

Behavior

of

Steel

Short

CoLumn

Subjected

to

Axial

Foice",

Transaetions

of

A.I,J.

No.229,

March

1975,

pp.67-76

{in

Japanese).

Shibata,

M.

_,

Nakamuia,

T.

and

Wakabayashi,

M.

_,

"Mathematical

Expression

of

Hysteretic

Behavior

of

Braces

Part

1".

Tiansactions

of

A.I,J.,

No.316,

June

1982,

pp.18-24

(･in

Japanese).

Takada,

M.,

"Static

_Analysis

on

Full-Scale

Six-Story

Steel

Test

Building",

Visiting

Researcher's

Report,

Building

Research

Institute,

Ministry

of

Construction,

March

]983

(in

Japanese).

Jain,

A,

K. ,

Goel,

S,

C.

and

Hanson,

R.

D.

_, "Hysteresis

Behavior

of

Bracing

Members

and

Seismic

Response

of

BTaced

FTames

with

Different

PrQportions",

Repert

No.

UMEE

78 R

3,

Civil

Engineering

Department,

University

of

Michigan,

Ann

Arbor,

Michigan,

July

1978.

Fukuta

T

"Seisrnic

_Perforrnance

and

Elastic

and

Plastic

Behavier

of

Steel

Frames

with

Concentric

K-Braces",

Doctoral

7tt

Thesis,

University

of

Tokyo,

july

1987

(in

Japanese},

-106

(9)

NII-Electronic Library Service

【

論

　

文

】

UDC ：524

．

014

．

2 ：624

．

0ア

2 ．

9 日本建築学会構造系論文報告集第 398 号

・

1989 年

4

月

鋼構

造

K

形筋

か

い

_{付骨組}

の

終局

水

平

耐力（

梗概）

正会員正会員

福

山

田

内

俊　　文

＊

泰

　　

之

＊＊

　

1．

序

　

地震危険

度

の

_高

い

地

域

では

，

建

物

の

地震動下

での

挙動

，

特

に

，

その

終局水平耐

力を設計時におおむね

推定

してお

く

こ

と

が

必要

で

あ

る。

1 層

1

スパン

K

形

筋

かい

付鋼構

造骨組

の

終局水平耐力を簡略

に

評価

する

方法

は

文

献

1 ）

と

2 ）

に

述

べられているが，それが

多層多

スパン

骨組

にも

適用可能

か

否

かは

確

かめられていない。また

，

多層

K

形筋

かい

付骨組

の

復元力特性

について

井上ら

が

提案

した

方法

は

，

複雑

で

，

_{中低層骨組}

の

_構

造設計

に

利用

し

得

る

も

のではない。このように，

多

層

多

スパン

K

形筋

かい

付骨組

の

終局

水

平

耐

力

に

関

しては

十分

な

情報

が

得

られているわけではない

。

　

本論

は

，

中低層多層多

スパン

K

形筋

かい

_付

骨

組

の

水

平力耐荷特性を明

らかにし

，

その

復元力特性評価

上

重要

な

筋

かい

_（

細長比

70 −

130 ）

の

座屈後耐力

と

筋

かいスパンのはりの

水

平

耐力寄与分を明確

にし

，

これらを

骨組

の

終局水平耐

力

評価方法

に

導

入する

。

最後

に

，

本法を

日

米

共同大型

耐

震

実験

研

究

で

得

られた

実験

結

果

に

適用

し，その

有用性

を述べるQ

　

2 ，

終局水平耐力

の

評価方法

　

2 −

1 　水平力耐荷特性

　

骨

組

の

終局水平耐力

を

適切

に

評価

するため

，K

形

筋

かい

_{付骨}

組

の

実験結果

を

用

いて_，この

骨組

の

_{水平}

力

耐荷

特性

を

明

らかにする

。

　

図

1

に代

表

骨組

（

骨

組

No ．

1 ）

の

立面

を示す

。

圧

縮側

筋

かい

〔

有

効

細

長

比

54 ）

は_，

層間変位

がごく

小

さい

時

に

座屈

し

，

層間

変位

の

_増

加

に

伴

い

_，

その

軸力を減じ

る。この

時

，

圧

縮側

と

引張側

筋かいの軸力の

大

きさは異なり

，

両筋

かいの

軸力

につり

合

う

鉛

直

力

が

筋

かいはり

交差部

に

作用

する

。

通常

，

はりの

強度

は

．

卜分でないので

，

はりは

降伏

し_，

筋

かいはり

交差部

は

鉛直方向

に

変位

する

。

この

鉛直変位

に

よ

り

，

圧縮側筋

かいの

_{軸変位}

はさらに

_{増加}

し，

引張側筋

かいの

軸変位

は

減少

する。

層間

変位

の

大

きい

範

囲で_，

筋

かいの

圧縮変位

は

降伏変位

の

10 倍

を超え

，

その

軸力

は

大幅

に

減少す

る

（図

2 ）

。事 _{建設省建築研究所第} 二研究部

　

主任研

究

員

・

工

博

＊＊ _{建設省建築研究所振動研究}_室_長

・

_工_博　（1988 年 8月30 口原槁受理

，

1989 年 2 月 7日採用決定〕

　

各繰返

し

変形下

で_，

骨組

No

．

1

と

No

，

3

の

_筋

かいに生じる

引張

力

の

_最

大

値

を

図

3

に

示

す

。

両骨組

の

違

いは

，

はりの終局

曲

げ

耐力

のみである

。

第

2

サ

イ

クル

時

に

，

両骨

組

とも

筋

かいが

_{座屈}

し，その

後

，

両骨組

の

引張側筋

かいの

軸力

に

差

が

生

じる

。

すなわち

，

両骨組

のはりのせん

断

力

の

_大

きさは

，

第

2

サイクル以降異なる

。

　

これら

骨組

のはりと

筋

かいの

_強度

の

_割合

は_，

通常

の

設

計

で

用

いられる

も

のであるので

，

_{実験結}

果

から

，

終局水

平

耐力

を

論

じるような

変形域

での

_筋

かいの圧

縮

耐力の

大

きさの

評価と筋

かいスパンのはりの

耐力

の

評価

が，

K

形筋

かい

付骨組

に

共通

の

問題

であることが

分

かる

。

2 −

2 終局

水平

耐力

の

基本式

　

地

震応答解析

や

構

造設

計

には，

単純

な

方法

が

有用

である。

例

え

ば

，X

形

筋

かい

付鋼

構

造骨組

の

荷重変形曲線

は

，

柱

はり

剛接骨組部分

と

筋

かい

_{部分}

の

_荷

重

変

形曲線を合成

す

るとい

う単純

な

方法

で

合

理

的

に

椎定

できる

。

この

方法

を

，

K

形筋

かい

付骨組

の

終局水平

耐

力

の

_評

_価

に

適用

する

。

　 K

形筋

かい

付骨組を

2

つの

_{部分}

に

分解

する。

柱

はり

剛接骨組部分

は

，

元

の

骨組

と

同

じ

断面

を

持

っ

_柱

_{とはりか} ら

成

る

。

筋

かい

部分

は

，

筋

かいと

筋

かい

交

差

部

に

接合

されるはりから

成

る

。

この

筋

かいは

，

元

の

筋

かいと

同

じ

座

屈

荷

重

，

降伏荷

重

およ

び終局圧縮耐力を有

する。また

，

はりは元の

_{骨組}

の筋かいスパンのはりと同

』一

の

材長

および

全塑性

モ

ー

メントを

有

する

単

純

ば

りである

。

元

の

骨組

の

終局水平耐力

は

両

部

分

の

終

局

水平耐

力

の

和

とする

（

式

1 ）

。

実際

には

，

降伏後応力

の

_{再配}

分が

行

われるが，

式

1

ではこの

効果

が

考慮

されていない。

次節

に

示

す

骨

組

解

析結

果を用いて

，

応力再配分

の

終局水平耐力

に

及

ぼす

影

響

の

程度を調

べた

。

その

結果

，

構造設計

の

観点

からみると

，

応力再配分

の

終局水平耐力

への

影響

は

無視

できることが明らかとなった

（

図

5

）

。

2 −

3 筋

かい

部

分の

終

局

水

平

耐力

の

評価

　筋

かい

部

分の

終局水平耐力

は_，

筋

かい

圧縮側

の

終局水

平耐

力と

引

張

側

のそれとの和とする

（

式

2 ）

。

前述

のように

_，

_全

_体

_骨

_組

の

_終

_局

_水平耐

力を議

論

するような

変形

の

範

囲で圧

縮側

筋

かいの

変形量

は

非常

に

大

きく

，

その

軸力

は降伏軸力の

20

から

30

％になる

。

この

時

の

軸力

を

筋

かい

終局

圧

縮耐力

とし

，

筋

かい圧

縮側

の

終局

水

平耐

力は

，

107 一

N工工

一

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