外圧を受ける周辺固定部分円筒殼の座屈と変形性状に関する理論解析的研究(梗概)

(1)

[wh

!1

Jou[nal

of

StrucLural

and

Construction

EngineerLng

UDC :624. o74. 43:624. o4 :s3g. 3s4

(Transactlons

of

AIj)

No.

364,

June,

lgB6

Htseca\2msinLxas

cvasst

ag

364 e-vatu

61 ij

6 fi

THEORETICAL

STUDIES

OF

THE

BUCKLING

AND

NONLINEAR

DEFLECTION

BEHAVIOR

OF

CLAMPED

PARTIAL

CIRCULAR

CYLINDRICAL

SHELLS

UNDER

EXTERNAL

PRESSURE

by

KAZUO

UCHIYAMA'

and

SEISHI

YAMADA'",

Members

of

A,

I,

J.

1. Introduction

Many

circular

cylindrical

roofs

have

collapsed

during

the

snow

in

Japan,

and most of

them

seem

to

have

buckled

generally]).

Thus

the

buckling

criteria

of

_partial

circular

cylindrical shells

have

required

for

the

designs.

In

this

paper,

to

make clear

the

basic

_general

buckling

behaviors

of cylindrical shell roofs, we shall

take

the

case of

_partial

cylindrical shells

submittecl

to

the

action of uniform normal

_pressure.

For

complete

cylindrical shells under external

_pressure,

the

bilckling

behavior

was

fully

explored

toward

the

beginning

of

the

1970

s2).

On

the

other

hand,

for

the

_partial

ones,

it

does

not seem

to

have

been

fully

looked

into

theoretically

in

_particular.

The

strongest reason might

be

that

the

_partial

cylindrical shells

have

not only

the

curved edges

but

also

the

straight edges.

Uchiyama3)

experimentally clarified

the

buckling

behavior

and expressed an approximate method of estimation of

the

buckling

stress

for

the

design,

Singer

et

al`)

obtained

the

buckling

loads

of simple-supported

_partial

cylindrical

shells

based

on

Donnell's

linear

equatiens neglected

the

_prebuckling

deflection

in

a wide range of

the

_geometrical

parameters.

Yokoo

and

NakamuraSj

theoretically

obtained some examples of

the

symmetrical noniinear

deflection

behavior

of

clamped

_partial

cylindrical

shells

using

the

finite

differential

method,

And

theyM

carried out experiments

on

the

pre-and

post-buckling

behavior

for

the

shells with

free

edges ancl represented an empirical

formula

for

the

buckling

loads.

Kokawa')

theoretically

analyzed

the

symmetrical and asymmetrical

_general

buckling

behavioi

of reticulated cylindrical

roofs

with

arbitrary

generator

edges.

Suzuki

and

OgawaS)

a]so analyzed

that

behavier

using

a simplified method which

dealt

bending

rigidity as rotational spring stiffness

of

member

joints.

The

authors")･'")

showed

the

experimental

data

of

the

overall

distribution

of

initial

_geometrical

imperfections

and

deflections

for

many accurate specimens of clamped

_partial

cylindrical shells, and

discussed

the

buckling

behavior,

The

object of

this

_paper

is

to

clarify

the

buckling

loads

and

the

nonlinear

deflection

behavior

of clamped

_partial

cylindrical shells under external

pressure

in

a wide range of

the

Batdorf

parameter

Z

and

of

the

aspect ratio a systematically.

We

obtain

not

only

the

symmetrical

snapthrough

buckling

loads

but

also

ttie

asymmetrical

bifurcation

ones

applying

the

Galerkin

procedure

to

Donnell]s

nonlinear

eqliations.

The

po$sibility

of asymmetricat

buckling,

which

have

not

experimentaly

been

established,

is

indicated

for

some shells.

The

effects of

the

_parameters

on

the

deflection

behavior

and

the

buckling

modes are also shown, ancl

the

characteristics are

in

_good

agreement with

the

authors',

experimental

results.

2. Method

of

Analysis

Let

us

consider

a

thin-walled

_paitial

circular cylindrical shell of

longitudinal

length

L,

wall

thickness

h,

radius

R

and central angle

ip,

which

is

made of

homogellerous,

isotropic

elastic

materiat with

Young's

modules

E

and

Poisson's

ratio

v.

The

middle surface of

the

shell

is

refeTred

to

cyiindrical coordinates as shown

in

Fig,

1,

Distances

'

_PTofessor,

_Department

_of

_{Architecture,}

_Faculty

_of

_Engineering,

_Tohoku

_Univeisity,

_Dr.

_Eng.

#

_Reseafch

_Associate,

_Department

_of

_{ATchitecture,}

(Manuseript

[eceived

October

11,

1985)

(2)

Architectural Institute of Japan

NII-Electronic Library Service

Arohiteotural エnstitute 　of 　Japan

from

the

　middle 　surface 　are 　measured 　

by

　a 　coordinate 　z

，

_positive

inward

．

Displacemeot

　components 　

in

the

　x

，

Y

，

i

．

and 　

z

directions

，

　 respectively

，

are　

deno1ed

by

u

，

v

，

　and 　

w

．

The

　shell 　

is

　considered 　

to

be

subjected

to

uniform 　external 　

_pressure

_q

_（

_positive

inward

_）

．

On

the

basis

　of　

the

Donnell

nonlineaf 　

theory

　of　

the

　cylindricaL 　shell

_，

the

basic

　equations 　are　

given

　as

fo

｝

lowsZ

川 1 ： ∂

nx

　　　

∂

nxy

　　十　　　；

0

∂

x

　　　

∂

y

∂πエン

　　　

∂πり ∂

x

＋

_可

一

＝

o

−

_・

（

房

∂2 ＋ ∂∂

y

z・

）

2 ・＋

争

・

導

…

巉

晶

…

穿

・・一・

Q

Fig

．

1 Sign

　convention 　and 　shell 　　　

geometry

．

　　　　

・

………・

・

……・

…・

・

………・

…

；

………・

（

1 ）

Eh3

where

　

P ＝

　　　　

12 （

1 一

ノ

｝

In　

equation

_（

1 _）

_，

n

エ

，

　

ny

，

　

and

　

nxy　

are

　

the

　

stress

　

components

　

per

　unit 　

length，

　

and

τ

　シ η 　 η 1 ーレ

0

−

　 2 　ん y 　

E

一

1

レー

01

，

］

　籌

＋

弖

儲

）

2

・

ll

一

詈

＋

去

（

迦

∂

y

）

z ∂u 　　 ∂

v

　　　 ∂

w

∂

w

∂

y

＋

『

認

＋

蕊

一

冴

The

following

　nondimensional 　

forms

　are　

introduced

　　

ξ

＝ _ユ：

／

_」_し

　

，

　

_η

＝

y

_／

R

_φ

，

　　

σ＝

Ru

_／

L

〜

L

，

V

二

v

_／

_φ

h ，

w

＝

_ω

_／

ん

，

　　Z ＝

s

／

i

’

＝

_」

L2

_／

R

_ん

_（

Batdorf

_parameter12

〕

_）

　　

α；

L

／

R

_φ

　　　

（

aspect 　ratio

）

　　

Q

＝

肌 ’

q

／

π2D

　

（

10ad

_parameter

_）

・

一・

・

一一・

・

…噛

曁

・

一

…

一

・

一

・

…

_（

₂

_）

・

……・

…………・

………・

……・

…・

・

…・

・

…

_（

₃

_）

　

The

　nondimensio _耳al

basic

equations 　are 　

der

玉ved 　

by

出e　

intrQduction

ofequatlons

（

2 ）

and

（

3 ）

into

　equation

_（1

）

，

as　

follows

_：・x

−

・

｛

離

・

1 −

・

喋

・

1

＋

）

・

諾

一

・・

剽

　

・

誓

｛

礎

・

面

・

劉

・

1

＋・

・

錯

諾

　＝

OLy

−

_・

_｛

標

・

（

≒

の

ぢ

i

￥

＋

〔

1

＋

の

冪

昜

一2

誓

｝

　

・

鰐

｛

・

襟

・

1 −

・

訓

珊

・

鐸

蠶

＝

0 し

一

誓

_・…

_β

・

（

　∂

u

　 ∂

v

P

　

∂ξ

　

＋

　

∂η

　

_w

）

一

去（

_妾

・・

券

）

2 ・・・

β

［

離

・・α

誓

笋

｝

嬰

・

1 −

・

舞

器

・

者

（

∂

w

∂

ξ

）

！

一

_・

畔

　

＋

｛

・

肇

＋

考劉

哥

刊

一

・）

言

嵩

弩

＋

氛

咢

）

w

鬱

］

・・

［

離

・・

聯

｝

闇

＋・

・

1 −

・

鑞

讐哥

　

＋・

（

Yge

＋・

穿

｝

（

Th

）

2

］

　＝

0 …・

…・

…………・

_（4

_＞

一 81 一

N工工

一

Eleotronio _Library

(3)

U=Z

£

AtJU}j=Z

£

AwsmminesinnJrrn

tj iJ

V=2]ZB"V}J=ZZBijsinmEngsinnjnn

IJ iJ

W=ZZC"VXJ=

£

Cvsinm`nesinnjnosinnesinnn

tJ IJ

where

mi

and

nJ

are

the

longitudina}

wave number and

the

application of

Galerkin's

method

to

equation

(

4 },

the

Galerkin

where a=a2 and

fi

=zlVi=-IJT

The

independent

paTameters

for

the

shell

_geometry

in

equation

(4)

are

the

Batdorf

parameter

Z

and

the

aspect ratio a

instead

of

the

four

geometrical

vaLues

R,

h,

L

and

ip

in

equation

(1),

The

bundary

conditions

are

aw

w==

ae=u=V==o,

at

e=O,i

l""

_-"".

_"""-.

_.

_..

_.

_(s)

W=aevro=U==V=o,

at

_n=o,1

l

The

displacement

functions

U,

V,

and

W

are

assumed

in

the

form

of

a

linear

combination

of

functions

multiplied

by

unknown coefficients

A",

B:J,

and

C"

which are not only

independent

of each other

but

also satisfy equation

(

s

},

and

they

are written as

---･---･--･---･---･-(6)

circumferential wave number, respeetively.

By

system of equations

is

_given

by

.('

f'

L. u},dpto==

.ti

-('

L. v,,dcao=.ti

Jf'

L. Viljdedo=o

'''''''''''''''''''''''''''''''''''''''''''''''''''''''''(

7 )

By

substituting equation

(

6 )

into

equation

(

7 ),

the

nonlineai algebraic equations,

involving

the

load

_parameter

Q,

for

the

unknown coefficients

AiJ,

Bw,

and

Cw

are obtainedia),

The

numerical selutions of

the

algebraic equations

are

_procured

step-by-step

by

both

a

limear

incremental

method

and

the

Newton-Raphson

method,

The

bifurcation

point

for

an asymmetrical

deflection

mode

is

obtained as

the

_point

at which

the

determinant

of coefficient matrix

in

the

linear

simultaneous algebraic equations

for

the

supposed asymmetrical mode

is

zero,

or

regarded as zeroi`).

The

computation

was

_performed

on a

digital

computer

(ACOS

99-NEAC

1000)

at

Tohoku

University

in

double

precision.

3. Numerical

Results

First,

we checked

our

numerical results with some

_pTevious

theoretical

results.

In

ref.

[15]

_,

the

comparison with

the

results of

Yokoo

etal.

obtained

by

using

the

finite

difference

method5i

was

described,

FiguTe

2

also shows

the

good

agreement with

Crisfield's

results

obtained

by

using

the

finite

element

method]6) as

to

nonlineat

iarge

deflection

behavior.

In

the

computation,

parameters

mt,

and

nj

in

equation

(6)

are

_given

as

follows:

for

symmetrical mode;

15

fn,=2,2,2,2,2,4,4,4,4,6,6,6,8,8,10

Ul

nj==1,3,5,7,9,1,3,5,7,1,3,5,1,3,1

mt=1,3,5,7,9,1,3,5,7,1,3,5,1,3,1

vl

nJ==2,2,2,2,2,4,4,4,4,6,6,6,8,8,10

m,=1,1,1,1,1,3,3,3,3,5,5,5,7,7,g

wl

nJ=1,3,5,7.9,1,3,5,7,1,3,s,1,3,]

(Ns=3

×

15)

for

asymmetrical mode;

m,=2,2,2,2,4,4,6,8

vl

nJ

==

2,

4,

6,

8,

n,2 n, n, n

m,=1,],1,1,3,3.5,5

vl

nJ=1,

3,

5,

7,

n-1,

n+1,

n-1,

n+1

mt=1,1,1,L3,3,5,7

Wl

nj=2,4,6,8,n,2n,n,n

(IVb=3

×

8)

-82-Q

10 5 o PRESENTCRISFIELD .

(FEM)

a=1.oZ=30.5

O

l

2 5

Wc

Fig.2

Comparison

of

load

Q

versus center

deflection

VVI

curves wlth

Crisfield's

results.

(4)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

CENTER

ARCH

(g=O,5)

a=1.0Z=200

-1.0

AS

'ttttts11tL-O.5

,ln'V'

o

O.2

'f,'fO.4ttl

KO,6no.s'''J-t-'xLO /-sK

hLX

'

ab

11S XxxK'

_..i

NSVxN't-i--t

O.5,x-LX1Xdzlij-f/c

ls,it' -'

1,o

'W,h..et--'''

(a)

Fig.3

Example

of

_pre-and

_{post-buckling}

deflection

behavior

for

the

(

a

)

Circumlerential

distJibutions

of

deflectien

W

along the

Cb)

Load

Q

versus center

deflection

iS(L

curves.

_Sym.

]o

'-.

c

dAsym.

e

20 Q

10 Ocr

b

a

Wc

-O.5

O O.5

{b)

shell

having

a=l.O and

Z=

200.

center arch at various

load

levels.

where

N.

and

Al.

are

the

total

numbers of

terms

for

symmetrical and asymmetrical

displacement

modes, respectively, and n

is

an even

number

of

circumferential

half-waves

of asupposed asymmetrical

deflection

mode,

On

the

basis

of

the

deflection

modes

in

the

longitudinal

direction

are

s]ighted

in

this

study.

The

difference

of syrnmetricaL snap-through

buckling

loads

calculated

by

two

adoptions which were

N.=3

×

15

and

tV.=3

×

21

_{was about}

O.5 %

in

the

_{case of}

th,e

_shell

having

Z==300

_and

a=1.o.

Poisson's'ratio

y was

taken

as

_O.3

in

_this

study.

Next,

we shall elucidate

the

variation of

the

distribution

of

_pre-and

_{post-buckling}

deflection

with applied

_pressure

Q

for

clamped

_partial

cylindrical shells.

Because

of space consideration,

the

results

foronly

the

shell

having

Z=20e

and a=1.0 are shown

in

Fig.3

(a),

In

this

figure,

solid and

clotted

lines

correspond

to

the

circumferential

distributions

of

deflection

W

at

_pre-and

_{post-buckling}

equilibrium

_points

a,

b,

c,

d,

and e

in

Fig,3{b).

The

nonlinear

deflection

behavior

is

remarkably recognizable not only after

btickling

but

aLso

before

buckling,

and

it

is

iound

that

the

efiect oi

_prebuckling

nonlinearity can not

be

omitted

in

this

Problem.

4. Effects

of

the

Geometrical

Parameter

Z

.

The

_geometrical

_parameter

Z

is

widely

known

as

the

Batdorf

_parameter

for

a complete circular cylindrical shell`Z) and

_participates

the

aspec,t ratio a

in

the

independent

_geometrical

parameters

for

a

partial

circular cylindrical shell as shown

in

eqdation

(

4 ).

Calculations

weie carried out

for

Z$loOO.

Table

l

shows

the

comparison

between

the

_present

buckling

loads

Q,.--RL2q,.ln:D

and

Yarnaki's

results of complete cylindrical shells subjected

to

lateral

_pressurei8].

The

buckling

behavior

of

_partial

cylindrical shells seems

to

differ

from

that

of

the

complete ones without straight edges.

However,

it

is

considered

that

the

dominant

component of

buckling

stress

is

the

circurnferential

Table1

Buck]ing

loads

for

partiar

circular cylindricat shells. membrane component

for

both

_the

_partial

ones and

the

cemplete ones, so

the

similarity

is

anticipated when

a

is

small

enough.

The

buckling

loads

Q,,

for

a=1.

0

and

a

=2,

O

lie

16-29

%

and

62-91

%

higher

than

those

_of

the

complete

shells,

respectively.

The

variation of

Q,.

with

Z

is

also shown

in

Fig.4,

where n,.

denetes

the

buckling

mode

number which

is

defined

as

the

circu'mferential

half-wave

nUmber of

the

dominant

component of

the

symmetrical

or

asymmetric-al

deflectien

at

the

buckling

_point

in

the

case of

the

syrnmetrica] or asymmetrical

buckling,

respectively.

The.

buckling

load

_Q..

and

the

buckling

rT{ode number

n..

become

large

as

Z

increases.

In

the

case of

a=1.

o

PRESF.Nl'RESULTS

.1a･=1,Oa=2

40----F----5013.72J-L

7017.oq2.a'

1oo20.30.29'

15023.B736' f

₂₀₀

300

400

500

7001000

26.9232,3835.7741,

l547. 2756,27E

4L49.53.S7.69,78.

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134240462362636062

UAMAKI,$

QoCCOMPLETE)

10. 6311.63'13,

44

, 15.

77.1

9,

O52L

8626, 6330, 6834,2B40.584a, 56

-83-NII-Electronic Mbrary

(5)

1OOQcr

50

20

10 20Fig,4

o

Fig.7

Effects

O.1

50of

the

O.2

100 geometrical

o.s

e

o.4

200 parameter

Z

en os 200

025

400 O.]O 700 O.50

Longitudinal

the

buckling

eedist[ibutions

points

in

the

o

O.5

W

LO

n=1.0Q=Ocrof

deflection

case of a=1W at.o

60 a=2.0

oJa=1.o

so

s

500 1000

Zthe

bllckling

loads

_Q.r.

40

50

20

10

o

Fig,5Loadcurves

O,1

O.2 VOI.UME

CHANGE

Ve

Q

versus voturne change

in

the case of a=1.0

O.3 va

-O,5

O

O.5 LO

1.5

2.0 -to

7o,s

o

O.5

1,O

1.5 2,O

CENTRAL

DEFLECTION We

(a)

CENTER

DEFLECT[ON

W,

{b)

Fig.6

(a,

b)

Load

_Q

versus center

deflection

WZ

curves

in

the cases of a=1.0 and a==Z.O,

respectively,

･

the

shells

for

2<4s

do

not

buckle,

ancl

the

non-buckling range expands

into

Z<60

in

the

case of

a=2.

o.

The

discrepancy

between

the

authors'

experiment9L'O}

denoted

by

the

triangle

symbols and

the

_present

theory

might

be

caused

by

the

initial

imperfections'9).

Figures

5,

6,

7

and

₈

are

the

variation of

the

deflection

behavior

with

Z. In

Fig.

s,

the

initial

stepe of

the

load

_Q

(6)

-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan oO.1O.2

nO.5

O.4O.5

1Z=700

_...7..

a=1.o

.-.'

Z=200

s''-LY.--Jrt!''

Z=70

LZ=100

Q=Qcrl

'

CENTERARCHCC=O.5)

z=4oe

(a)

HO.5

o O.5w1,O

o

.5

5

o

5

Fig.

8 (

a,

b

)

Variatiens

ofcircumferential

distributions

of

deflection

along the center aTch with

Z

in

the cases of a=1.

0

and a=

2. 0,

respectiveLy,

versus

the

volume change

_V}

curve

increases

as

_Z

increases,

where,

from

equations

(3)

and

<6),

V}=

v.lhLRip=Jgr]

Jl']

wa6du=

_C.!4

-･･---･---･-･-･･--:----･-････-･･-･----･i･-･･-･----･-･(

8 )

'

The

circles and

the

triangles

represent

the

syinmetrical

buckling

_points

and

the

asymmetrical

bifurcation

_points,

[espectively.

In

this

figure,

the

even nombers accompanied

by

the

triangles

are

the

bifurcation

mode numbers.

When

a shell

has

an asymmetrical

bifurcation

point

before

syinmetrical snapthrough

buckling,

it

is

common

for

the

load

after

buckling

to

decrease

in

the

same manner as

the

curve

ce

in

Fig,

3(

b

),

However,

in

the

case

of

the

shell with

Z

=::40 and

a

=:

1,

O

which

does

hot

have

a,symmetrical

buckling

_point,

the

secondary equilibrium

_path

with

_n;2

has

a

positive

sLope,

Therefore

we understood

that

it

does

not

buckle.

According

to

the

results of

the

calculation, asymmetrical

buckling

(n..=even)

occurs

for

_Z==

(80-400)

and

Z=

(100-250,

700-1000>

in

the

cases of

a=1.0

and

a=2.0,

respectively.

And

it

should

be

more emphasized

that

the

'

asymmetrical

buckling

_points

on

the

load-deflection

curves are very close

to

the

symmetrical

buckling

_points

in

all

'

cases as shown

in

Figs.5

and

6,

'

Figure

7

shows

the

distributions

of

W

in

the

longitudinal

direction

through

the

maximum

_points

at

the

buckling

loads

_Q...

For

all

values

of

the

geometrical

parameters,

the

dominant

component of

the

longitudinaL

distributions

of

W

was one

half

wave mode

in

this

study.

It

is

in

_good

agreement with

the

_previous

experimenta] results9)･iO)･t7).

On

the

other

hand,

the

distribution

of

VV

in

the

circumferential

direction

at

Q=Q..

becomes

of a

high

order as

Z

increases

as shown

'in

Fig.8.

Figure9

indicates

that

the

distribution

of asymmetrical

buckling

deflection

in

the

¢

ircumferential

clirection

becomes,

of

a

high

order

as

Z

keeps

increasign.

5. Effects

ef

the

Geometrical

Parameter

a

In

this

chapter we shall

discuss

the

'effect$

of

the

_geemetficat

_parameter

a,

which

is

defined

as

the

aspect [atio

/.'Lxx

a=2.0

'li

ttO

.i

O,1

SLo,5x o.s T]

s

O.4[

-o

,

O.5

o

5.0

N

L

i

N s Z=200 ,

Z=700

'

s

N

t

Ii L i ! _r CENTER

ARcH

X

₁

cf=o.s)

N...-../

ASYM.BUCKLINGMODES

Fig.9

Examples

of asymmetricat

modes

for

a=2.0

7

/Jlr',r

O.5 w

1.0buck]ing

20

oox L uo

10

2.0

4.0

O.5 1,O

a

Fig,

le

Effects

of

the

geometrical

parameter

a on

the

buckling

Lead

Q,.

in

the case of

Z=zoo.

(7)

--Table2Comparison

between

the authors' experimentai results and

the

present

theoreticai fesults

for

the

buckling

loads

and medes.EXPERIMENT THEORYQcr aoqcrQcrncrQerncvEXXTH dee･"Pg

%

O.72so1.1729.0636.7579

O.82701,2731.5537.25S5 O.95601,2731.5539.45so L15501.2631.3442.2474 1,43401,6340.3347.6485 i,91301.9949.3254.63go

Z=477.

60Q

50

40 h=O,3mm,

R=L=1

Z=200

50mm,

Et5.

6GPe,

a=5.0 a=2.0

fi

2

ASYM,BfFURCATION

S\M.

BUCKLING

a

±

1.5

a=Lo

4

atO.8

'

: a=O.6

' '

',/-KS"t.compLete

stselL

under LateraL Pressure

NG

TION

'

50 2o

10 o'j'''''-O.1

O.2

O.3

VOLUME CHANGE V,

versus yolume change

VL

curves

in

of

z=zoo.

-1,O

-O,5

O

O.5

1.0

1.5 CENTER

DEFLECTiON

W,

Fig.11

Loacl

_Q

versus center

def]ection

_VVZ

curves

in

the

case of

Z=200,

-O.5

Z=200

a[o.s

o

o.1

e.2

o.s

n

o.4 os

Q=Qer

CENTER ARCH

{C=O.5)

Fig.13

Variatien

of circumferential

deflection

along

the

case of

Z=ZOO.

o O.6

O.5

1.5

w

].o

1.0 distributions

centeT arch with a

o s

o

ofin

o

Fig.12Load

_Qthe

case

ASYM.

BUCKLiNG

MODES

Hg.14

Examples

of asyrnmetrical

for

z=zoo.

O.5 w

.o

buckling

modes

LIR

¢

in

equation

{

3 ),

on

the

buckling

load

and

the

deflection

behavior

on

the

basis

of

the

results

for

Z=2oo

and

Z

=477.

In

this

study, a was varied

from

O.6 to

4.0. Figure

lo

shows

the

variation of

the

buckling

load

with

a.

The

ordinate

is

the

ratio

of

the

buckling

load

for

_partial

cylindrical shells

Q..

to

those

for

complete cylindrical shells

Q,

obtained

by

Yamakit8).

The

buckting

load

Q..

increases

with an

increase

in

a, and

that

of

a=4.

0 is

about

three

times

as

laTge

as

Qo.

As

a

keeps

decreasing,

Q..

decreases

like

approaches

Q,,

and

the

buckling

mode shows more and more waves

in

the

circumferential

direction.

(8)

-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

In

Table

2,

the

_present

results are compared with

the

_previous

experimental results9)

in

the

case of

Z;477.

The

present

buckling

loads

are

10-26

%

higher

than

the

experirnental ones,

HosyeveT,

the

agreemen,t

between

the

theory

and

the

experiment as

to

the

effects

of

the

_parmamete[

a on

the

buckling

loads

and mpdes

is

obtained.

The

load-deflection

behavior

is

shown

in

Fig.11.

In

the

initial

loading

stage

_the

nonlinearities are not very recognizable,

however,

these

are remarkably recognizable

in

the

vicinity of

the

buckling

load..

The

initial

slopes of

Q-

VL

curve

in

Fig.

12 have

a

constant

gradient

of

294fl

when

Z

=200,

and

this

value

is

larger

than

the

gradient

26011

{denoted

by

a

bioken

line)

for

the

complete cylindrical shell.

The

difference

is

caused

by

the

clamped

longitudinal

straight edges _of

the

partiaa

_cylindrical

_,shells.

The

results _of

Figs,

5

and

12 indic4te

that

the

initial

_slope of

Q-

V,

curve

depends

not on

the

_parameter

a

but

on

the

_parameter

z.

Figure

13

shows

that

the

symmetrical

deflection

mode

in

the

circumferential

direction

has

more and more waves as a

keeps

decreasing.

The

circumferential

distribution

of asymmetrical

buckling

deflection

becomes

ef a

high

order as

a

keeps

decreasing

as shgwn

in

Fig.

I4.

6. Conclusiens

,

'

ApplyLng

the

Ggterkin

procedure

to

Donnelt]s

nonlinear equations,

the

buckLing

and nonlinear

dgflection

behavior

of

_partial

circular cylindrical shells under external

_pressuTe

is

the6retically

clarified.

We

treat

the

fully

clamped

beundary

conditions so as

to

compare our results with

the

_previousi

experimental results,

Main

contents are

summarized

as

follows

:

(

1 )

The

load-deflection

relationships

in

the

initial

loading

stage are almost

linear.

'However,

the

nonline'arities

are remarkably recognizable

in

the

vicinity of

the

buckling

_point.

(

2 )

The

nondimensional

buckling

load

Q..

increases

with an

{ncrease

in

the

Batdorf

_parmeter

g,

and with an

increase

in

the

aspect iatio a.

This

is

in

good

agreement with

the

(3)

It

is

possible

to

buckle

asyrnmetrically as well a,s symmetrically.

However,

when

the

shell

has

an

asymmetrical

bifurcation

_point

before

symmetrical snap-through

buckling,

the

bifurcation

_point

is

very

close

to

the

symmetrical

buckling

_point.

'

(

4 )

The

initial

slopes of

the

nondimensionaHoad

Q

versus

the

nondimensional vDIume change

_V,

curves

aTe

almost constant

irrespective

of

a

if

Z

is

fixed.

(

5 )

The

dominant

cornponent of

the

longitudinal

distribution

of

deflection

is

one

half

wave mode

in

each case.

The

circumferential

distribution

of

buckling

deflection

has

more and more waves with an

increase

in

Z

and

with a

decrease

in

a.

These

are also

in

_good

agreement with

the

_previous

experimental results.

This

study $howes

that,

by

using

this

analytica] rnethod, reasonably accurate solutions can

be

systematically

ebtained

for

the

buckling

loads

and

the

deflectien

behayior

of

_partial

cylindrical shells,

in

the

range of values of

Z

and

a

analyzed

in

this

paper.

The

present

parametical

resutts may

be

available

for

the

design

of

gylindrical

shetl roofs.

Reterences

･

1}

Uchiyarna,

K.,

et al

:

Snow

DaTnage

to

S{eel

Structures

in

Tohoku

and

HokuTiku

District,

Dec.19BO-Feb,

]981,

Jollrfial

of

Society

of

Steel

Construction

of

Japan,

Vel.17,

No.185,

pp.1-4g,

1981,

(in

Japanese}

2)

Yarnaki,

N.

:

EIastic

Stabi]ity

of

Circular

Cylindrical

She[Is.

North-Holland

Series

in

App!ied

Mathematics

and

Mechanlcs.

Vol.27,

North-Holland.

Amsterdam,

1984.

3)

Uchiyarna,

k,

:

The

Buckling

of

Cylimdricai

Shell

Roofs,

Laige

Span

SheLls,

Proceedings,

of the

_I.A.S.S,

_Congress,

pp.743-758,

1968.

4)

Singer,

J.,

Meer,

A.

and

Baruch,

M,

:

Buckling

of

CyLind[ical

Panels

under

Lateral

Pressure,

Aeronautical

Journal

of

Royat

Aerenautical

Society,

Vol,73,

pp.169-172,

1969.

5)

Yokoo.

Y.,

Nakarnura,

T.

and

Kanoh,

A. :Numerical

AnaLysis

en the

Snap-Threugh

Buckling

of

Clamped

Partial

Cylindrical

Shells,

Technical

Papers

of

Annual

Meeting,

A.I.J.,

pp.1069-1068,

1971.

_(in

_Japanese)

6)

Yokeo,

Y.

,

Nakamura,

T.

and

Miyazuki,

J. :

A

New

Method

of

Experimental

Buckting

of

Circular

Cylindrical

Roof

Shetl

Models

with

Free

Edges

subjected to

External

Pressure,

Int.

J,

Non-Linear

Mechanies,

Vol.6,

_pp.633-647,

1971,

7)

Kokawa,

T. :

Gene[al

Buckling

of

Cyllndrical

Reticulated

Shell

Reofs

with

Arbitrary

Generator

Edges,

Part

1,

2

and

3,

T[ans,

of

A.I.J.,

No.280,

pp.53-65,

1979;No.283,

pp,76-86,

]979;Ne,295,

pp.51-61,

1980.CinJapanese)

8)

Suzuki,

T.

and

Ogawa,

T,

:

Buckling

Analysis

of

Reticulated

Cylindrica[

Shel[

Roofs,

Part1

and

2,

Trans.

of

A.I.J.,

(9)

-87-9)

10)

]])

12)l3)

14)

l5)

16)

]7)

18}

19)

No.288,

pp.29-37,

1980:No.295,

pp.45-50,

1980,

(in

Japanese)

Uehiyama,

K.

and

Suzuki,

A. :

The

Buckling

ofthe

CLamped

Cylindrical

Sheii

Roofs

under

External

Pressure,

Proceedings

of

the

I.A.S.S.

Congress,

pp.133-147,

l979.

Uchiyama,

K. Yamada,

_S.

and

Minami,

H. :

Experimental

Studies

of the

Buckling

ef

Clamped

Cylindrical

Panels

under

ExtemaL

Pressure,

Technical

Papers

of

Annuat

Meeting,

A.I.J.,

pp.1103-ll04,

1985.

(in

Japanese}

Donnell,

L.H.

:A

New

Theoiy

for

the

Buckling

ef

Thin

Cylindrical

SheHs

uncler

Axial

Compression

and

Bending,

A,S.

M.E,

Trans.,

Vol.

56,

pp,

795-806,

1934,

Batdorf,

S.B.

:

A

Simplified

Method

of

Elastic

Stabllity

Analysis

for

Thin

Cylindrical

Shells,

NACA

Rep,879,

1947.

Yarnada,

S.

and

Yamada,

M. :

Buckling

and

Postbtckling

Behavior

of

Half-Loaded

Shallow

Sphe[ical

Shetls,

Int.

J. Non-Llnear

Mechanics,

Vol.20,

pp.239-248,

1985.

Uchiyama,

K.

and

Yamada,

M. :

BuckLing

of

Clamped

Impe[fect

Thin

ShaL]ow

Spherical

Sheils

under

External

Pressure,

Partl

and

2,

TechneL

Rep.

Toheku

University,

Vel.39,

_pp.101-l32,

1974;

Vol,40,

_pp.l-23,

l975.

Yamada,

S.

and

Uchiyama,

K. :Theoretical

Studies

of

tlte

Buckl{ng

of

Clamped

CyLind[lca[

Panels

under

External

P[essure,

Part

1,

Technical

Papers

of

Annual

Meeting,

A.I.J.,

_{pp.ll05-ll06,}

1985.

(in

Japanese)

Crisfield,

M.

A. :

Solutien

Procedures

for

Non-Linear

StructuraL

Problems,

Recent

Advances

in

Non-Linear

Computational

Mechanics,

Pineridge

Press

Lirnited,

pp,2Z-23,

1982.

Yokoo.

Y.,

Nakarnu[a,

T.

and

Matsuda,

A. :

ExpeTirnental

lnvestigation

of the

Stabi[ity

of

Clamped

Partial

Cifcular

Cylindrical

Shells

subjected

to

External

Pressure.

Recent

Researches

of

Structural

Mechanics,

Uno

Shoten,

Tokyo,

pp.333-343,

1968.

Yamaki,

N. :

Influence

ofPrebuckling

Deforrnations

on

the

Buckling

of

CircularCylindrica]

She]Ls

undef

External

Pressure,

Rep.

Institure

of

High

Speed

Mechanics.

Tohoku

University,

Vol.21,

pp.81-104,

1970.

Yamada,

S,

,

Uchiyama,

K.

and

Yamada,

M. :

Experirnental

InyestLgatlon

of the

Buckling

of

Shal;ow

Spherical

Shells,

Int,

J. Non-Llnear

Mechanics,

Vol,18,

_pp.37-54,

1983,

(10)

-88-Architectural Institute of Japan

NII-Electronic Library Service

Arohiteotural エnstitute 　of 　Japan

1

論

　

文】

UDC ：624

．

074

．

43

：

624 ．

04 ：539

．

384 日本建築学会構造系論文報告集第

364

号

・

昭和 61 年 6 月

外

圧

を受

け

る

_周

辺固定部分円

筒殻

の

座屈と変形性状

に

関

する理

論解析的研

究（

梗

概

）

τ

E

会員正会員

内

山

田

和

聖

夫

＊

志

＊＊

　

1 ．

序

　

従来

，

円

筒

形屋

根

の

積雪

による倒壊災

害例

は

非常

に

多

い1 ）。これらの

倒壊

の

多

くは

屋根面

全

体

の

座屈破壊

と

考

えられ

，

設

計

に

当

たって全

体座

屈に

対

する criteria が必

要

とされている

。

本文

では

，

円

筒

形屋

根

の

基礎

的

な

全体

座

屈

性状を明ら

かにするため，

荷重

は

面

にノ

ー

マルな

一

様

分

布

の

部分円筒殻

の場

合

を

取

り

扱

うこととする

。

　

外

圧を

受

ける完全円筒

殻

の

座

屈

性状

については

，

1970

年代初頭

にほぼ十分に

解明が

な

さ

れている

％

しかし_，

部分円筒殻

については

，

境界条件

の

扱

いがより困

難

なこと

も

あって

，

特

に理論面で

不十分

な

状

況といえよう

。

　

内

山町

ま_，

一

様外圧を受

ける

部分円筒殻

の座

屈

実

験

によっでその

座屈性状を明ら

かにし

，

設

計

用

座

屈応力を

求

めるための

近似的手法を提案

した

。

Singer

ら4 ｝_は

_，

_{周辺}

単純支

持

の

部

分円

筒殻

の

座屈荷重を

，

座屈前変形

を無

視

した

Donnell

型

の

線形座屈条件

式を

解

くことで，

形状

係

数の

広

い

範囲

で

系統的

に

求

めた

。

横

尾

・

中村

5｝_は

_，

周

辺

固定部分円筒殻

の

_{飛移}

り座

屈付

近の非

線

形

変形性状

を

，

差分法

を

用

いて

数値解析

を

行

った

。

さらに

彼

ら 6）_は

_，

母

線縁自

由

の

部

分円

筒殻

の

座屈実験を行

い

，

座屈荷

重

に

関す

る

実験式

を

求

めた

。

粉川

71は_，

任意母線縁境界

をもつ

_{円筒形網目}

シェル屋

根

の

全体座屈

についての

_{理論解析}

を

行

い

，

対

称

ならびに

非対称座屈性状

を

明

らかにした

。

鈴木

・

小河

S）

CS

，

円

筒形

トラスシェル

屋根

の座

屈性状

を

，

ト

ラス

_部材

の剛

接度

を

バ

_{ネ剛性}

で

評価

し

有限要素法

で

解

析

した。

筆

者

ら 9）

・

le）

e

：

_，

周

辺

固定部分円筒殻

の

座屈

実験

を行

い

，

種

々の

形状係数

を

有

する

試験体

について

，

初期

不整

と

座屈前後

の

変

位

分

布を殻全面

にわたって

詳細

に

測

定

し座

屈

性

状

を明らかにした

。

　

本論文

は_，

外

圧を

受

ける周辺固定部分円筒殻の

座

屈

荷

重

と

非線形変形性状

を

，Batdorf

の

係数

Z

と辺

長比

a の広い

範

囲で

系

統的

に

解

明すること

を

目

的

としている。良く知られている

Donnell

型

の

非線形方程

式に

Galerkin

法

を

適

用することにより

，

対称飛移り

座屈値

　

本論文の

一

部は

，

日

本

建

築

学会大会学術講演梗概集（昭和

60

年

10

月）に発表済

。

　

＊東北大学

　

教授

・

工博

　

＃東北大学

　

助手

・

工

博

　　〔昭和

60

年 10 月ユ_{1 日原稿}_受理）ならびに

非対称分岐座屈値

を

も求

めている。さらに

，

それらの

形状係数

が

変形性状

ならびに座

屈

モ

ー

ドに

及

ぼす

効果

を

明

らかとし

，

その

_特性

が

筆者

らの

_{実験結果}

と

良

く

対応

するこ

とを示

している。

　

2．

解析方法

　

母線長

さ

L

，

殻厚

t

，

曲率半径

R

，

開角 φ

なる

等

方

性弾性体

の

_{薄肉部分円筒殻}

を

考

える。 x

，

y

，

z

方向

の

変位

を

各

々

u

_，

v

_，

ω とする

（

図

一1

）

。一

様外

圧

q

は

内

向きを正とした

。

円筒殻に関する

Donnelli1

〕_の非線形理

論

に基づいて，式

（

1 ）

の

平衡方程

式を

基礎

式として用いたZLiU

．

式

（

2 ）

は

応力

・

変

位

関

係式

である

。

式（

3 ）

なる無次元量を用いて式

く

ユ

_）

を整理すると

，

無次元

化

された

基礎式（

式（

4 ））

が

得

られる。すると

，

形

状

に

関

する

独

立なパラメ

ー

タは

，Batdorft2

〕_の

係数

Z

と辺

長

比 αの二つであることがわかる

。

境界条件

を式

（5 ）

とする_。

_{変位係}

数

〔／_，

V

_，　

W

には_，

式

（5 ）

を満足

するように

式（

6 ）を仮定

した

。

ここで

，

mt

と

nj

は

各

々母

線方向

と周

方向

の

波数

を

表

す

。Galerkin

条件式

は

式

（

7 ｝

で

与

えられる

。

式（

6 ）

を

式（

7 ）

に

代

入し

積分

を実行す

れば

，

変位関数

の

未定係数

Ai

」

，

B

“

，

CiJ

に

関

する

（

荷重

Q

を

含

む

｝非線形

連立

代

数方

程

式が

得

られる 13）。この

代数方程式

の

数値解

は

，

増

分

法

と

Newton

−Raphson

法

を

併用

すること

』

で逐

次

求

められる

。

非対称分岐

座

屈

点

は

，

仮定

した

非対

称

変

形

モ

ー

ドに

対

する

線

形同次代数方程式において

，

その係数マトリックスの

行列式

の

符号

が

変化

する

点

として

得

られるH ）。

数

値計

算

は

，

東北大学大型計算機

センタ

ー

ACOS

99 −NEAC

ユ

OGO

を

使用

し

，

倍精度

で

行

った

。

　

3．

数値解析結果

　

ま

ず

，

既往

の

数値解析結

果との

比較を行

った

。

前報（

15 ）

では

，

差

分

法

による

横尾

らの

結果

51との

比較的良

い

対応

を示

した。

図

一

2

は

，

有限要素法

に

よ

る

Crisfieldiel

の

結

果

との

極

めて

良

い

一

致

を

示

したものである。

本研究

では，

数値解析

に

際

して

，

式

（

6 ）

の

仮定

変位

関

数の

項数

は

_，

対称項

Ns

＝

3

×

15 ，

_{非対称}

N

。；

3

×

8

としている

。

なお，既

往

の

_実

験

結

果

Y）

・

1°）

・

17）を

参考

として，母

線

方向

に沿う

非

対称

なたわみモ

ー

ドについては

考

慮

しないこととした

。

N

。＝

3

×

15

と

3

×

21

との

差

は

，

（

Z

＝

300，

α

＝

1 ．

0 ）

の

対

一

₈₉

一

N工工

一

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