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12
ᛂ⏝ຊᏛㄽ㞟 Vol.13 (2010 8 ) ᅵᮌᏛ ホ౯ࡓࡢ㠀⥺ᙧゎᯒἲ࠸ࡘ A static non-linear analysis method for redundancy evaluation of steel truss bridges ዲᓫኵ*㸪すᮧᐉ⏨**㸪❧▼ඃ***㸪ዉⰋ **** Takao Miyoshi, Nobuo Nishimura, Yuichi Tateishi and Satoshi Nara *()㸪㜰ᏛᏛ㝔ຓᩍ㸪ᕤᏛ◊✲⛉ᆅ⌫⥲ᕤᏛᑓᨷ㸦565-0871 㜰ᗓ྿⏣ᕷᒣ⏣ 2-1**ᕤ༤㸪㜰Ꮫᩍᤵ㸦565-0871 㜰ᗓ྿⏣ᕷᒣ⏣ 2-1㜰ᏛᏛ㝔༤ኈ๓ᮇㄢ⛬㸪ᕤᏛ◊✲⛉ᆅ⌫⥲ᕤᏛᑓᨷ㸦565-0871 㜰ᗓ྿⏣ᕷᒣ⏣ 2-1****ᕤ༤㸪㜰ᏛᏛ㝔ᩍᤵ㸪ᕤᏛ◊✲⛉ᆅ⌫⥲ᕤᏛᑓᨷ㸦565-0871 㜰ᗓ྿⏣ᕷᒣ⏣ 2-1This paper proposes a static non-linear analysis method for redundancy evaluation of steel truss bridges. Since many main members of the steel truss bridge consist of spatial frames with closed cross-sections, accurate evaluation of the torsional rigidity should be required in the elasto-plastic state. Especially, the implicit stress integration method and the evaluation method of the elasto-plastic stiffness are proposed in this study, in order to improve accuracy and efficiency. Validity and reliability of the proposed method are clarified through some numerical examples. As a result, it is shown that the proposed method has higher accuracy and more excellent efficiency than conventional ones. Finally, a numerical example for redundancy of a truss bridge is demonstrated. Key Words : steel truss bridge , thin-walled closed cross-section, implicit stress integration, redundancy ⫗㛢㠃㒊ᮦ㝜ⓗᛂຊ✚ศἲ 1ࡌࡣ 2007 ⡿ᅜࢱࢯࡅ࠾ I-35W ቯᨾ㸪 ᮌᕝᶫ㸪ᮏⲮᶫࡅ࠾ ᶫᮦ◚ࡢᨾ௨㝆㸪 㒊ᮦ◚ホ౯┠ⓗࡓࡋ ✲◊ࡢࡃ1) 3) ሗ࿌✲◊ࡢ㸪ᗋ∧㸪ᗋ⤌ ࡀࢺ1) 㸪㠃ຊ↷ᰝᘧ⏝ࡓ࠸ホ౯ἲ 2) 㸪㒊ᮦ◚⾪ࡢᙳ㡪 3) ホ౯㸪⑂ປ ⭉㣗క㒊ᮦ◚ᶫᱱᔂቯ㛵ಀᢕᥱ 㸪㔜ⅬⓗⅬ㒊ᮦඃඛ ࡀࡅฟ᮶㸪⥔ᣢ⟶⌮ ࡀ࡞ࡘᮇᚅ⯡ⓗ㸪 ᘻᮦ㸪ᅽ⦰ᮦ➼ᵓ㒊 㸪⫗㛢㠃㒊ᮦከ⏝㸪㠀 ⥺ᙧゎᯒ㒊ᮦ◚ ホ౯㸪㍈ຊḟ᭤┤ᛂຊ ࡌࡡᛂຊᙳ㡪⪃៖㍈ᛂຊሙᙎረᛶၥ㢟ࡀࡢ࠺㸪㒊ᮦ◚ᚋ㠃ຊศ㓄ᣲ ᔂቯ㐣⛬㏣㊧㸪㒊ᮦ◚๛ ᛶపୗ⾪ᧁⓗゎᨺ㠃ຊ⪃៖࠶ࡀ┤ᛂຊ㸪 ࡌࡡᛂຊ㍈ᛂຊ ≧ែ⪃៖㸪㛢㠃㦵⤌㒊ᮦᙎረᛶၥ㢟 ゎᯒἲ࠸ࡘ㸪᪤ࡘࡃ࠸ᡭἲ 4)8) ࡕ࠺ࡢࡇ㛢㠃㦵⤌㒊ᮦ㸪⊩ 4), 5)ᐃᘧᇶ⤌せ⣲㐺⏝⪃ࡓ࠼ሙ㸪ᙎረᛶゎᯒࡅ࠾ຊቑศ⟬ᐃ᪉ἲ㸦௨ୗᛂຊ✚ศἲ⛠㸬㸧 ࠸ࡘ㸪⊩ 7)㝧ⓗᡭἲࡢࡢ㸪ኚᙧᚋ㝆అ᮲௳ࡉࡓ‶ࡀ㝜ⓗ࠸࡞࠸㸪⊩ 4), 5)ᐃᘧᑐ㸪ゎᯒ⢭ᗘຠ⋡ᡭἲᥦ࠸࡞࠸ࡇࡑ㸪ᮏㄽ㸪㗰ᶫ㒊ᮦ◚ ⏕ࡀ- 965 - 応用力学論文集 Vol.13, pp.965-976 2010 8 月) 土木学会

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Page 1: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

Vol.13 (2010 8 )

A static non-linear analysis method for redundancy evaluation of steel truss bridges

* ** *** ****

Takao Miyoshi, Nobuo Nishimura, Yuichi Tateishi and Satoshi Nara

* ( ) 565-0871 2-1 ** 565-0871 2-1

565-0871 2-1 **** 565-0871 2-1

This paper proposes a static non-linear analysis method for redundancy evaluation of steel truss bridges. Since many main members of the steel truss bridge consist of spatial frames with closed cross-sections, accurate evaluation of the torsional rigidity should be required in the elasto-plastic state. Especially, the implicit stress integration method and the evaluation method of the elasto-plastic stiffness are proposed in this study, in order to improve accuracy and efficiency. Validity and reliability of the proposed method are clarified through some numerical examples. As a result, it is shown that the proposed method has higher accuracy and more excellent efficiency than conventional ones. Finally, a numerical example for redundancy of a truss bridge is demonstrated. Key Words : steel truss bridge, thin-walled closed cross-section, implicit stress

integration, redundancy

1

2007 I-35W

1) 3)

1)

2) 3)

4) 8)

4), 5)

7)

4), 5)

- 965 -

応用力学論文集 Vol.13, pp.965-976  (2010 年 8 月) 土木学会

Page 2: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

4), 5)

2

4), 5)

4), 5) 8) 2 1

4), 5)

eSSf pe (1)1

Txyz MMMNf (1)2

Tyze 0 (1)3

GJsymEIEIEI

ESESEA

Sy

yzz

yze

e

.000

(1)4

p

pp

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S

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.

(1)5

i O

x y z(1) 2 N

zM yM z yxM

(1) 3 0

z y z yx

u 1 y zv w 3

(1) 4 5 [Se][Sp]

E G

tdsAJ

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dAzSdAySdAA

s

eyzeyez

eyezee

2

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,,,

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244

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534

21

25

533

21

424

21

54623

21

24

422

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314

21

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21

43212

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23

111

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,,2

,,

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,,

CCGJCS

CCACS

CCCDS

CCACS

CCCCDS

CCCDS

CCACS

CCCCDS

CCCCDS

CCCDS

psp

psp

pp

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dsrAdsEHzEC

dsEHyECds

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22

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(2) *y z e ds p ds

srs

t H

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xn

yn

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i

x, u j

y, v

z, w

O r s

- 966 -

Page 3: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

nn+1

u fn 8)

pudxffe nTl nT 10 (4)

l pn 1

n+1 (4)u

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wvuu (5)

(5) iu ju iv jv iw jwi j x y z

i j yi yj zi zj

x y z (4) f (1)

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lxSSlxSSSD eei

ei

ej

ei

e (6)1

lxSSlxSSSD ppi

pi

pj

pi

p (6)2

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e] [Sip] [Sj

p] ij

(4) f

eDDf pe (7) (7) (4)

2 2

NNs ttq 11 (8)

sq k

ktN

(8)

NNs ttq 11 (9)

skk As 2 (10)

k

ks

4) 5)

k

k

(10)(9)

k

9), 10)

11), 12)

(10)

pk sq

z

y yk*

zk*

tk

sk

rsk

O

- 967 -

Page 4: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

k sq pk

pk sq (8) (10)

sq pk

k sq pk

1p

k sq (1)

ek

ek k

kE k kG

kkekkk

ek GE , (11)1, 2

von MisesYkf

2121212 3 kn

kn

kn

Ykf (12)

kn 1

kn 1

kn 1

n+1

(12) von Mises

pk

pk

kn

knp

kp

kkn

knp

kp

k1111 3, (13)1, 2

k k (11)(13)

kn

kn

pk

k

kkk

n

kn

pk

k

kk GE

11

11 3, (14)1, 2

kn 1

kn 1

kkn

kn

kkn

kn 11 , (15)1, 2

kn

kn

(15) 2 (14) 2 (8)

(9)

k

ssn

kn

pk

kk

sk t

qqtG

q13 (16)

(16) sn q

kn 1

kn 1

pk

pk

nkkYYk

nk

n H011 (17)

Ykn 1

kY 0 kH pk

n

k sq

pik (16) (17)

(2)

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kn

pk

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s

s

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q

1

1

3

32 (18)

(18) p

xM sqi j xiM xjM

siq sjq

sjsxjsisxissx qAMqAMqAM 2,2,2 (19)1~3

xiM xjM xM(19) sq sq

2,2 sjsissjsis qqqqqq (20)1, 2

(20) 2 (19)s

nsj

nsi

n qqq

sqi jp

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i j

p kikn

kp

ik

kk

k

p kjkn

kpjk

sn

js

p kjkn

kpjk

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k

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kp

iks

nis

p kjkn

kpjk

kk

k

p kikn

kp

ik

kk

ks

ts

tGs

ts

qA

ts

tGs

tsqA

ts

tGs

ts

tGsq

G

11

11

11

332

332

332

0

(21)

(3)

(11) ik ik

p

ikikkik E (22)

(22) (15) 1

- 968 -

Page 5: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

pikk

Tikik

n E1 (23)1

ikkiknT

ik E (23)2

(23) 2 Tik

(23) 1 (13) 1

i k ikn 1

iknp

ikkT

ikikn E 11 1 (24)

i kik

n 1 (8) sn q sq

kssn

ikn tqq1 (25)

von Mises (12) (24) (25)i k

pik sq

031

212

2

21

2

ikn

k

ssn

iknp

ikk

Tik

ik tqq

EF (26)1

j k

031

212

2

21

2

jkn

k

ssn

jknp

jkk

Tjk

jk tqq

EF (26)2

(4)

13)

i jk T

ikT

jk

2222 3,3 T

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Tjk

Tik

Tik

Tik (27)1, 2

jkkjknT

jkikkiknT

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k

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pjk

kk

sjk

k

ssn

ikn

pik

kk

sik

tqq

tGq

tqq

tGq

1

1

3

,3

(27)1, 2

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ikT

jk nYik

nYjk

n

(5)

pik

pjk sq (21)

(26)

pik

pjk sq Newton-Raphson

pik

pjk

pik

pjk sq

sq

TjNjiNi

Ts

pjN

pj

piN

pi

spjN

pj

piN

pi

s

jNpjN

jN

s

jpj

j

s

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s

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i

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q

qGGGGGq

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11

11

1

1

1

1

1

1

000

000

000

000

(28)1

(28) 1(21) (26) p

ikpjk sq

p kikn

kp

ik

p kjkn

kpjk

kk

k

p kjkn

kpjk

p kikn

kp

ik

kk

k

s

ts

ts

tGs

ts

ts

tGs

qG

11

11

2

3

92

(28)2

0006

63

36

1

1

1

211

pik

pikjs

p kjkn

kpjk

kk

ks

n

p kjkn

kpjk

kk

ks

ikn

kp

ik

ikn

k

k

pik

forforA

ts

tGsq

ts

tGsq

Hts

G (28)3

0006

63

36

1

1

1

211

pjk

pjkjs

p kikn

kp

ik

kk

ks

n

p kikn

kp

ik

kk

ks

kjk

n

pjk

jkn

k

k

pjk

forforA

ts

tGsq

ts

tGsq

Hts

G (28)4, 5

- 969 -

Page 6: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

0600

2 pikkss

n

pik

s

ik

fortqqfor

qF (28)6, 7

0600

2 pjkkss

n

pjk

s

jk

fortqqfor

qF

(28)8, 9

02

1

12

01

1

11

31

2

pikkik

n

ikn

pik

kik

nk

iknp

ikk

Tik

pik

pik

ik

forH

HE

E

for

F (28)10, 11

02

1

1

201

1

11

31

2

pjkkjk

n

jkn

pjk

kjk

nk

jknp

jkk

Tjk

pjk

pjk

jk

forH

HE

E

for

F(28)12, 13

(28) pik

pjk sq

rpik

rpjk

rsq

Ts

pjN

pj

piN

pi

Tr

srp

jNrp

jrp

iNrp

i

Tr

srp

jNrp

jrp

iNrp

i

q

q

q

11

1111

111

11

(29)

(29) 1rpik

1rpjk

1rsq

(28) (21) (26)

(21) (26)

Yikn 1

Yjkn 1

ikn 1

jkn 1 (17)

ikn 1

jkn 1 (24)

ikn 1

jkn 1 (25) ik jk

(16)

3

i j1P

ik

1Pjk

START

(18) i jqsi

(1) qsj(1)

qs(1)=( qsi

(1)+ qsj(1))/2

(13) 2(16)

ik(1)

jk(1)

i

j (27) T

ik

? Yes

No

YiknT

ik ? Yes

No

No Yik

nTik ?

Yes

(17) i jr

ikn 1 r

jkn 1

(21)G(r) (26) i j

Fik(r) Fjk

(r)

Fik(r)=0 Fjk

(r)=0

Fik(r) Fjk

(r)

G(r)

? No

Yes

(28) i jrp

ikrp

jk

qs(r) p

ikp

jk q’s

(29) i jrp

ikrp

jk qs(r)

(13)2 (16)

i jik

(r)jk

(r)

i j

(n+1)qs

RETURN

- 970 -

Page 7: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

Step1

{Qfm} Step2

Step1

0fmnnn QQPUK (30)

[(n)K]n

{ U}{(n)P}

{(n)Q}

Step3Step1

fmrn

fmr

Pnrrn QQQiPUK 11 (31)

(31)n+1 (r)

i1 p

(r)

0 p(r) 1

4

23

4 1

(1)

PT

Y0

H’ P T PT /

3 Path-1 / Y0=1.2

32.1 0Y Path-2 Path-1

x

Node-B

y

z

O

L=200mm

Node-A ( )

170mm

T P

170m

m

E=200000 N/mm2 G=77000 N/mm2

(SM400) Y0=245 N/mm2 ET=E/100

170mm

150mm10mm

170m

m

150m

m

10m

m

O

y

z

2@80mm=160mm

160m

m=2

@80

mm

Step1

Step2

Step3

- 971 -

Page 8: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

221

20

00

0

YY

Y

forHE

forE (32)1,2

663

60

00

0

YY

Y

forHG

forG (32)3,4

Path-2

02.1

44.131ln31

5

2.1

0

02

0

20

000

0

Y

Y

Y

YYY

Y

forHE

forHE

forE

(33)1~3

02.13tan32.13

0

10 forHHG Y

Y (33)4

Path-1 kNP 6.1181mkNT 9.86 100

Path-2 kNP 6.1181 50P

mkNT 9.86 501

8(34)

10

11100.1torT

rTr

cBUBU (34)

rU rB r

Path-1 2HY

p03

HYp

0 (32) (33)

(2)

4)

PT

43.0 4) SB-4

24

1kNP 6.326

mkNT 3.5 20(34)

YTT

Y

x

Node-B

y

z

O

L=400mm

Node-A ( )

80mm T P

80m

m

E=205800 N/mm2 G=76440 N/mm2

Y0=271 N/mm2 ET=0

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08

Path-1( )Path-1( )Path-2( )Path-2( )

37.5

mmO z

y

- y0/6

7 y0/8

-y0

/6

-y0

/3

14.1mm4.7mm

37.5mm

4.7m

m14

.1m

m

5 y0/83 y0/8

y0/8

- y0/3

7 y0/8

5 y0/8

3 y0/8

y0/8

y0

y075

mm

10mm

10m

m

75mm

- 972 -

Page 9: 5ð º Û « « b Ü ² å ² å © î0Û o b S u b8)z g0 Ò 2 X 8 Zlibrary.jsce.or.jp/jsce/open/00561/2010/13-0965.pdfL S7 b 8 b Ì (4Ä ò · x>* o4#&ì b q _ 7 K Z>*5ð4 _ e"©4),

(3)

2Step1 P = 1600kN 100

Step2

AP U

4 2

(1)

3), 4) A14)

B

8mm100

(34)

505

A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8/ Y

T/T

Y

E=200,000 (N/mm2) G=77,000 (N/mm2)

(SM400) Y0=245 (N/mm2)Tangent modulus ET=2000 (N/mm2)

P = 1600kN

1

X (U)

A

1000mm

Y (V)

1000mm

Step 1

Step 2 2

2

1 2 y

160m

m=4

@40

mm

O

160mm=4@40mm

10mm

10m

m z

P

A

0

500

1000

1500

2000

0 1 2 3 4U (mm)

P (k

N)

Step2

Step1

x

Node-By

z

O

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55.4mm129.2mm240mm

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(1)

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1)

pp.1-4 2009. 2)

A Vol.65No.2 pp.410-425 2009.

3)

64I-113 pp.225-226 2009.

4) Komastu S. and Sakimoto T. Elasto-plastic behavior of thin-walled steel tubes under combined forces Proc. of JSCE No.235 pp.125-136 1975.

5) Komastu S. and Sakimoto T. Nonlinear analysis of spatial frames consisting of members with closed cross-sections Proc. of JSCE No.252pp.143-157 1976.

6) Li, X. S. and Goto Y. A three-dimensional nonlinear seismic analysis of frames considering panel zone deformations Journal of Structural Mechanics and Earthquake Engineering No.605

-45 JSCE pp.1-13 1998. 7)

305 pp.47-58 1981. 8)

1998. 9) Simo, J. C. and Taylor, R. L. A Return Mapping

Algorithm for Plane Stress ElastoplasticityInternational Journal for Numerical Methods in Engineering Vol.22 pp.649-670 1986.

10) Simo, J. C. and Govindjee, S. Exact Closed-form Solution of the Return Mapping Algorithm in Plane Stress Elasto-Visco PlasticityEngineering Computation Vol.5 pp.254-2581988.

11) Krieg, R. D. and Krieg, D. B. Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model Journal of Pressure Vessel Technology Transacions ASMEpp.510-515 1977.

12) Schreyer, H. L. Kulak, R. F. and Kramer, J. M.Accurate Numerical Solutions for Elasto-Plastic Models Journal of Pressure Vessel TechnologyTransactions. ASME Vol.1.101 pp.226-2341979.

13) M, Kojic and K, J, Bathe Inelastic Analysis of Solids and Structures Springer 2005.

14)

No.689 I-57pp.85-100 2000.

15) III 2002

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