chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~...

14
.(vt:J "';'J 4S tA cJ.i>.,J"'t.!"? \.:,,1.* o\!..!..it.:> - -0 1 p O!;.l [email protected] .ir f'} )JA-""'c4 0 1 * o\!..!..it.:> - - .jl ...... [email protected] it W>c.i..a. \.0 ii A. ""'MY r jjN) 0'* o\!..!..it.:> - - OjL.... t;1.9 (A,(IHI\t:J 6;)li .A'(I"-" , c)L.1 6;}.i .A,(I\ln 6;)li) ..s.).J.r> ..sjW..l..o .;"lS:..1 4$ .).)}' ..r ":I} .:.:i-IojL. j1 J..l..o I.J"!<A, .),..I:>u ..s4-'WI d;.i "y,,) ,-",L.Ir,.utA.... ,:r.I).) r} }:a),;?}' ».i).) I) ..;.)),> S;.i jl....,-t.s""W ..slA}a, 4"lS" , "'Ili .1l.)";';' .)),>r, , Yl.J"!<A ..s4-'Wl ..;.,) 0.))'> S;.i ).) . ..:.....1 0.)lG::;..,1 '.:,.$.;1) ..;.)),> S;.i ;t".... ;1 .:.:i-I ..sjW..l..o ..sIr, . .)jL. ..r .r..L. ...,:..;S";..,.;I t.Y f.r....i.: js:.:..)./J- r, .,'Js:. .),..I:>u .1r.-I..slA .;"WI.;"l;.) f,-",W-.¥lS:.. .,:r.l ;1 o..l..ol ..r ..sjL. I; .)..01 }a) (.),.:. ..r ........ ,-",W -.¥lS:..;.) 4$) 4-'WI . 4-.ttA... Ci!:!t;; ..;.,; s jI };;j) , I..s" ..,..t:;.:;.;! ..;.)),> , .,;.,) .1t ..... ,; ..;.)),> . SjW..l..o U"'W S .;.,) U"'! .... I;! ):a) s.).).&. SjW..l..o )4 0&,1 f ..; f>} ):..9) .:?./ }:U).) 4 tl S .;,) U"'!....I;! .[0]<' Rashid .1t ..... ,; .c .ul;! Hillerborg S J..l..o ).) .;,) 0t1.[ n'].).r.S' I..s" }:U ).) I) ...:.-S:..:. I .)}-4 U"'! .... I;! .:........s:.:. SI;! Cervenka .1t ..... ,; , '\ '\ f J! .... ,'\ '\0 J! .... ).) .[ II , I ...s.;A sLI>.)}-4 ).) Aliabadi , Saleh .1t ..... ,; Cervenka .;,) jI .;,) 0t1 ).) .[' o.)I.A::....I sj.;A II ..l..o ).) ..; f>} , ...:.-S:..:. sj}1 , SIA OJ!'''').) ..;.)),> S 0,..9..l..o slA OJ!'''' .1A.l.w t;l- I..?".JI> slA}.iS')4 }I ).) .... , 1.51 slA),:;51) )b),>;! ($.1.;-. jl , jl ).) .c 1.5.)"""'-:'" , .)y.-, . ..:......1 .)),1;! ..u...!.4 I..s" };-.. IA> OJ!'''' 0t' ):;..9) .ul) I.lJ .r-..l.; ..;.)!.... s);.!) .c IS"'W s)·;;j) s4:J..l..o j1 o.)1.A::....1 4 , OJ!'''' ..;.)),> S , IS"'W ?).1.i1 f>j'Y slA,.r.-> , W I) .)),>;! .)S ..::..I}I jl ;!} )t.;;.>1).) I) oj!"" s;!)4 J? U"'!....1 ;! 0t 1 ).) Sly. s)\.::a-o .:........s:.:. , .')1;

Transcript of chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~...

Page 1: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

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J ~~~ 4StA CjlwJ~ cJigtJt -~Jl5~ ~

~4S~W1

4S~Jtt~

1 oitgt - ~o~b - 0 1p iS~ Ol )4~l1

smohamutac ir

f ~)JA-c4

01 oitgt - ~o~b - jl 1SS~ 1S~b

maramutac it bullWgtcia0 iiA

MY r jjN)

0 oitgt - ~ o~tgt - OjL l~)~ iS~)S ~I t19 (A(IHItJ -~ 6)li A(I- o~ c)L1 ~I) J~)) 6i A(Iln j~)) 6)li)

s)Jrgt sjWlo lS1 4$ )) r I ~ i-IojL j1 ~ Jlo ~ IJltA )Igtu s4-WI di y) -LIrutArI))

i-I~ r a) raquoi)) ~ I) ))gt Si jl-tsW slAa 4lS o~s~i~) )~IlA Ili1l) ))gtr

YlJltA s4-Wl ) ~I) 0))gt Si ~ )) 1 ~ 0)lG1 $1) ))gt Si t 1 i-I sjWlo sIr )jL r ~I) rL ~sW SI tY ~~)gt fri js)J- r Js )Igtu 1r-IslA WIl) 4$~~r ~b f-W-yenlS

rl ~ s)Jrgts~ 1 olol ~ Cit~ r sjL ~ I )01 ~ a) () r ~ -W -yenlS) 4$) 4-WI

I~ 4-ttA ~L) Cit ~

s ~ jI ~ ~ j) )~ Is t ))gt

~~ ~ S~WI ) ~I) 1t ))gt

)~ SjWlo UW

S ~ ) UI ~ )a) s))amp SjWlo

)4 0amp1 f ~ ~ fgt )9) U)) 4 tl-~

S ~ ) UI [0]lt PM~ fgt~1 Rashid 1t c ~ ul Hillerborg ~~o~ S ~ Jlo vl~

)) ) 0t1 [ n])rS Is U )) I) ~ -S I )-4

UI s SI Cervenka 1t f J

0 J )) [ ]~) ~ II I sA sLIgt)-4

S~WI )) Aliabadi Saleh 1t Cervenka )

jI ~IS ~~ ) 0t1 )) [ f]~ o)IAI sjA

II lo )) ~ fgt ~li -S sj1 ~ ~Ia

~ SIA OJ)) o1i~ ))gt S ~ ~bo

09lo slA OJ middot4cAlSA4middot~ 1Alw ~ tl-~ IJIgt slAiS)4 I )) 151~ slA)51) bull

)b)gt ($1- ~ ~I jl ~4) ~l1 jl ~I )) c 15)- JoI~ ~~ )y- ~ ~ 1

~ ~) ))1 bullu4 Is - IAgt OJ 0t )9)

~) ul) IlJ middot~4 ~ r-l 0~1 ) ~ s))

0~1 c ISW s)middotj) s4Jlo 0t~~~ j1 o)1A1 4 OJ ~-4 ))gt S ~ ISW )1i1 ~t

~ fgtjY slAr-gt ~W ~ I) ))gt )S II

jl ~)A) )tgt1)) I) oj s)4 ~J J U1 ~ 0t1 )) ~~ s)~ ~

s~ Sly s)a-o s ~~ cI~~ ~~ )1

Administrator
Stamp
Administrator
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fYP

o~ ylgtrl Aj~ W ~J )~ )1 ~ j~l 5~ o)lo ~ ~ )~ -r1 d-lb 1

fft tSlgt )li ~ ~ JI)lAgtlS1 J oS ))H

~) )~ ~J

0)

1 )1JI~I )j r) ~li J 2 W1))oS

~~ )leampo ~ oS lAgt)5 w~ tS)~ ltIJ tS)i3)~ ~lo)

~li ltIJ jllS-j oS 1 ~ cb ~ tSl))

tSrol)4 )~ )~ jl 0)Li1 ~ cb -r1 ~~ 1 I I

I) d-Lb -r~ SJ) I~ tSrol) ~ tS

tSlJ 1 )IAo ~ lol ))b ~~lo) ~l ~

-r1IJLIJ tS)~ ~ 401J j~ )) tSk rol)4

I) jSo -r1 ~I) o)L )~ ~~5 jSo k)~

~ ~ ) ~ oS J$5 w~ o4 W5 wLl

J ~ olA )IAo ~ rslgt ~I W~)

~) ~ )) ~J 4 ~j tS~ ~ ~ wl J )~ ~ ltIJ )~ -r1 ~W )5 ~bshy

IJLI J cb -r1 n Js)1 ~I tSk)~

)5 ~~ xj ) ~ tS~L

fc~AlI) =2JiJ7CosO + Z - 3J =0

0)

~ )bJ ~Ij 0 )j JI ~li II oS

S tSr Js tStl) ~~ 5 01 ~I)~

)5 Jgtj rsllgt ~~I ~ J )~

01

3

y~ J~o )Iii)

~~ r ~~1lgtoS 41 j1

Jw ~oS iSk))) wi jl W51 o~

~I) ~ Belytschko Lin AV JL )) 1 j~ ))

04~ ~)JI (Eroding Element) ~L j tS~WI

~jx ))gtJ ~I ~~ ~ ~ ) ))gt J tSlJ

tSjWJo tS~ ltIJ ) ~ I) ~ JI) ~

) jl o)LiI ~ i JL )) Morikawa n o]~ jl ~I )4 ))gt S (DEM) Ir--o tS~WI

1[V])f ~ I) ~~ tS4)b ~ ~lix ))gtJ

1 o)f 0)Li1 ltSW tSk particle 0041jl

~ -r1 )) o~ ~I) ~~L JJo 6 ~I)J )) ~ oS )r 5 ) o)Li1 )) ~)

S jl ~ ltSW tSk)~ ~L ~~ 9)

~j o)~ Lts J tS~) ~ ~ ))gt

-r1 )) ))br I) 0)S tSk ))gt S ~ ~ ~ jl I~ tS~WI ~ 0))gt S ~ ) ~ xl Js 9) ~ ~WI -r1 jl ~ yIb oS )~

~ tS~WI xL ~ ltSW tS~)J1 tSr)S~ ~

))b ltSW )I)gt jr

~~~)ti) Jlo bull f ohl) ~~ Jow ~ tS)9) JJo

~~ ))gt S jl ~ )9) ~ )~

tSkrol)4 ~ ~ tS)9) JJo ~ Owen Figueriras

Jow )19 )) f ~ Jots ~ ~

~1 ~ ) ~ )1 ~b Jgt~ ~ cb Bathe [ ] n AA) 1 o)b ~I) 0)S

J )) J o~ oS 10)1) ~1))Jo Ramaswamy

-r1 )) middotn 7])r 5 )) r I) ~ 1 ~f ~ olA w)y ~~ ~ tS)9) JJo ~

tSjL ~ Of ~ tSj1 J ~ tS)~ S

~L ~ ) ~ ~ jl J ~ ) 5

~ ~ ~I ~ j1 ~S 5 )Lj) ~~11

~j~ j~1 iS~ ~ j ~f ~ 1 )Lj)

)) 5 ~ S ~1 ~ ) ~ ~ o~ ~ )lo) ojgt

~ ~ )~ tS~WI tSjWJo tS)1

~f ohl) )~ Jgtj Jots tS)1 ~ ~1

))gt S wl Jots ~~ -11 bull )1

)~ ) 5 Jr ~y 1 jl ) 5 Jgtj

YVV )gt S JlgtU

0 W1 Jgtr )I) Ie gt IJIgtIgt ~L gt~ Iy-I 0 W1

0 W1 ~ ~) Ie -L ouj s 4 ~ gtS A~ )j ) ~ (V) 0 W1 ~ 4(A)

I=A2 for2D e

(f)

=P3 for 3D e

(0)

l1 ( bgt ~ VLl ) rgt l1 ( Jgto

oIl S -1 )j ) ~ (f)~l) O)~

E = Ie

t 2G

(Y)

~L u 4 l1 ( 4iL -s) 4bA Jgt )gt eJL 0l1 ~)l 0r-o bull 4bA fb )gt Ii ~

I 6 )gt ~L 0- IJIgtgt S 0 W ) 4bA 0 1

1 s S l1j4 j~i Jlf l1gtygt

)~i- j) 0raquogt Sr-J~)gt (I~)

Gf FIfIOOlre Eoefgy

)u )gt 41 ~Yl bgt -s) ) ) )

16 tL (I) 1lt l1 Jgto - 0 r-ltS ( ~ ~ r

dAL tL let Ili L 4- 1100 lgt - j- - bull )7 ~- (j bull ~ y

-slet oIAJ jI ~ 6 e ~S CF 61 ~L bull lt bullbull -~ Yf I bull bull rr ~

IV CAU) 11-tJNf ~ - ~ -s))lf 0L ) s Ii~r -sr-fJI~I )

~lo L) (13 fb Jb )gt ) 0-1 ) 6 1

JgtI ~L ) 0-1 ) 1 jt ) ~)~

4 ampr ~I ouj ~ 01S I) ~~gt

J~ )h-o ~ S Jgt IIgt~ -s4Lliu )1 olA1

6 bull ~s lt ogtlA1 ~L ~ s~V) ltT - r S shy

~I) IT5))A6 c Vf JL ) )4 0l1 -sl)

( S ~I) s 6 1 VLI 0-1 ) ol1

0-) ~ S ) ~ -sj1 ~ ~I l1

gtS A~ ~ r1- )b ~ ~~~ )

~ s -s10l1 -SjL~ )~ 1 J-o Ii JLd III II I -slctgt JL gt ~ j1 ~Y

) IJIgtIgt r-S S l1) )gt 6 ogtL )~ s ol1 gt1) -sj1 ~ 0t-~~1 s ~ Ii1S

gt~ Iy-I ) ) 0-1 igt gt ~I) let ~

~ l1 ~~ ~~ -SjL~ )gt rgt- -s~L 1 o~) ~I ~ ~ ampr 4 raquogt) 1 )gt eJL

6 1 lbo 0-1 olAgtgt 0W G=~t Jraquo L ~~ s tL bull lt bull bull bull bull I~ rr ~ ltS ( w11

1 ~I gt~ Iy-l f- ~ 6 Ie ~ Jb

raquogt~

~ r1- ~ s ~ -sj1 ooll 0-1)~

-slct~ 4 6 IJIgtS ~I) JgtL l1 ( ~I) -sl)

4- __ - ~ )gt ~S l1 ( ~ -S~

1 lA ~~ -Slet~ ~

0 S ogtL ) s ~ wi ) 6 Jgto s If

Jgto bullgtS -SjL ~ I) S l1 ~ ~li ~ 10Jgto -- -sl)gt 61 ~1)l1 ( 4- _ _ - ~

(1 ~)I~gt~l1(

)j ~ JI~I ) ~ ~ 0l1 gt1) -sj1

~S A~ S l1 ( 4iL ~

G =[~(ampII - amp)] (1)

~~ ~S amp amp bull rs IA

middot~4S l1 ~ Jgtr Ie ~~ Jgto ~ -sj1 0S 0~ ~ le)~ A~

~gt J-a- lIr--o S Jgto ~ ~~ S ~ -sj1 )I) Ie Ir--o S Jgto ~ -sj1

I) (w)lr--o S s Jgtr I)j 1 ~~ S Jgto

S Jgto )gt Ie ol1 ~ Jgtr 4 JgtL 0 1 S

~ bY Ie )5 16)gt (i~) Jraquo ~~

4 6 ~ ~ ~Ui jI 1 0 W1 s )~ ~ ~ gtS Ir--o S Jgto ) S ~I

- -- -- -- -

(VA

Ilf J~ ))b I~ ltS4cWI J) )) I) wloj ~)Pgt

0- ))gtft 4 W jl Jlgt sUbr-i Jw1 1)

~ o~ 1 Ilf )) ~~~ ~ 4 UbWWI

gt~ ))gt S ~ ~ UbS J-f

Jw1 wW1 ~ ~~ sUboJ ~~ S

)~ ltS~ ~~ sUboJ )) s)lfjL ~Ir-

)r~ lS)ft ))yen ~~ 4 lI~1 JlAi1

Crack

) Jlgt )) )~ ltS~ s1 ltf) jS ))

o~) jS )) IS )hgtt 1 o~ 0)1) ~W s~

l) ~ slgt 4 A Jo ltSI ~ )) ~~ )~

A 01 ) 4 A oJ~) b ~~ ))gt SIS

~ wW1 ) 4 S ltSlgt wWI o~ ~~ A

JA )~ 4 )~ ~~ ~~ ~ ) o~

10 o~ ~~ wW1 s) B ~~ 01 s)lfjL

)~ o)b ~s~ wi ~

-W ~JJil

jfgtlt-gt lS)ft I~ wWI ) 0- W u1 jI ~

W sr-i ~ IS l)~ wi 4 ~ ~)J

)~ Ir-I i)t 4 L ~ ~ Jw1 b

IIio ~ ~ )) ISW ~ ~ ))1 Jw1

0- b4gt1 ~ 4 ltS)I) ~)) ~ sIft ~

1 4 I) ~ Ibt middot~4~ AI ltS)ljl 40)) wL ltlt) ~I) )P 4 wI~ )S Jlgt )) ~4)

) CUQ

n-1I2 n+ 112 blAi )) sr ~I~ ~ 9)

)~ ~bu n ~)) vyl

v = U n+1 - Un V =Un - Un_I n+112 III n-112 III

n n-I

v = r+12 - r-112

n III n

(V)

~ J)W 6I)t )) r)lio 0-t1 s)l40 4 gt01 ~Igt 1J i s)Ij1~)) sIft yl~ )S jS

P11+12 =Pn-12 + I1lnmI(ut)n + von)n unt)n)

(A)

n n-l ~loj s+olf ~~ I1ln ~I) 0-t ))

ltSr-i W jI ~L ltS-)~ i-- tnt hi l0n

0)lA1 1) 4 ~ 4 ~~ raquo1) sr-i I)Igtshy

ft ~~~ )) 5 )~ ~loj ltS+olf jI

~ J) )~4 o)~ )) o)~ bull W sLSo

bulll) ~Igt er ~~)

--- -- ---I

On On

wIshy1~~~s~~~Jof ~

~ ~lAuW ~jWJo

4$4)) Lal Ilf ltl 4 I~ sUbwWI J) S ~) nw 5)li1 rmiddotw 1 4$Ub1 ~l) Jlllf jI J~ )~ s~~

~~ 4 IS 4$UbWWI )S)gt ~ 4 IS))Ii

lS)ft sIft middot~4 ~ W )) 4 J$~ ))gtft

J) ~ ~ ltSUbJ) J- sUbW

4$13) raquo) ltSjLor-gt) J) ~c ltS~

0-t~ ur 0-t1bull1 o~ ~I) slai -YU J)

[W] 1 ol )~ 05E lt a lt 2E

~IS ) ~~ )) ~ I)) ~o -4 JgtI)) ))

)1) ~w on ~I) ) ~

on I) ~ I)) ~ IJw SIx )t cr1 ~ f

) ~~ 01) ~I) ) ~ 01~1S

g =[x2 Xl]n 0 rc SI))

( f)

~ X2 I) Xl ) )IS Ool-ol gap ~l g 6

~ x JL )bX n I) r I) 141 ci4 j5 ~

(0 j5 ) l41S IJW 0 jp reI) IJW ~

1) lt n )t I~ f 9 ~)u ~ ~~ 4 )1) ~W xj f ~ 01~IS r- lAo ~x

owcon = J agog(u)dare

(0)

01~1S ~W SI)y-i )bx ~ x ) ( 0) ~I)

xj ) ~

oWcon = J fcon Og ouda re ou

oy)

) ~01~1S nV) ~I) k~ I) fcon 6

fcon = ACoc

(W)

OA)

-4 --I~ ~rL a I) IJw ~ pound 6

f ~ 01~1S I) S 0 SIx J)t SI)y-i )bx l41S

( ) ~I) ~Iil

Ii s

lt ) 1 i 4lJ u I) i xy k nt ~I) cr1 )) 6

g = (gngt) )bx middotl41S S 0 )) 015 ~ )bx

middotl41S gap)bx

Ii~ ~ ~) I ~ IJW ~15 )) Ip))px

)) x Q I) S)I) ~)) )bxU ~ ~rL e 6

S)~ ~)t ~ I4 ~I) cr1 middotl41S ~11 x)LA

Jgt ~I 4 0L~ ) ~ I) )) 0))1 ~

~)t ~ cr1Jgt SIx ~ S1pJgtI)) ))

0)f ~ Jgt)) J J)) ~ 1 ol) f~

fj~ ~)t 0))1)1 lJ SIx Jr J-~ S31

IJW ~ blr ~ J~I SIx Jgt) cr1)) middotl41S con - I It rmiddot 1 lIo 7r H l s

~ v- r--- bull )y ~ ~ bull

))IS 0))I

con ( )7r = 7r(II) + 7r e u

f~ ~l I~ ~ xlx7r I~ ~l I~

l41S ol

( )

~ SI)L 07r I~ J-~ ~l I~ Jlgt ~ ~ IJW ~ ~~lJ I) )IS o)b )

~ra ~ j~ Jgt)) cr1 ~)X 1)1 )) IS ~ con

))) 7r ~ ~l ) e ~

It II 4JNJ

)~61 )~I J-1cr1x -4 Jgt) S~

~6 IS~ I) l41S ~~~ o)~ eTc ~) )~ 1 ~ xix 01 I~ )IS L)

o(CrC) =0

xj ) ~ 01~1S I) ~ ~l Jgt) cr1 ))

Jcrcon 1 C -T=2 a

on 7r (II) (II)U

j1 )IS ool-ol -4 --~ a 0 n ~I) )) ~ ~ j~ ~ J~I 6 1 01J) cr S1p~j-o

I) O)1 lt$H -4 Jgt) b a~ ~L )t

-4 --~ )IA ~ 0 1 jI olol lJ S~~

)YI Slbgt --~ cr1 )IA ~l) 4 ))b ~

S)j1 )L- x)LA Sr)~ I) lJ4 IS ~I oloi

~ lo~1 IS Jbu S)Jamp S)I~li ~ 01 SIx

u-19 )) a SIx Jbu S1l~1 )) ~ 1)1 1 )IA

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 2: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

fYP

o~ ylgtrl Aj~ W ~J )~ )1 ~ j~l 5~ o)lo ~ ~ )~ -r1 d-lb 1

fft tSlgt )li ~ ~ JI)lAgtlS1 J oS ))H

~) )~ ~J

0)

1 )1JI~I )j r) ~li J 2 W1))oS

~~ )leampo ~ oS lAgt)5 w~ tS)~ ltIJ tS)i3)~ ~lo)

~li ltIJ jllS-j oS 1 ~ cb ~ tSl))

tSrol)4 )~ )~ jl 0)Li1 ~ cb -r1 ~~ 1 I I

I) d-Lb -r~ SJ) I~ tSrol) ~ tS

tSlJ 1 )IAo ~ lol ))b ~~lo) ~l ~

-r1IJLIJ tS)~ ~ 401J j~ )) tSk rol)4

I) jSo -r1 ~I) o)L )~ ~~5 jSo k)~

~ ~ ) ~ oS J$5 w~ o4 W5 wLl

J ~ olA )IAo ~ rslgt ~I W~)

~) ~ )) ~J 4 ~j tS~ ~ ~ wl J )~ ~ ltIJ )~ -r1 ~W )5 ~bshy

IJLI J cb -r1 n Js)1 ~I tSk)~

)5 ~~ xj ) ~ tS~L

fc~AlI) =2JiJ7CosO + Z - 3J =0

0)

~ )bJ ~Ij 0 )j JI ~li II oS

S tSr Js tStl) ~~ 5 01 ~I)~

)5 Jgtj rsllgt ~~I ~ J )~

01

3

y~ J~o )Iii)

~~ r ~~1lgtoS 41 j1

Jw ~oS iSk))) wi jl W51 o~

~I) ~ Belytschko Lin AV JL )) 1 j~ ))

04~ ~)JI (Eroding Element) ~L j tS~WI

~jx ))gtJ ~I ~~ ~ ~ ) ))gt J tSlJ

tSjWJo tS~ ltIJ ) ~ I) ~ JI) ~

) jl o)LiI ~ i JL )) Morikawa n o]~ jl ~I )4 ))gt S (DEM) Ir--o tS~WI

1[V])f ~ I) ~~ tS4)b ~ ~lix ))gtJ

1 o)f 0)Li1 ltSW tSk particle 0041jl

~ -r1 )) o~ ~I) ~~L JJo 6 ~I)J )) ~ oS )r 5 ) o)Li1 )) ~)

S jl ~ ltSW tSk)~ ~L ~~ 9)

~j o)~ Lts J tS~) ~ ~ ))gt

-r1 )) ))br I) 0)S tSk ))gt S ~ ~ ~ jl I~ tS~WI ~ 0))gt S ~ ) ~ xl Js 9) ~ ~WI -r1 jl ~ yIb oS )~

~ tS~WI xL ~ ltSW tS~)J1 tSr)S~ ~

))b ltSW )I)gt jr

~~~)ti) Jlo bull f ohl) ~~ Jow ~ tS)9) JJo

~~ ))gt S jl ~ )9) ~ )~

tSkrol)4 ~ ~ tS)9) JJo ~ Owen Figueriras

Jow )19 )) f ~ Jots ~ ~

~1 ~ ) ~ )1 ~b Jgt~ ~ cb Bathe [ ] n AA) 1 o)b ~I) 0)S

J )) J o~ oS 10)1) ~1))Jo Ramaswamy

-r1 )) middotn 7])r 5 )) r I) ~ 1 ~f ~ olA w)y ~~ ~ tS)9) JJo ~

tSjL ~ Of ~ tSj1 J ~ tS)~ S

~L ~ ) ~ ~ jl J ~ ) 5

~ ~ ~I ~ j1 ~S 5 )Lj) ~~11

~j~ j~1 iS~ ~ j ~f ~ 1 )Lj)

)) 5 ~ S ~1 ~ ) ~ ~ o~ ~ )lo) ojgt

~ ~ )~ tS~WI tSjWJo tS)1

~f ohl) )~ Jgtj Jots tS)1 ~ ~1

))gt S wl Jots ~~ -11 bull )1

)~ ) 5 Jr ~y 1 jl ) 5 Jgtj

YVV )gt S JlgtU

0 W1 Jgtr )I) Ie gt IJIgtIgt ~L gt~ Iy-I 0 W1

0 W1 ~ ~) Ie -L ouj s 4 ~ gtS A~ )j ) ~ (V) 0 W1 ~ 4(A)

I=A2 for2D e

(f)

=P3 for 3D e

(0)

l1 ( bgt ~ VLl ) rgt l1 ( Jgto

oIl S -1 )j ) ~ (f)~l) O)~

E = Ie

t 2G

(Y)

~L u 4 l1 ( 4iL -s) 4bA Jgt )gt eJL 0l1 ~)l 0r-o bull 4bA fb )gt Ii ~

I 6 )gt ~L 0- IJIgtgt S 0 W ) 4bA 0 1

1 s S l1j4 j~i Jlf l1gtygt

)~i- j) 0raquogt Sr-J~)gt (I~)

Gf FIfIOOlre Eoefgy

)u )gt 41 ~Yl bgt -s) ) ) )

16 tL (I) 1lt l1 Jgto - 0 r-ltS ( ~ ~ r

dAL tL let Ili L 4- 1100 lgt - j- - bull )7 ~- (j bull ~ y

-slet oIAJ jI ~ 6 e ~S CF 61 ~L bull lt bullbull -~ Yf I bull bull rr ~

IV CAU) 11-tJNf ~ - ~ -s))lf 0L ) s Ii~r -sr-fJI~I )

~lo L) (13 fb Jb )gt ) 0-1 ) 6 1

JgtI ~L ) 0-1 ) 1 jt ) ~)~

4 ampr ~I ouj ~ 01S I) ~~gt

J~ )h-o ~ S Jgt IIgt~ -s4Lliu )1 olA1

6 bull ~s lt ogtlA1 ~L ~ s~V) ltT - r S shy

~I) IT5))A6 c Vf JL ) )4 0l1 -sl)

( S ~I) s 6 1 VLI 0-1 ) ol1

0-) ~ S ) ~ -sj1 ~ ~I l1

gtS A~ ~ r1- )b ~ ~~~ )

~ s -s10l1 -SjL~ )~ 1 J-o Ii JLd III II I -slctgt JL gt ~ j1 ~Y

) IJIgtIgt r-S S l1) )gt 6 ogtL )~ s ol1 gt1) -sj1 ~ 0t-~~1 s ~ Ii1S

gt~ Iy-I ) ) 0-1 igt gt ~I) let ~

~ l1 ~~ ~~ -SjL~ )gt rgt- -s~L 1 o~) ~I ~ ~ ampr 4 raquogt) 1 )gt eJL

6 1 lbo 0-1 olAgtgt 0W G=~t Jraquo L ~~ s tL bull lt bull bull bull bull I~ rr ~ ltS ( w11

1 ~I gt~ Iy-l f- ~ 6 Ie ~ Jb

raquogt~

~ r1- ~ s ~ -sj1 ooll 0-1)~

-slct~ 4 6 IJIgtS ~I) JgtL l1 ( ~I) -sl)

4- __ - ~ )gt ~S l1 ( ~ -S~

1 lA ~~ -Slet~ ~

0 S ogtL ) s ~ wi ) 6 Jgto s If

Jgto bullgtS -SjL ~ I) S l1 ~ ~li ~ 10Jgto -- -sl)gt 61 ~1)l1 ( 4- _ _ - ~

(1 ~)I~gt~l1(

)j ~ JI~I ) ~ ~ 0l1 gt1) -sj1

~S A~ S l1 ( 4iL ~

G =[~(ampII - amp)] (1)

~~ ~S amp amp bull rs IA

middot~4S l1 ~ Jgtr Ie ~~ Jgto ~ -sj1 0S 0~ ~ le)~ A~

~gt J-a- lIr--o S Jgto ~ ~~ S ~ -sj1 )I) Ie Ir--o S Jgto ~ -sj1

I) (w)lr--o S s Jgtr I)j 1 ~~ S Jgto

S Jgto )gt Ie ol1 ~ Jgtr 4 JgtL 0 1 S

~ bY Ie )5 16)gt (i~) Jraquo ~~

4 6 ~ ~ ~Ui jI 1 0 W1 s )~ ~ ~ gtS Ir--o S Jgto ) S ~I

- -- -- -- -

(VA

Ilf J~ ))b I~ ltS4cWI J) )) I) wloj ~)Pgt

0- ))gtft 4 W jl Jlgt sUbr-i Jw1 1)

~ o~ 1 Ilf )) ~~~ ~ 4 UbWWI

gt~ ))gt S ~ ~ UbS J-f

Jw1 wW1 ~ ~~ sUboJ ~~ S

)~ ltS~ ~~ sUboJ )) s)lfjL ~Ir-

)r~ lS)ft ))yen ~~ 4 lI~1 JlAi1

Crack

) Jlgt )) )~ ltS~ s1 ltf) jS ))

o~) jS )) IS )hgtt 1 o~ 0)1) ~W s~

l) ~ slgt 4 A Jo ltSI ~ )) ~~ )~

A 01 ) 4 A oJ~) b ~~ ))gt SIS

~ wW1 ) 4 S ltSlgt wWI o~ ~~ A

JA )~ 4 )~ ~~ ~~ ~ ) o~

10 o~ ~~ wW1 s) B ~~ 01 s)lfjL

)~ o)b ~s~ wi ~

-W ~JJil

jfgtlt-gt lS)ft I~ wWI ) 0- W u1 jI ~

W sr-i ~ IS l)~ wi 4 ~ ~)J

)~ Ir-I i)t 4 L ~ ~ Jw1 b

IIio ~ ~ )) ISW ~ ~ ))1 Jw1

0- b4gt1 ~ 4 ltS)I) ~)) ~ sIft ~

1 4 I) ~ Ibt middot~4~ AI ltS)ljl 40)) wL ltlt) ~I) )P 4 wI~ )S Jlgt )) ~4)

) CUQ

n-1I2 n+ 112 blAi )) sr ~I~ ~ 9)

)~ ~bu n ~)) vyl

v = U n+1 - Un V =Un - Un_I n+112 III n-112 III

n n-I

v = r+12 - r-112

n III n

(V)

~ J)W 6I)t )) r)lio 0-t1 s)l40 4 gt01 ~Igt 1J i s)Ij1~)) sIft yl~ )S jS

P11+12 =Pn-12 + I1lnmI(ut)n + von)n unt)n)

(A)

n n-l ~loj s+olf ~~ I1ln ~I) 0-t ))

ltSr-i W jI ~L ltS-)~ i-- tnt hi l0n

0)lA1 1) 4 ~ 4 ~~ raquo1) sr-i I)Igtshy

ft ~~~ )) 5 )~ ~loj ltS+olf jI

~ J) )~4 o)~ )) o)~ bull W sLSo

bulll) ~Igt er ~~)

--- -- ---I

On On

wIshy1~~~s~~~Jof ~

~ ~lAuW ~jWJo

4$4)) Lal Ilf ltl 4 I~ sUbwWI J) S ~) nw 5)li1 rmiddotw 1 4$Ub1 ~l) Jlllf jI J~ )~ s~~

~~ 4 IS 4$UbWWI )S)gt ~ 4 IS))Ii

lS)ft sIft middot~4 ~ W )) 4 J$~ ))gtft

J) ~ ~ ltSUbJ) J- sUbW

4$13) raquo) ltSjLor-gt) J) ~c ltS~

0-t~ ur 0-t1bull1 o~ ~I) slai -YU J)

[W] 1 ol )~ 05E lt a lt 2E

~IS ) ~~ )) ~ I)) ~o -4 JgtI)) ))

)1) ~w on ~I) ) ~

on I) ~ I)) ~ IJw SIx )t cr1 ~ f

) ~~ 01) ~I) ) ~ 01~1S

g =[x2 Xl]n 0 rc SI))

( f)

~ X2 I) Xl ) )IS Ool-ol gap ~l g 6

~ x JL )bX n I) r I) 141 ci4 j5 ~

(0 j5 ) l41S IJW 0 jp reI) IJW ~

1) lt n )t I~ f 9 ~)u ~ ~~ 4 )1) ~W xj f ~ 01~IS r- lAo ~x

owcon = J agog(u)dare

(0)

01~1S ~W SI)y-i )bx ~ x ) ( 0) ~I)

xj ) ~

oWcon = J fcon Og ouda re ou

oy)

) ~01~1S nV) ~I) k~ I) fcon 6

fcon = ACoc

(W)

OA)

-4 --I~ ~rL a I) IJw ~ pound 6

f ~ 01~1S I) S 0 SIx J)t SI)y-i )bx l41S

( ) ~I) ~Iil

Ii s

lt ) 1 i 4lJ u I) i xy k nt ~I) cr1 )) 6

g = (gngt) )bx middotl41S S 0 )) 015 ~ )bx

middotl41S gap)bx

Ii~ ~ ~) I ~ IJW ~15 )) Ip))px

)) x Q I) S)I) ~)) )bxU ~ ~rL e 6

S)~ ~)t ~ I4 ~I) cr1 middotl41S ~11 x)LA

Jgt ~I 4 0L~ ) ~ I) )) 0))1 ~

~)t ~ cr1Jgt SIx ~ S1pJgtI)) ))

0)f ~ Jgt)) J J)) ~ 1 ol) f~

fj~ ~)t 0))1)1 lJ SIx Jr J-~ S31

IJW ~ blr ~ J~I SIx Jgt) cr1)) middotl41S con - I It rmiddot 1 lIo 7r H l s

~ v- r--- bull )y ~ ~ bull

))IS 0))I

con ( )7r = 7r(II) + 7r e u

f~ ~l I~ ~ xlx7r I~ ~l I~

l41S ol

( )

~ SI)L 07r I~ J-~ ~l I~ Jlgt ~ ~ IJW ~ ~~lJ I) )IS o)b )

~ra ~ j~ Jgt)) cr1 ~)X 1)1 )) IS ~ con

))) 7r ~ ~l ) e ~

It II 4JNJ

)~61 )~I J-1cr1x -4 Jgt) S~

~6 IS~ I) l41S ~~~ o)~ eTc ~) )~ 1 ~ xix 01 I~ )IS L)

o(CrC) =0

xj ) ~ 01~1S I) ~ ~l Jgt) cr1 ))

Jcrcon 1 C -T=2 a

on 7r (II) (II)U

j1 )IS ool-ol -4 --~ a 0 n ~I) )) ~ ~ j~ ~ J~I 6 1 01J) cr S1p~j-o

I) O)1 lt$H -4 Jgt) b a~ ~L )t

-4 --~ )IA ~ 0 1 jI olol lJ S~~

)YI Slbgt --~ cr1 )IA ~l) 4 ))b ~

S)j1 )L- x)LA Sr)~ I) lJ4 IS ~I oloi

~ lo~1 IS Jbu S)Jamp S)I~li ~ 01 SIx

u-19 )) a SIx Jbu S1l~1 )) ~ 1)1 1 )IA

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 3: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

YVV )gt S JlgtU

0 W1 Jgtr )I) Ie gt IJIgtIgt ~L gt~ Iy-I 0 W1

0 W1 ~ ~) Ie -L ouj s 4 ~ gtS A~ )j ) ~ (V) 0 W1 ~ 4(A)

I=A2 for2D e

(f)

=P3 for 3D e

(0)

l1 ( bgt ~ VLl ) rgt l1 ( Jgto

oIl S -1 )j ) ~ (f)~l) O)~

E = Ie

t 2G

(Y)

~L u 4 l1 ( 4iL -s) 4bA Jgt )gt eJL 0l1 ~)l 0r-o bull 4bA fb )gt Ii ~

I 6 )gt ~L 0- IJIgtgt S 0 W ) 4bA 0 1

1 s S l1j4 j~i Jlf l1gtygt

)~i- j) 0raquogt Sr-J~)gt (I~)

Gf FIfIOOlre Eoefgy

)u )gt 41 ~Yl bgt -s) ) ) )

16 tL (I) 1lt l1 Jgto - 0 r-ltS ( ~ ~ r

dAL tL let Ili L 4- 1100 lgt - j- - bull )7 ~- (j bull ~ y

-slet oIAJ jI ~ 6 e ~S CF 61 ~L bull lt bullbull -~ Yf I bull bull rr ~

IV CAU) 11-tJNf ~ - ~ -s))lf 0L ) s Ii~r -sr-fJI~I )

~lo L) (13 fb Jb )gt ) 0-1 ) 6 1

JgtI ~L ) 0-1 ) 1 jt ) ~)~

4 ampr ~I ouj ~ 01S I) ~~gt

J~ )h-o ~ S Jgt IIgt~ -s4Lliu )1 olA1

6 bull ~s lt ogtlA1 ~L ~ s~V) ltT - r S shy

~I) IT5))A6 c Vf JL ) )4 0l1 -sl)

( S ~I) s 6 1 VLI 0-1 ) ol1

0-) ~ S ) ~ -sj1 ~ ~I l1

gtS A~ ~ r1- )b ~ ~~~ )

~ s -s10l1 -SjL~ )~ 1 J-o Ii JLd III II I -slctgt JL gt ~ j1 ~Y

) IJIgtIgt r-S S l1) )gt 6 ogtL )~ s ol1 gt1) -sj1 ~ 0t-~~1 s ~ Ii1S

gt~ Iy-I ) ) 0-1 igt gt ~I) let ~

~ l1 ~~ ~~ -SjL~ )gt rgt- -s~L 1 o~) ~I ~ ~ ampr 4 raquogt) 1 )gt eJL

6 1 lbo 0-1 olAgtgt 0W G=~t Jraquo L ~~ s tL bull lt bull bull bull bull I~ rr ~ ltS ( w11

1 ~I gt~ Iy-l f- ~ 6 Ie ~ Jb

raquogt~

~ r1- ~ s ~ -sj1 ooll 0-1)~

-slct~ 4 6 IJIgtS ~I) JgtL l1 ( ~I) -sl)

4- __ - ~ )gt ~S l1 ( ~ -S~

1 lA ~~ -Slet~ ~

0 S ogtL ) s ~ wi ) 6 Jgto s If

Jgto bullgtS -SjL ~ I) S l1 ~ ~li ~ 10Jgto -- -sl)gt 61 ~1)l1 ( 4- _ _ - ~

(1 ~)I~gt~l1(

)j ~ JI~I ) ~ ~ 0l1 gt1) -sj1

~S A~ S l1 ( 4iL ~

G =[~(ampII - amp)] (1)

~~ ~S amp amp bull rs IA

middot~4S l1 ~ Jgtr Ie ~~ Jgto ~ -sj1 0S 0~ ~ le)~ A~

~gt J-a- lIr--o S Jgto ~ ~~ S ~ -sj1 )I) Ie Ir--o S Jgto ~ -sj1

I) (w)lr--o S s Jgtr I)j 1 ~~ S Jgto

S Jgto )gt Ie ol1 ~ Jgtr 4 JgtL 0 1 S

~ bY Ie )5 16)gt (i~) Jraquo ~~

4 6 ~ ~ ~Ui jI 1 0 W1 s )~ ~ ~ gtS Ir--o S Jgto ) S ~I

- -- -- -- -

(VA

Ilf J~ ))b I~ ltS4cWI J) )) I) wloj ~)Pgt

0- ))gtft 4 W jl Jlgt sUbr-i Jw1 1)

~ o~ 1 Ilf )) ~~~ ~ 4 UbWWI

gt~ ))gt S ~ ~ UbS J-f

Jw1 wW1 ~ ~~ sUboJ ~~ S

)~ ltS~ ~~ sUboJ )) s)lfjL ~Ir-

)r~ lS)ft ))yen ~~ 4 lI~1 JlAi1

Crack

) Jlgt )) )~ ltS~ s1 ltf) jS ))

o~) jS )) IS )hgtt 1 o~ 0)1) ~W s~

l) ~ slgt 4 A Jo ltSI ~ )) ~~ )~

A 01 ) 4 A oJ~) b ~~ ))gt SIS

~ wW1 ) 4 S ltSlgt wWI o~ ~~ A

JA )~ 4 )~ ~~ ~~ ~ ) o~

10 o~ ~~ wW1 s) B ~~ 01 s)lfjL

)~ o)b ~s~ wi ~

-W ~JJil

jfgtlt-gt lS)ft I~ wWI ) 0- W u1 jI ~

W sr-i ~ IS l)~ wi 4 ~ ~)J

)~ Ir-I i)t 4 L ~ ~ Jw1 b

IIio ~ ~ )) ISW ~ ~ ))1 Jw1

0- b4gt1 ~ 4 ltS)I) ~)) ~ sIft ~

1 4 I) ~ Ibt middot~4~ AI ltS)ljl 40)) wL ltlt) ~I) )P 4 wI~ )S Jlgt )) ~4)

) CUQ

n-1I2 n+ 112 blAi )) sr ~I~ ~ 9)

)~ ~bu n ~)) vyl

v = U n+1 - Un V =Un - Un_I n+112 III n-112 III

n n-I

v = r+12 - r-112

n III n

(V)

~ J)W 6I)t )) r)lio 0-t1 s)l40 4 gt01 ~Igt 1J i s)Ij1~)) sIft yl~ )S jS

P11+12 =Pn-12 + I1lnmI(ut)n + von)n unt)n)

(A)

n n-l ~loj s+olf ~~ I1ln ~I) 0-t ))

ltSr-i W jI ~L ltS-)~ i-- tnt hi l0n

0)lA1 1) 4 ~ 4 ~~ raquo1) sr-i I)Igtshy

ft ~~~ )) 5 )~ ~loj ltS+olf jI

~ J) )~4 o)~ )) o)~ bull W sLSo

bulll) ~Igt er ~~)

--- -- ---I

On On

wIshy1~~~s~~~Jof ~

~ ~lAuW ~jWJo

4$4)) Lal Ilf ltl 4 I~ sUbwWI J) S ~) nw 5)li1 rmiddotw 1 4$Ub1 ~l) Jlllf jI J~ )~ s~~

~~ 4 IS 4$UbWWI )S)gt ~ 4 IS))Ii

lS)ft sIft middot~4 ~ W )) 4 J$~ ))gtft

J) ~ ~ ltSUbJ) J- sUbW

4$13) raquo) ltSjLor-gt) J) ~c ltS~

0-t~ ur 0-t1bull1 o~ ~I) slai -YU J)

[W] 1 ol )~ 05E lt a lt 2E

~IS ) ~~ )) ~ I)) ~o -4 JgtI)) ))

)1) ~w on ~I) ) ~

on I) ~ I)) ~ IJw SIx )t cr1 ~ f

) ~~ 01) ~I) ) ~ 01~1S

g =[x2 Xl]n 0 rc SI))

( f)

~ X2 I) Xl ) )IS Ool-ol gap ~l g 6

~ x JL )bX n I) r I) 141 ci4 j5 ~

(0 j5 ) l41S IJW 0 jp reI) IJW ~

1) lt n )t I~ f 9 ~)u ~ ~~ 4 )1) ~W xj f ~ 01~IS r- lAo ~x

owcon = J agog(u)dare

(0)

01~1S ~W SI)y-i )bx ~ x ) ( 0) ~I)

xj ) ~

oWcon = J fcon Og ouda re ou

oy)

) ~01~1S nV) ~I) k~ I) fcon 6

fcon = ACoc

(W)

OA)

-4 --I~ ~rL a I) IJw ~ pound 6

f ~ 01~1S I) S 0 SIx J)t SI)y-i )bx l41S

( ) ~I) ~Iil

Ii s

lt ) 1 i 4lJ u I) i xy k nt ~I) cr1 )) 6

g = (gngt) )bx middotl41S S 0 )) 015 ~ )bx

middotl41S gap)bx

Ii~ ~ ~) I ~ IJW ~15 )) Ip))px

)) x Q I) S)I) ~)) )bxU ~ ~rL e 6

S)~ ~)t ~ I4 ~I) cr1 middotl41S ~11 x)LA

Jgt ~I 4 0L~ ) ~ I) )) 0))1 ~

~)t ~ cr1Jgt SIx ~ S1pJgtI)) ))

0)f ~ Jgt)) J J)) ~ 1 ol) f~

fj~ ~)t 0))1)1 lJ SIx Jr J-~ S31

IJW ~ blr ~ J~I SIx Jgt) cr1)) middotl41S con - I It rmiddot 1 lIo 7r H l s

~ v- r--- bull )y ~ ~ bull

))IS 0))I

con ( )7r = 7r(II) + 7r e u

f~ ~l I~ ~ xlx7r I~ ~l I~

l41S ol

( )

~ SI)L 07r I~ J-~ ~l I~ Jlgt ~ ~ IJW ~ ~~lJ I) )IS o)b )

~ra ~ j~ Jgt)) cr1 ~)X 1)1 )) IS ~ con

))) 7r ~ ~l ) e ~

It II 4JNJ

)~61 )~I J-1cr1x -4 Jgt) S~

~6 IS~ I) l41S ~~~ o)~ eTc ~) )~ 1 ~ xix 01 I~ )IS L)

o(CrC) =0

xj ) ~ 01~1S I) ~ ~l Jgt) cr1 ))

Jcrcon 1 C -T=2 a

on 7r (II) (II)U

j1 )IS ool-ol -4 --~ a 0 n ~I) )) ~ ~ j~ ~ J~I 6 1 01J) cr S1p~j-o

I) O)1 lt$H -4 Jgt) b a~ ~L )t

-4 --~ )IA ~ 0 1 jI olol lJ S~~

)YI Slbgt --~ cr1 )IA ~l) 4 ))b ~

S)j1 )L- x)LA Sr)~ I) lJ4 IS ~I oloi

~ lo~1 IS Jbu S)Jamp S)I~li ~ 01 SIx

u-19 )) a SIx Jbu S1l~1 )) ~ 1)1 1 )IA

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 4: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

- -- -- -- -

(VA

Ilf J~ ))b I~ ltS4cWI J) )) I) wloj ~)Pgt

0- ))gtft 4 W jl Jlgt sUbr-i Jw1 1)

~ o~ 1 Ilf )) ~~~ ~ 4 UbWWI

gt~ ))gt S ~ ~ UbS J-f

Jw1 wW1 ~ ~~ sUboJ ~~ S

)~ ltS~ ~~ sUboJ )) s)lfjL ~Ir-

)r~ lS)ft ))yen ~~ 4 lI~1 JlAi1

Crack

) Jlgt )) )~ ltS~ s1 ltf) jS ))

o~) jS )) IS )hgtt 1 o~ 0)1) ~W s~

l) ~ slgt 4 A Jo ltSI ~ )) ~~ )~

A 01 ) 4 A oJ~) b ~~ ))gt SIS

~ wW1 ) 4 S ltSlgt wWI o~ ~~ A

JA )~ 4 )~ ~~ ~~ ~ ) o~

10 o~ ~~ wW1 s) B ~~ 01 s)lfjL

)~ o)b ~s~ wi ~

-W ~JJil

jfgtlt-gt lS)ft I~ wWI ) 0- W u1 jI ~

W sr-i ~ IS l)~ wi 4 ~ ~)J

)~ Ir-I i)t 4 L ~ ~ Jw1 b

IIio ~ ~ )) ISW ~ ~ ))1 Jw1

0- b4gt1 ~ 4 ltS)I) ~)) ~ sIft ~

1 4 I) ~ Ibt middot~4~ AI ltS)ljl 40)) wL ltlt) ~I) )P 4 wI~ )S Jlgt )) ~4)

) CUQ

n-1I2 n+ 112 blAi )) sr ~I~ ~ 9)

)~ ~bu n ~)) vyl

v = U n+1 - Un V =Un - Un_I n+112 III n-112 III

n n-I

v = r+12 - r-112

n III n

(V)

~ J)W 6I)t )) r)lio 0-t1 s)l40 4 gt01 ~Igt 1J i s)Ij1~)) sIft yl~ )S jS

P11+12 =Pn-12 + I1lnmI(ut)n + von)n unt)n)

(A)

n n-l ~loj s+olf ~~ I1ln ~I) 0-t ))

ltSr-i W jI ~L ltS-)~ i-- tnt hi l0n

0)lA1 1) 4 ~ 4 ~~ raquo1) sr-i I)Igtshy

ft ~~~ )) 5 )~ ~loj ltS+olf jI

~ J) )~4 o)~ )) o)~ bull W sLSo

bulll) ~Igt er ~~)

--- -- ---I

On On

wIshy1~~~s~~~Jof ~

~ ~lAuW ~jWJo

4$4)) Lal Ilf ltl 4 I~ sUbwWI J) S ~) nw 5)li1 rmiddotw 1 4$Ub1 ~l) Jlllf jI J~ )~ s~~

~~ 4 IS 4$UbWWI )S)gt ~ 4 IS))Ii

lS)ft sIft middot~4 ~ W )) 4 J$~ ))gtft

J) ~ ~ ltSUbJ) J- sUbW

4$13) raquo) ltSjLor-gt) J) ~c ltS~

0-t~ ur 0-t1bull1 o~ ~I) slai -YU J)

[W] 1 ol )~ 05E lt a lt 2E

~IS ) ~~ )) ~ I)) ~o -4 JgtI)) ))

)1) ~w on ~I) ) ~

on I) ~ I)) ~ IJw SIx )t cr1 ~ f

) ~~ 01) ~I) ) ~ 01~1S

g =[x2 Xl]n 0 rc SI))

( f)

~ X2 I) Xl ) )IS Ool-ol gap ~l g 6

~ x JL )bX n I) r I) 141 ci4 j5 ~

(0 j5 ) l41S IJW 0 jp reI) IJW ~

1) lt n )t I~ f 9 ~)u ~ ~~ 4 )1) ~W xj f ~ 01~IS r- lAo ~x

owcon = J agog(u)dare

(0)

01~1S ~W SI)y-i )bx ~ x ) ( 0) ~I)

xj ) ~

oWcon = J fcon Og ouda re ou

oy)

) ~01~1S nV) ~I) k~ I) fcon 6

fcon = ACoc

(W)

OA)

-4 --I~ ~rL a I) IJw ~ pound 6

f ~ 01~1S I) S 0 SIx J)t SI)y-i )bx l41S

( ) ~I) ~Iil

Ii s

lt ) 1 i 4lJ u I) i xy k nt ~I) cr1 )) 6

g = (gngt) )bx middotl41S S 0 )) 015 ~ )bx

middotl41S gap)bx

Ii~ ~ ~) I ~ IJW ~15 )) Ip))px

)) x Q I) S)I) ~)) )bxU ~ ~rL e 6

S)~ ~)t ~ I4 ~I) cr1 middotl41S ~11 x)LA

Jgt ~I 4 0L~ ) ~ I) )) 0))1 ~

~)t ~ cr1Jgt SIx ~ S1pJgtI)) ))

0)f ~ Jgt)) J J)) ~ 1 ol) f~

fj~ ~)t 0))1)1 lJ SIx Jr J-~ S31

IJW ~ blr ~ J~I SIx Jgt) cr1)) middotl41S con - I It rmiddot 1 lIo 7r H l s

~ v- r--- bull )y ~ ~ bull

))IS 0))I

con ( )7r = 7r(II) + 7r e u

f~ ~l I~ ~ xlx7r I~ ~l I~

l41S ol

( )

~ SI)L 07r I~ J-~ ~l I~ Jlgt ~ ~ IJW ~ ~~lJ I) )IS o)b )

~ra ~ j~ Jgt)) cr1 ~)X 1)1 )) IS ~ con

))) 7r ~ ~l ) e ~

It II 4JNJ

)~61 )~I J-1cr1x -4 Jgt) S~

~6 IS~ I) l41S ~~~ o)~ eTc ~) )~ 1 ~ xix 01 I~ )IS L)

o(CrC) =0

xj ) ~ 01~1S I) ~ ~l Jgt) cr1 ))

Jcrcon 1 C -T=2 a

on 7r (II) (II)U

j1 )IS ool-ol -4 --~ a 0 n ~I) )) ~ ~ j~ ~ J~I 6 1 01J) cr S1p~j-o

I) O)1 lt$H -4 Jgt) b a~ ~L )t

-4 --~ )IA ~ 0 1 jI olol lJ S~~

)YI Slbgt --~ cr1 )IA ~l) 4 ))b ~

S)j1 )L- x)LA Sr)~ I) lJ4 IS ~I oloi

~ lo~1 IS Jbu S)Jamp S)I~li ~ 01 SIx

u-19 )) a SIx Jbu S1l~1 )) ~ 1)1 1 )IA

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 5: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

[W] 1 ol )~ 05E lt a lt 2E

~IS ) ~~ )) ~ I)) ~o -4 JgtI)) ))

)1) ~w on ~I) ) ~

on I) ~ I)) ~ IJw SIx )t cr1 ~ f

) ~~ 01) ~I) ) ~ 01~1S

g =[x2 Xl]n 0 rc SI))

( f)

~ X2 I) Xl ) )IS Ool-ol gap ~l g 6

~ x JL )bX n I) r I) 141 ci4 j5 ~

(0 j5 ) l41S IJW 0 jp reI) IJW ~

1) lt n )t I~ f 9 ~)u ~ ~~ 4 )1) ~W xj f ~ 01~IS r- lAo ~x

owcon = J agog(u)dare

(0)

01~1S ~W SI)y-i )bx ~ x ) ( 0) ~I)

xj ) ~

oWcon = J fcon Og ouda re ou

oy)

) ~01~1S nV) ~I) k~ I) fcon 6

fcon = ACoc

(W)

OA)

-4 --I~ ~rL a I) IJw ~ pound 6

f ~ 01~1S I) S 0 SIx J)t SI)y-i )bx l41S

( ) ~I) ~Iil

Ii s

lt ) 1 i 4lJ u I) i xy k nt ~I) cr1 )) 6

g = (gngt) )bx middotl41S S 0 )) 015 ~ )bx

middotl41S gap)bx

Ii~ ~ ~) I ~ IJW ~15 )) Ip))px

)) x Q I) S)I) ~)) )bxU ~ ~rL e 6

S)~ ~)t ~ I4 ~I) cr1 middotl41S ~11 x)LA

Jgt ~I 4 0L~ ) ~ I) )) 0))1 ~

~)t ~ cr1Jgt SIx ~ S1pJgtI)) ))

0)f ~ Jgt)) J J)) ~ 1 ol) f~

fj~ ~)t 0))1)1 lJ SIx Jr J-~ S31

IJW ~ blr ~ J~I SIx Jgt) cr1)) middotl41S con - I It rmiddot 1 lIo 7r H l s

~ v- r--- bull )y ~ ~ bull

))IS 0))I

con ( )7r = 7r(II) + 7r e u

f~ ~l I~ ~ xlx7r I~ ~l I~

l41S ol

( )

~ SI)L 07r I~ J-~ ~l I~ Jlgt ~ ~ IJW ~ ~~lJ I) )IS o)b )

~ra ~ j~ Jgt)) cr1 ~)X 1)1 )) IS ~ con

))) 7r ~ ~l ) e ~

It II 4JNJ

)~61 )~I J-1cr1x -4 Jgt) S~

~6 IS~ I) l41S ~~~ o)~ eTc ~) )~ 1 ~ xix 01 I~ )IS L)

o(CrC) =0

xj ) ~ 01~1S I) ~ ~l Jgt) cr1 ))

Jcrcon 1 C -T=2 a

on 7r (II) (II)U

j1 )IS ool-ol -4 --~ a 0 n ~I) )) ~ ~ j~ ~ J~I 6 1 01J) cr S1p~j-o

I) O)1 lt$H -4 Jgt) b a~ ~L )t

-4 --~ )IA ~ 0 1 jI olol lJ S~~

)YI Slbgt --~ cr1 )IA ~l) 4 ))b ~

S)j1 )L- x)LA Sr)~ I) lJ4 IS ~I oloi

~ lo~1 IS Jbu S)Jamp S)I~li ~ 01 SIx

u-19 )) a SIx Jbu S1l~1 )) ~ 1)1 1 )IA

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 6: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

U)JSU~

4 ~~ lr ~)amp J1o 60 I~I u~ 0-1 )~

JI ~~ ~ Jr-s sAgt~L) ~l

~)gtr u- 60 ~ ~~IS ~ ~~

) ~ ~)T ~rS oli) ~ ~~)gt S

lAr-ol)4 l 0 1r-o ~~ 0-1)~ ~rIS

~ ~I~ r iJw ~ts ~I) )~ ~lgt ~ bulllil ci ) 6JJlb ~)T

~ o)~ Jto )~ o~ o~b 0L ~~I4- ~ y ~ J1o 0-1

0 ) ~lI)I~~s ~~ (Y)~

I) sr lA~T )~)4 I )~ ~~ 1

() J~ )~ eJLa ~ [ ]a~ IS 0L

~l1 ~ jL ~)~ ~ 1 o~ 0~)1

~~~ ~ ~ r s o~1i1 ~)T ))1 f 4 IS ~) ISI) ~ 1) r-)lgt )4 xS IS ~ ~~

~ ~ s ~) li) (A) (V)~ ~

~~)gt S)I ~~- d~ 4ol~ ltli

)~ s )blA gtJb~ IS 0L I) ~r - ~j~

~ ~~)gt s ~ ~~ oal () ~

)I 0101 ~ ~l o~lgtlto )~ IFL J~ 4 0101

[ ]~~ ) sAgt~L) J1o 0-~

time=t

n1 V

I j I ysI

g j

t r f

(UJI) ()

~ 0 ~)gtr -r 0 ~ 0 ~)gtr - )u~ s ~lgtlto Ir-I - ~ Ir ~ ~ ~ ~)gtr -i ~

JL iJw ~) 4 y-~ ~ 0 ~)gtr Jlgt )~

~ J~ I)~ )IfjLri- Jlgt ~ ~ s Yf)lfjL

~ ~w IS r-ol I) ~ ~)~ ~I~ 1

(n) (rmiddot) lA ~I) )- ~ iJw ~~L

[f] 101 algt ~

fcon =agn[-~Dl agnNs -N2D

K COI1 a[N NT - g(TNT +N TT C ss Ism ms

+gN NT)I

(n)

~~ j i ~ ~ lt-I~ N 2 tV s li~ IS ~~ (rr) ~I) )- ~ Ts N

(rr)

-~lgtlto Ir-I ))1 ~ )~ 0101 ~ ~I) ~

~Jt 4LJgtU Ir ~4 J-r-- Ir--o ~WI

lil o~~ o~Ll

- - -Pc =2400 kgm 3 Ec =38GPa vc=015

a COl1loCt = 25 GPa t=30MPa Gf =100Nm

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 7: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

fA ~))s~

o D=150mm

p

04 3012 02 20 04

CMOD controlling signal [] ~ o)~ Jlo )~I ~

-c~ J))s 1 ~)~s~ ~s ~J OJs~ ~ s~1 V ~

114e+06 S42e+OS 142e+OS S4le+OS 33ge+OS 130e+OS -630984 -l6e+OS -41e+OS -61e+OS -81e+OS -1le+06 -13e+06

150 ~-----~----r-

125

100 - _ _ _ ~Ul bullI

~LojE 75E

50

25

0+-------------1 mm -10 0 10 20 30 40 50

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 8: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

I -l5 lt~ bull _9 ~ Qgt - L~ IAgt~ ~ LY V ~~f u----

-S~~ JIo bull 0~L ~~ Lr- jiJ 0~ loW (~ ~ ~~~

)~ I) (~~~ r ~~)~j -s4S ~~ ~ J-= 0151

~l5 ~ Qgt -L Qgt ~I 1 ~IIi - Lo Cl - -- 7 r- U - -~

09 lt 19 [j - 1( [jmiddotL ~ - r-) r- r-s-) LY ~

1

4JJ O)~ Jlo 15~ ~~I ~ lt19~ ~ Jb ~ JIo r1

~ 4S L (1) J lgt _1La u~ lt r) 1( - ~ C ~ ~

)~ Jb Lr1 [V] 1 0 JIo -S~~~ ~ ~ u)

0~)gt5 m1SW~~1 ~j-wJ ~ ~[j ~ ~)gtr1

~[jr ~~ ~ JS ~ ~~ 01 Jgtb ~ ~[j ~

ygt )~ ylb p~li lt~) ~)~ 0Lj ~ ~

o)Ii1 ~ ullAgtL ~ u -s~~ k-Gu ) Jolgt ~l

1 ~~L) -s~r5

~I tL ~)gtr ~L )~ -sUzrol)4 ) ~ L r 1_ _ ro 1( shy ~ ~-s~ ~ s--S~

) o)~ Jlo I 110 - cUz - lt ~ c~~ I 1_7r-- ~ - r- 15r cs------ - ~

1 IS-Uz J ~ -Sl)~ ~r -SUz~yen ~yfr )~

u~ ~ ~ LY- JaAo ~~~ ~ ~ JIo r1)~ 0 I 1-71 C~l1 1gt L 4S (r) J lgt ~ ~ IS-) ~

-S~~~ ~~ ~ u) ~ Uz)~L) 0- [A] 1

Iil 0 -SjUIo ~ -S~WI ~~ ~ o~~

~ 1 0 jgt Jol5 ~~~ ~ LY- Lr- J ~ h~ 015 ~~ IAgt~ t) ~1 Lr- ~jiJ

)~ ~ J~)gt 5 ~ ~ p )~ ~r ~ ~Uz~

ygtylb 1 0 ~ lt1) ~ lt ) -S~ bullIAgtIo li I J~ ~lLmiddot1 ~l L I middot1ltraquo- u ) ltY~) IS- -) -- ~ Tmiddot

1iJ 1 09 1-l5 lt ~ b [j I 1_7 ~ r ~ V cs------ r ~

1middot-1 ~ ~ lt ~ b 0 lAo JI ~-~- ~IS- ~ - -) cs------ r ~ ~

~~) ~ )lb1 lt ) ~~ -s4S ~~ ~ olAgtL 4S )~W (H) ~)~

~r -~ ~ ~ -s4S)d oloI~

~ ~~ )~ ~j )~ ) 4S ~IIo1 JI~I 1

~ b 1 r-oS r ) PJ I 4S IS-4- ~ olf

Pc = 2400 kgm 3 Ec = 28GPa vc=015 J=24MPa Gf =100Nm

Ps = 7850 kgm 3 Es = 210GPa Vs = 03 ~ = 4400MPa aeonact = 25 GPa

80

70

60

50

~ 40 ir

30

20 1 10

0

(mm) 0

middotmiddotmiddot middotmiddot~rrerical

--Experirrental

2 4 6 8 10 12

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 9: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

- -Pc = 2400 kgm 3

Ec 24GPa Vc =017 ~ = 210GPa

alt(COltfoCI) = 2100GPa 1 26MPa Gj =llONm PI = 17850kg m3

400 ------------------ fItlrmrical

350 --Exprirmntal shy

300

250

i 200 T150

Bcm 100

050

)) ~tJ ~~ 4 ol~ gt)gt oSi Jr-f ~4 LS~ ULI ) oS ~ ~loj LS4-0l3

bull1 ogt o)b 0li ~I 0101 ~ ~

)~ o)~Jlo

~ ))gt) I oS ~ - ~ JIo r-I ))

1 ogt ~ cif ) 400m S ampr 4 ~tJ

)) ~tJ ~~ Jr-o oSi Jr-f LS~I n A~) ~ ~ J(j ~ ogt o)b 0li ()~

))gt) 4~1 jl middotgt4 IS JIo r-I ~~~

IS ot~ ~loj)) )4j)~ LSI )~I 0jL 4 ~tJ

0)lgtt4 )) )) ~ oS ) IS 4 r- ~

~ ~ )jb~ oli ~ ojL 4 ~tJ ))gt)

4 ~tJ ))gt) ~ )) middot~4 c1~ I) 0jL ~

~~ 4 IS~ t ~ LSI ~I ~ ojL

~~~ ulalo ~~lo) ~l

oS1 ~~ [rmiddot A]l) IS ~~)gt~

gt algt ij4 ~b~ ~4 LS~r))

010)T 0)lAli rA cili )gtL 0 ylgtjl -kJ~

0 0 WI LSUtjr )) 4Si gt) )~I 0lSo1 ~

IS oalo ( )~ ~ l )) 0)A fl ~~I 41

)

) IS f~1 ~ LSI~I )) )~ ~ LS~

~ 4 3)lt LS~ ~ )~ u) )) bi9

ogt cif )~ )I f )) ) IS )P 0)4) 6-JIJb

Global Search ~li tLz i~ Zone ~Jr-I))

4 middot~4 IS Local Search ~li tLz i~ Field

yl~ )1gt LS)l JIo ~ ~)j-i l$)) ~ ~0)

0WI 0 jI~1 )1gt )) ~r-t Field Zone LSI) l

LS))gt) 0 WI 0 jI~1 middotr )1gt )) LS))gt)

LSI) l y-)li ylgtjl r-I ) o~ Iol~ Gf (a) -sJ~ ~ro ~ jI ~ LSUt~1)4 y-L

LS)~4 4r1~ 0 )) l LS~ 0 ~

ul )) a LSI))~r-~ I)j$ r-1t ~ J(j )~ ~I) -r1 lol middotgt4 1S05E lt a lt 2E

01 LSI) rV~4 ampr L ))gt) JLo )) ~ )))IS

- H ~) ) ) Ja ~ )) o~ ))gtr ~

(-sJ~ ~ro) Uw LSj~ ~ 1 fj~ (iJ1

) IS oalo uL r-I )) oS )~w ~4 v411

0) )1gt -sJ~ ~ro l yl~ ~ 0~) LSI)

4 ~)~ 1 o~)f ylgtjl ~tJ 34 JIo )I)

)) ))gtr LS4-0rlSo lli -sJ~ ~ro ylgtjl

(~-H) (y-H) LS~ )) ) IJb1~ Js -sJ~ ~ro LSI) )4j )~ ylgtjl jI ~li LS)~4li

LS~I (W) ~ )) 1 o~ o)b 0li Field

000 -yen------------------r- shy

o 01 02 03 04 05 t(ms)

Dloj ~ J ~ix iIa ~ A ~

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 10: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

fAf

(I) ( )

W ~tso soJIt sly ~toli )io -Iiil 1 ~Li sJII1t li l sz

t = 00001 original mesh

t = 00003

Pc =2400 kgm 3 Ee =23GPa vc =015 Gj =11ONm

P =7850kgm3 Es =210GPa v =030 t=20MPa

a eonfael(n) =28000 GPa a eonloel(t) =28 GPa a e(n) =28000 GPa V=400mls

20m

t=00003 t=OOOO2

t = 00004 t=OOOO5

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 11: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

middot u~jgts~

-ix ~ ~)gt~ jl ~li ~~~ s)lf)4 ~ (0)

9 =20middot ~)gt~ ~Ij V = 20mls y1 ~r 4

Ib~ sA 0L (1) ~ d-Ib I) ~19 )~ - ~

~)l s1A)s) ~b ~)~ ~ JIo 0-11 [1]

Ib~ sA 0L I) oliM s4 ~) ~~I o~

~19 0~ rziU )1 oliM ~~)gt S ~ lil 0101 (H) (n) s~ )~ ~ ltli1A~ 6

I4j ~r ~ uL 0-11 )1 1 ~ - rJY

imiddot - ~I ~)gt~ 16 yIJ~ ~~ bullI)gt~

1 bullbullbull jI ~ ~ ~ [1 -ix amp4 JIo ~l~ to -J~ ~~ ~r ~IS I ~~Lj Ilf

~ o~b ~IS -ix amp4 JIo ~I~

~))WJo

J~ -~ 4iS~~ ~~ ~ JIo 0-11

[r] ~ II)~ JoW ~1Ao ISVJ 0 J~

Ee = 23GPa vc=0152500kgm3 G r II0NmPc

Ps 7850kgm3 ft 10MPaEs =210GP VS = 030

(lCN 250GPa (lSN = 15e + 4 V=20 mls

t 00019

t=OOOI bull

bullt=0002 S t=OOO1 S 1S~IoJ)5~ ~-1 1S~1 n ~ Disp (m) UB-4

148-4

128-4

1(1114

8IB-S

6OB-S

4OB-S

lIB-S

oGB+G Time(s) 0G02

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 12: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

OjL (I~I ~W )~ Jlc ~I ) iS )~

0)~ 4[ ~r~ -st~ ~ ~~~

~ 0 JiJ1 j (IS rmiddot li 41 Jgt) i-

lil~ iS 4li Ilblgt cbr ~r ~ )1 )

(~W)A OjL) 1l0)~

~ I bullIS I(~ lt1gt06 11(lS ) ltS~ ) ~~ ~r ltS~

~o)~JM

) I) ~ o Jlc ~~~ ~j Jlc ~I (r-)J~ 0~ ~ S6- ) )slc JLgt

~r ~ ltSI 4lix )gtr ) ~I ~~ ltS)l3)1

ltSlA olj- []Ilb~)~ (n) ~ V =20ms

raquo6-) lil 0 ~ X J~ ltStI) ) CD AB

A

3m

1m

1670NODES 3155ELEMENTS

15m

D I I I I

2m 30m 2m

~~ iS r ~ Jkgt ltSlAyol)4 rL

Zone =00017 Field =000035 Time step = 10e - 08

an(contact) =20000GPa at(contact) = 20GPa

t=00023 s t=00013 S lS+iloj))s~~- lSsJl rf ~

iAY

1 o)b 0L ~~L) ~li 4sp y1b ~I) ~) )~I O~ Q) ~I O~ ~ 01) ~ ))

1 1_7 LA-gt 1061 1l9 Imiddot I I ( If IS L ~ -T ltoS - if) Y U) LY- Igtr-) bull bull C ~ ~lgt ISItt ~l )) ~ I) O~~ 1S4S

0L~ )~I )) Itt OjL liAI slrgt ~) ~ )) S ~) ltI]rv A-i 4 Jb) IS 0L

)rf IS -6 6J ciL A Imiddot~I 0) S ~I_I ) ~ Lbull lr )r Y) V-shy

))lio -Gwl ~ 1S4r1~ lt s~) ~ IS~ )) ~ )i ~ ) IS ~) ~

~ ~lS UW ~lS ISIttr-ol)4 1S1y -1 lJL HI ~ ) ~f_l) I ~ Jl~A IS lr ~ ) lS IS

~Ilraquo~ IgtjL -1 IS~ ~ ~ ~4 IS JlIgttS ~ ~y ~I~ b J lt I]

C lt ) r1~ 0lS1o~1 IS~ )) 1

))b )~ )j i~iSft shy 1S-41)4 ~ ~))p S 1S1tt)~ 0)y5 ~LI bull (DEM) lr=-o IS~WI ) ULI r 4llio r1))

S ~ )) ~U~ r ~ II 0) 1W UW ~lS 1S4r) ) 0)1Q1 4 -_ ~ U L ltLgt lio ) ---gt00 ltLo I 6J If)-c ~ _ ~ y v IS )r ISjLJl4 1S1r ~I 1$))0 ) ~ ~lS

~ IttLL 11_71$)_ ~ 4i1 ~ ~ ISItt Igt jL o)-f ~))p s

~ 0 jL )~ )5 IS~ ~ 1lS1 Lh1 bull Itt~ ~ ~ ISItt OjL)) ))p s ~oo~ ~4-~1 )I ~I 1$))0 1S1ttLkgt ~ fb JlIgttS ~~) 1S~~I~s Uigt ISItt 0 jL ~~4

(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

t ~ ~ )ls ~ ~ ~I~ ~I )ls 11lt -~ If I ) Ls)l 11_7 A

IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 13: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

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(~ )lgtto I-I ~) ~IIS~WI lgt )I ~ ) ~ ~4) ~I 1S1tt)4 I )) bull - If~ If- lj ~ - bull ~ I) IS ~~) ) -S)r - r1 ~gt ) middot~4 IS r~1 ~l9 o~ 4i1 1$))0

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IS~~ U-r r) ~U)

bull10gt) IS IS ~))p S) ~ ~ ~y5 ~~ r9 S)j

~~ ISItt 0jL ~))p S IS~ d- ~ bull I 7 Lu ~ b1 11 1_7 li ~Loo ~ ) _ _ 0 roo IS~~

1 - Chen W F and Han D J (1988) PlasticityforStntctural Engineers

2 - Owen D R 1 and Hinton E (1968) Finite Elements in Plasticity

3 - Shah S P and Swartz S E (1995) Fracture Mechanics ofConcrete

4 - Mohammadi S (2003) Discontinuum mechanics by finite and discrete elements WITPress

5 - Rashid Y (1968) Analysis of prestressed concrete pressure vessels Nuclear Engineering and Design

Balkema Ed PP 265-286

6 - Pearce C 1 (1996) Computational Plasticity in Concrete Failure Mechanics PhD Thesis

7 - Morikawa H and Sawamoto Y (1993) Local fracture analysis of a reinforced concrete slab by the

discrete element method Proceeding if the Second International Coforence on Discrete Element

Modeling (DEAf)

8 - Grzegorz Gajer and DUX P F (1991) Simplified nonorthogonal crack model for concrete J of

Structural Engineering Vol 117 No1 ASCE

9 - Owen D R J and Figueiral J A (1984) Ultimate load analysis of reinforcet concrete plates and shells

including geometric nonlinear effects Finite Element Softwarefor Plates andShelif by E Hinton and D R

1 Owen Pineridge Press London

10 - Cendon D A Galvez 1 C and Planas 1 (2000) Modeling the fracture of concrete under mixed mode

loading ECCOMAS

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured

Page 14: chpc.ut.ac.ir fracture analysis of... · fYP . o~ yl.:>r.:;,;l . Aj~ W"; ~J"! ...;.;; )~ .)'1"; ~ j~l \.5"'~ fft . o.)lo ~ ~')~ -:r.1 d-!lb... ...:......1. tS..l.> )..li.. ~ ~ JI)..l.A>l:S1

fAA

11 - Galvez 1 C Tork B Cendon D A and Planas 1 (2000) A plasticity model for bond and splitting in

prestressed concrete ECCOMAS

12 - Hillerborg A Modeer M and Petersson P (1976) Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements Cement and Concrete Research Vol 6 PP

773-782

13 - Cervenka J (l994) Discrete crack modeling in concrete structures PhD Thesis University of Colorado

14 - Saleh A and Aliabadi M (1995) Crack growth analysis in concrete using boundary element method

Engineering Fracture Mechanics Vol 51 PP 533-545

15 - Belytschko T and Lin J 1 (1987) A three- dimensional impact-penetration algorithm with erosion Int

J Impact Eng Vol 5 PP 111-127

16 - Bathe K J and Ramaswamy S (1979) On three dimensional nonlinear analysis of concrete structures

Nuclear Engineering amp Design Vol 52 No3 PP 385-409

17 - Peric D and Owen D R J (1992) Computational model for 3-D contact problem with friction based on

the penalty method Int Jfor Numerical Methods in Engineering Vol 35 PP 1289-1309

18 - Polak M Modeling punching shear of reinforced concrete slabs using layered finite elements ACI

Structural Journal VoL 95 No1 PP 71-80

ltloU 044 l~ s4WI ) ~ ~~ s1A 0)1 )J IA ~li))~ o~~ ))gt S j~ middot)middotr r) - ( rAmiddot ) bull l) (S1)5

ltloU 0~4 ~ j- JoI~ (S)j )j (41)) ampr) s1A ~I))~ J)gtj ~ I middot0~~L rmiddot n rAgt bull l) (S1)5

~~ sl ~ s1A)4 I SjU )~ s1A ojL ~ I c)lj )~ J slogtu 1 0~~L - n ( r A gt ~1r0 bull -n ll- ~) SjU )~ s1A 0 jL u-ilp -r-JI 8lAa

1- Rankine 3 - Contact Mechanics 5 - Smeared Crack 7 - Discrete Crack 9 - Failure Criterion 11- Willam-Warnke 13 - Hillerborg 15 - Rankine Softening Plasticity 17 - Explicit Transient Dynamic Analysis 19 - Finite Difference 21 Contact Interaction 23 - Node to edge 25 - Global Search 27 - High Velocity Impact

2 - Discrete Element Method 4 - Deformable 6 - Strain Softening 8 - Von Mises 10 - Hsieh 12 - Softening Behavior 14 - Localization Zone 16 - Fracture Indicator 18 - Explicit 20 - Contact Detection 22 - Functional 24 - Consistent Normal Contact 26 - Local Search 28 - Unstructured