analysis of frames containing cracks and resting

8/3/2019 Analysis of Frames Containing Cracks and Resting

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International Journal of Fracture 45: 81-102, 1990.1990 Kluwer Academic Publishers. Printed in the Netherlands. 81

Analysis of frames containing cracks and resting onelastic foundationsM . H. E L - H A D D A D , 1 0 . M . R A M A D A N and A .R . B A Z A R A ADepartment of Structural Engineering, Cairo Un iversity, Egypt; ~and University of Qatar, QatarReceived 20 June 1988; accepted in revised form 3 April 1989

Abstract. A fracture mechanics model has been implemented in the stiffness matrix m ethod to analyze skeletalstructures resting on elastic foundations and containing cracks at superstructure and substructure elements.Stiffness matrices for cracked superstructure and substructure elem ents have been developed based o n fracturemechanics techniques and the stiffness matrix m ethod.The present model has been applied to investigate effects of crack size and location, type of loading, s oil sub-grade modulus, foundation rigidity, and geometry of the structure o n the behaviour of cracked structures takinginto consideration the soil-structure interaction effect.

1. IntroductionI n r e c e n t y e a r s c o n s id e r a b l e e f f o r t s h a v e b e e n m a d e t o s t u d y t h e b e h a v io u r o f s t r u c tu r e sc o n t a in in g c r a c k s . Ba s e d o n f r a c tu r e m e c h a n i c s t e c h n iq u e s a n d s t i f f n e s s m a t r i x m e th o d t h er e d i s t r i b u t i o n i n i n t e r n a l f o r c e s d u e t o c r a c k p r e s e n c e i n s t a t i s t i c a l l y i n d e t e r m in a t e b e a m sa n d f r a m e s h a s b e e n o b t a in e d [ 1 - 4] . Th e s e i n v e s t i g a t i o n s c o n c e n t r a t e d o n t h e a n a ly s is o fs u p e r s t r u c tu r e s c o n t a i n i n g c r a c k e d m e m b e r s a s s u m i n g t h e s t r u c tu r e s t o m e e t t h e f o u n d a -t i o n s o r o th e r s u p p o r t i n g e l e m e n t s a t i d e a l s u p p o r t s a s t o t a l f i x a t i o n s a n d p u r e h in g e s . N o n eo f th e s e i n v e s t i g a t i o n s h a s c o n s id e r e d t h e s o i l - s t r u c tu re i n t e r a c t i o n e f f e c t o n t h e s t r u c tu r a lb e h a v i o u r o f c r a c k e d s t r u c tu r e s .

I n m a n y s t r u c tu r e s , c r a c k s o r f l aw s m a y i n it i a t e a n d p r o p a g a t e a t s u b s t r u c tu r e a n ds u p e r s t r u c t u r e e l e m e n t s d u e t o s e v e r e l o a d in g s , c o n s t r u c t i o n e r r o r s o r e n v i r o n m e n t a la t t a c k s . Th e r e f o r e , a c o m p le t e a n a ly s is o f b o t h s u p e r s t r u c tu r e a n d s u b s t r u c tu r e i s c e r -t a i n ly n e e d e d w h e n e v a lu a t i n g t h e s a f e ty o f s t r u c tu r e s c o n t a in i n g c r a c k s a t s u p e r s t r u c tu r ee l e m e n t s o r a t f o u n d a t i o n e l e m e n t s . I n f a c t , s t r u c t u r a l d a m a g e i n t e r a c t i o n m a y o c c u rd u e t o c r a c k in g a t b o th t y p e s o f e l e m e n t s . I n t h e p r e s e n t p a p e r , a f r a c tu r e m e c h a n i c sm o d e l h a s b e e n d e v e lo p e d t o a n a ly z e s k e l e t a l s t r u c tu r e s r e s t i n g o n e l a s t i c f o u n d a t i o n s a n dc o n t a in in g c r a c k s a t v a r i o u s l o c a t i o n s t a k in g i n to c o n s id e r a t i o n t h e s o i l s t r u c tu r e i n t e r a c t i o neffect .

2. Method of analysis2 .1 . S t i f f n e s s m a t r i c e s f o r u n c r a c k e d e l e m e n t sTh e s t if f ne s s m a t r i x o f a n u n c r a c k e d p l a n e f r a m e e l e m e n t ( s u p e r s t r u c tu r e e l e m e n t ) o f l e n g thL , c r o s s s e c t i o n a r e a A , m o m e n t o f i n e r t ia I a n d m a te r i a l m o d u lu s o f e la s t i ci t y E i s g iv e n b y


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82 M.H . EI-Haddad, O.M. Ramadan and A.R. Bazaraa(1) as follows [5]:

[K]0

- E AL

00

EA 0 0L

12EI 6EI0 L 3 L 2

6EI 4EIL 2 L0 0

- 12EI - 6EI

L 3 L 26EI 2EIL 2 L

- EA 0 0

L- 12EI 6EI

0 L 3 L 2-6EI 2EI0 L 2 L

EA 0 0L1 2 E I - 6EI

0 L3 L2-6EI 4EI0 L 2 L

(1)

whereas the s ti ffness mat r ix for un cracke d fo unda t ion e lements ( subs t ruc ture e lements) is theone der ived f rom the exact so lu t ion o f the d i f ferent ial equat ion of the e las tic line of a be amrest ing on inf inite num be r o f vert ical springs [4] an d [6] and is given by (2) as fol lows:

[K]

EA -EA0 0 0 0L L0 SQ1 SQ2 0 SQ4 SQ50 SQ2 SQ3 0 - SQ5 s o 6

- EA EA0 0 0 0L L0 804 -- SQ5 0 sO1 - SQ20 SQ5 SQ6 0 - S Q 2 SQ 3

(2)

where var ious e lements in the above mat r ix are def ined in Appendix I .2.2. Stiffness matrices fo r cracked elementsConsider the cracked sup ers t ruc ture and crac ked subs t ruc ture e lements shown in Figs. 1-aand 1-c respect ively. Both elements are divided into three segm ents as shown in Figs. 1-b and1-d. The f ir s t an d the th i rd segments g iven in both cases are the co mm on un crack ed e lementswith lengths, uL and (1 - u)L respect ively. The st if fness matr ices c orres pon ding to the f i rs tand the th i rd segments for the supers t ruc ture e lement can be obta in ed by replac ing the lengthL in (1) by uL and (1 - u)L respect ive ly . Whe reas those cor re spond ing to the f i rs t and thethird segments in the substructure elem ent can be obtaine d fro m (2) by replacing the length Lby uL an d (1 - u)L respect ively. The m iddle segmen t of both superstructure an d substructure


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Analysis of fra me s containing cracks an d resting on elastic founda tions(~4~, g l ) _(I,4 Z . 0l l

, . ~ ~ , ! . l , ,~ uc, .~ i ( " " ' ~ i ~'~ " ~ )(a) The forces and d isp lacements at the ends of the cracked f rame e lement .

3

I

Q (i)

~ t t . . . . . [ , c, - jJI L i"1 " "|

(b) Three e lements represent the cracked e lement

83

Hodo

,(N ~ e X

a ~ L ( I ' ~ 1 8 L

kI

(c) Basic degrees of f reedom for a cracked beam on e last ic foun dat io n.~2 Se~ent , I l l ~ lSe fment (2 l~Se l l~o nt 131~5

I USL ! zmr a , | l -u | "L :(d) Basic segments of a cracked beam on e last ic foundat ion.

Fig. 1. Representa tion of cracked superstructure and substructure elements.elements s imula tes the cracked sec t ion which connects the lef t and the r ight segments andi ts s t if fness matr ix has bee n deve loped base d on f rac ture mech anics techniques [2] . Thecracked sec t ion is s imula ted by an imagina ry co mbin ed spr ing wi th spr ing constan ts s imula t -ing the def orm at ions a t th is sec t ion. The s t if fness matr ix for th is spr ing can b e obta in ed us ing(3) as follows:

Spp 00 0

{Kc] = Smp 0-S~,~ 0- S,~, 0

0 0 0s~,,, - s~ m -sm,~

-s~m s,, s~o(3 )

The va r ious e lements of the abo ve matr ix are def ined in Append ix I I . I t should be not icedthat the e lements ly ing on the second row and second column are taken as zero s ince the


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84 M. H. EI-Haddad , O.M. Ram adan and A .R . B azaraaeffect of crack on shear deformation is neglected. The st i f fness matr ix given by (3) is val idfor both supers t ruc ture and subs t ruc ture e lements shown in Fig . 1. The s ti ffness mat r ix ofeach of the cracked e lements shown in Fig . 1 can n ow be assembled as an overa ll s ti ffnessmat r ix for i t s three segments and has the d imens ion of e leven by e leven. The sys tem ofnum ber ing o f the degrees of f reed om at the fo ur nodes of the e lements shown in Fig . 1 i staken such tha t the degrees of f reedo m at the two in terme dia te nodes have the las t num ber-ing. This is useful wh en con den sing three no des to obta in a matr ix o f size s ix by six. The threemat r ices cor respon ding to the three segments of each of the cracked e lements shown in Fig. 1are al l located in the overal l s t i f fness matr ix in the corresponding posi t ions referred to thechosen sys tem of number ing for the degrees of f reedom. Thus , the f ina l form of thesupers t ruc ture an d subs t ruc ture cra cked e lements mat r ix [S]c is g iven by the fo l lowing form[ASc] = B r (4)where, the sub matr ices [A], [B], and [C] are given in Appe ndi x II fo r both cases of superstruc-ture and subs t ruc ture e lements . I t should be noted tha t the two in termedia te nodes of thecracked e lements shown in Fig . 1 are f ic t i t ious because they are external ly unloaded.There fore, the st i ffness matr ix , [S]s relat ing the end forces and end displacem ents at theexter ior nodes o f these e lements can be obta ined by the cond ensat ion of the two in termedia tenodes in the fo l lowing form

[S]s06 = [AI - [B] [C1-1 [B] r (5)The st i f fness matr ix given by (5) is s imilar in dimensions to the st i f fness matr ix of thecom mo n unc racke d e lements g iven by (1)- (2) . So, th i s mat r ix can be d i rec t ly employed inthe st i f fness method of analysis .

3. Examples and discussionsA computer program has been developed to analyze skele ta l f rames res t ing on e las t icfounda t ions and subjec ted to var ious types of loading ac t ing a t the f rame nodes . Ho wev erin cases of appl ied d i s t r ibuted loads or concent ra ted loads not appl ied a t nodes , the loadvecto r is the resul t ing react ive forces at the en d joints of elements af ter intro duc ing f ixat ionsa t these jo in ts . F or c om mo n uncrac ked supers t ruc ture and subs t ruc ture e lements, th i s vec toris know n [5, 7]. On the o ther ha nd, a com puter su brout ine has been deve loped to ca lcula tethese forces for c racked me mbers us ing the e leven degrees-of - f reedom mod el sho wn inFig . 1. I t should be em phas ized tha t the present mode l is va lid for any nu mb er o f c rackedsections a t supers t ruc ture or a t subs t ruc ture e lements provided tha t c racks are assume d a tthe tens ion s ide of the e lement c ross sect ion . Cracks are assume d th roug h th ickness for a l lexamples cons idered.

The descr ibed method of analys i s was used to s tudy three types of f rames subjec ted todi f ferent loadings and cracked a t v ar ious loca t ions as shown in Fig . 2 . The mater ia l Yo ung 'smo dulu s, E, is assum ed to be 200 t cm -z unless otherwise indicated. Th e red istr ibut ion ofinternal forces and soi l pressure are ob taine d du e to crack d epth rat ios equal to 0.1, 0.2, 0.3,


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Analysis of fram es containing cracks and resting on elastic foundatio ns31 /m ,

I I | I l [ I I I l I I -I I I 1

" ( 3 0 1 1 2 0 ) c m4. O0)cm

I ( l O 0 . 5 0 ) m ' ~ ' ~ l ~ i

5 1 0 6 0 1 3 .6 0 m 5 = 0 6 0

85

FRAktEF , KO : 1.5 kg /cm 3

I 0 !

Ls_O:_So) __I i lO , ,40 ) eM

6 t / ~1 i ! i 1

/

(30 RQO) / /

( r iO=SO) |

. , o

?

I~ 0

i -

LI rnAhl l [ ~ ' ~= 3.6 ~/ m 3,

I [ =2SO t /m I

6t |Or 15! 15f lOt 6 f0 .51

0 .6

O i l

L, , h "2

3 . 0

3 0

3 0

3 .00 . ~

3.0 , , =._So_~+__3 :oo _.+ ~ _ ~ o ~LSO 4 .0 4. 0 4.0 LI)O ,

Fig. 2. Problems analysed.0 .4 and 0 .60. Co mpa r ison s betw een the orig inal bend ing mom ents and soil pressures for theuncracked f rames g iven as sol id l ines and those g iven by dot ted l ines corresponding tocracked f rames due to crack depth ra t io equal to 0 .60, a re g iven in Figs . 3-12 where valuescorresponding to the cracked cases are wr i t ten between brackets .

Figure 3 i l lus t ra tes the redis t r ibut ion of bendin g mo me nt and soil pressure due to twosymm etr ic through cracks located a t the bo t to m f ibres of foot ings a t the outs ide sec t ionsadjacent to the r ight and left columns of f rame F t . The crack causes d iscont inui ty in thefoot ing e las tic l ine , excess ive deform at ions , and ro ta t ions aro und the cracked locat ion. Asa resul t so i l pressure increased a t c rack locat ion and whi le bending moment decreased a tcracked sec t ions it increased a t uncrack ed foot ing sec tions. I t i s in terest ing to no t ice tha t theredis t r ibut ion of bendin g mo me nt i s negl ig ib le due to the fac t tha t the uncra cked f ramebehaves l ike a two-hinged f rame because the su bgrad e re la t ive s ti f fness i s low. Therefore , loss


6/22

86 M.H. El-Haddad, O.M. Ramadan and A.R. Bazaraa45.84 ~ / ~ 45.84(45.71) (45.71)

4~.84 45,84

: ?-:., .~..;~' , , , . , , ,0 . 8 6 x , , ~ ~ .609~0.797 Q797(0,9161 (~9 16 }

Fig. 3. Effect of cracks on bending moment (re.t) and soil pressure (kgcm- ~), Frame F~.

43194 45.84152".53) (5~86|/ / /

45.84 / //(52,551 / / / ~/ ; ~

~

~ ,.,, ,,,, ~~ : " / , , . ~~ 5.20 / / / 5 ~0 ~1~77 (4.~01 / 0 . ~ 0.68~ / (6.961 13.77~.9s ) /~ (o.~s~ " ~ . / (~o,ze~~ (0,965) I /Uncrockedo,ese "~ " = o~o . . . . . . . ~ o.8~/~/ ~ . ~ 10,465)(0.997) 0.797

( 0 , 9 5 6 )Fig. 4. Effect of cracks on bending moment (m,t) and soil pressure (kgcm-~), Frame F = .

45. 94139,88)

in footing stiffness due to crack presence does not much affect relative foundation subgradestiffness.

This frame is also analyzed assuming two cracks as shown in Fig. 4. The first is assumedat the bottom fibres of the right footing just to the right of the column while the second isassumed at the top fibres of the horizontal girder at its connection with the left column. Aredistribution in soil pressure and bending moment occurs due to crack presence leading toa reduction in bending moment values at cracked sections. A reduction in bending momentvalue at the uncracked right corner occurs due to crack presence at the right footing. Also,notice the increased values of bending moment at the uncracked sections of the footings.


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Analysis o f fram es containing cracks and resting on elastic foundations~ 31.57, (52.01)

87

[4 .t6 ~ / / I 51.57(15.67) (52.0N14.16(15.671

MOMENT ,/__UNCRACKED

16.47( i3.01 )

22.59 ] ~/ 2~ / 21.68 r5.21(25.61 ) (18.701 (5.6 9)

22.59(23.61)

0.15 "~',(o.o5)

S OIL P RE S S URE 2.70(5.021Fig. 5. Effect of cracks on be nding mo me nt (m. t) an d so il pressure (kg cm 2), Frame F 2.

Figures 5 , 6 show the redis t r ibut ion of bo th bend ing m om ent and so il pressure in f rameF 2 due to a top crack a t the r ight foot ing and a bo t to m c rack a t the le f t foot ing respect ive ly .The f ram e was analyz ed by Ting e t al . [6] assuming no cracks a nd the i r resul t s a re in comple teagreeme nt wi th the resul ts of the present mo del . I t should be ment io ned tha t in i t ia l ana lys isof th is f rame y ie lds some tens ion on the subgrad e . Th erefore , t r ia l and er ror p roced ure i sused to obta in the no tens ion so lu t ion by us ing supers t ruc ture e lement s t i f fness mat r ix fore lements in the tens ion zone . Due to c rack presence , excess ive d isp lacements take p lace a tthe crack loca t ions . Hence , bending moment decreases a t the c rack loca t ion due to loss ins t if fness and i t increases a t o ther loca t ions . Th e abov e resul ts sh ow tha t the e f fec t of c rackis more pronounced in the second case g iven in Fig . 6 . This may be due to the fac t tha tor ig ina l , uncracked, bending s t ress a t the second crack loca t ion i s h igher in th is casecom pare d w i th the f i rs t case g iven in Fig . 5 . Redis t r ibut ion of bending mo me nt and so ilpressure due to a c rack loca ted a t the upper le ft corner of the above f ram e i s g iven in Fig . 7 .I t shou ld be no ti ced t ha t r ed i s t r ibu t ion o f bend ing m ome n t occu r r ed s imu l t aneous ly i n bo thsubs t ruc ture and supers t ruc ture e lements due to c racking o f e i ther of these e lements , asshown in Figs . 5-7 .


8/22


9/22

Analysis of fram es containing cracks and resting on elastic foundations 89/'. // /

14.16(IO.26) // / j /

,:2: -> ...../

Yi//

//2 2 . 5 9123.2012 2 , 5 9(25 .20 )

B E N D I N GMOMENT

/fl

Lzt z

/ / (23.77123.771

3 J 157(32.77) i

31.57(52 .77 )

7 /

- - U no ro ke d= r f ' =o. 6

16,47( I 8 .43)

5.531

0.15(0.19)

S O I L P R E g S U ~ I E 2 . 7 0( '2 .77 )

Fig. 7. Effect of cracks on bending moment (m.t) and soil pressure (kgcm-2), Frame F 2.

stresses are induced at uncracked sections due to the redistribution of bending moments. Inaddition to these two reasons failure may occur in soil at locations where soil pressure isincreased due to cracking. Th erefore, proper repair of cracked structures shou ld involve asimultaneous check of stresses and soil pressures at various locations.

4. ConclusionsBased on the stiffness matrix method and fracture mechanics techniques, a mathematicalmodel has been developed to analyze plane frames resting on elastic foundations andcontaining cracks at superstructure and/or substructure elements. This model simulates thecrack size and location, type of loadings, soil structure interaction effect and relativestructure subgrade stiffness.

Based on the results of analysis of three frames cracked at various locations, severalconc lusio ns o f impor tance to the practical designers and researchers can be drawn as follows:


10/22

90 M.H. El-Haddad, O.M. Ramadan and A.R. BazaraaA 3~.57(23 .561

14.16( 1 8 . 7 9 )

14.16 ///(16.79) i / / /

31.57( 2 3 . 5 6 );'-7

///

22 .59( 2 7 . 6 7 )

BENDINGM O M E N T

/ / 2 2 5 9( 2 7 . 6 7 1

__ UNCRCKED- - - , ' n " = 0 . 6 0.f

16.47 ,281

21.68 ~ 5.21( 1 9 . 9 8 ) ( 5 . 7 0 )

v

( 0 .0:9 ) xx,x

2 . 7 0S O I L P R E B S U R E ( 3 . 0 3 )Fig. 8. Effect of cracks on bending m ome nt (m.t) and soi l pressure (kgc m 2), Frame F 2.

(1) A d is t r ibut ion of bending mo me nts occu rs in f rame mem bers due to c rack presenceleading to a decrease in bending moment a t c racked sec t ions and an increase in bendingmomen t a t unc racked sec t i ons .

(2) Red is t r ibut ion o f so il pressure occurs under f rame fou ndat io ns du e to c rack presenceat var ious loca t ions .

(3) Fo und at io n cracks can grea t ly a f fec t the redis t r ibut ion o f in terna l forces a long mostof the s t ruc ture mem bers wh ereas supers t ruc ture c racks hav e only loca l e f fec t on mem bersclose to the cracked sect ion.

(4) The ef fec t of c racking o n the var ia t ion of in terna l forces and so il pressure i s mo repro nou nce d as the crack depth ra t io i s increased and as the r ig id ity of the founda t ion i sincreased.

(5) Prop er repa i r o f c racked s t ruc tures should involve a s imul taneous check of st resses a tvar ious sec t ions and so i l pressures under var ious foundat ion e lements .


11/22

Analysis of frames containing cracks and resting on elastic foundations5.8552 2 3,02 16.51t

(5,261 (3.03) 0,89 238(0,22) [ t ,541

\~'~, ~.oo ~ ~"%7,~ .~ "" ~ ' .~ ,Z/. _ ~ o . _ ~ . ~ o . ~ , ~g/~ ~ ~ : /I ' ~ ' , , . ~~ 5 , 3 9 ~84] ~ 8V " '" ,~(3.7~ (1.~)

t xx' ON 9 ~ lx / ~ ~C ~D .~/ ---"n ".o.~o g.~~4.S2 ~4.7~ 5.00 I . ~~ ( 4 ~ 9 )~., ' ~ ~


12/22

92 M,H. EI-Haddad, O.M. Ramadan and A.R. Bazaraa5 . 2 2 5 . 8 5, =, ,~ ~ 5.0;~ ( 5.70 1

5 . 5 4 ~ U P~c~'~cked1 5 .5 5 ) ----- ~"~"=0.6

Fig. 1I. Effect of cracks on bending moment. (m.t), F rame F~. Values written for horizontal members.

( 2 2 . 0 5 }

I0.87J~*~'~- ~-10.9950 . 9 5 9t0 .8 6 4 { ' 0 .9 5 2 1 (0 .9 ~1 )

I I

, l~gJ ~, .5( 4 1 .4 6 )

, . ,~~~!,.~{ 1,184 } ( I, 3~,~| ( ,3~6}

-;._/i.......,~,.~4~' ,~,~,,,

( 4 2 . T ~ 1 U n c r ~ c ~ e d

. . . . . " q ~ ' = 0 , 6 0

I I

>21.~ (I ,0~9~ t . t ~ 5 1 { LO~4Il h ~ O 0 ~( I ,~4 t ) h ~T ~ i h 3 ~)~ 1 . 3 ~ 3 )Fig. 12. Effect of cracks on bending moment (re.t) and soil pressure (kgcm-2), Frame t b.


13/22

Analysis o f fra me s containing cracks and resting on elastic foun datio nsAppendix IUncracked substructure element matrixElements of matrix [K] given by (2)

S Q I = E [ f l 3 ( Z t Z 2 + 4 . Z 3 2 4 ) / ( Z 3 " Z 3 - - Z 2 2 4 )

SQ2 = El f l2 (Z , . Z 3 + 4" Z4" Z4) / (Z3 " Z3 - Z2" Z4)

S Q 3 = E 1 f l ( Z 2 Z 3 - - Z l Z 4 ) / ( Z 3 Z 3 - - Z 2 Z 4 )

SQ4 = 4E I f13 . Z2 _ 4 f t . Z4" SQ2 - z I SQ~

S Q 5 = - 4 E 1 1 3 2 . Z 3 - Z 1 .S Q 2 + Z 2 " S Q I/ 1 3

S Q 6 = - 4 E I 1 3 " Z 4 - Z , . S Q 3 + Z 2 .S Q 2 / 1 3

13 :

And,Z 1 = COS ( i lL) cosh (ilL)

Z 2 = 0.5 [sin (ilL) cos h (ilL) + cos (ilL) sinh (ilL)]Z 3 = 0.5 [sin (13L) sin h (13L)]Z4 = 0.25 [sin (13L) co sh (ilL) - cos (13L) sinh (ilL)]B = width of the sect ion.k0 = Soi l mod ulus of subgrade reaction.

93

Appendix IICracked section matrixThe compliance concept can be employed to obtain the relat ion between forces and displacements for crackedmembers in the form:

G = ( 1 - v 2 ) . K ~ . g ~ j = P ~ ' P j 0 20E 2 OA (n-l)

Where 2~j is the displacement in direct ion of the i th degree of freedom due to uni t force appl ied in direct ion of thej t h degree of freedom, Pi , ~ are the loads appl ied in direct ion of the i th degree and j th degrees of freedomrespectively, K~i and K~j are the stress intensity factors f or mo de I "o pen ing m od e" corr espon ding to loads ap pliedin direct ion of the i th, j th degrees of freedom, respect ively. A is the area of cracked surface and G is the energyrelease rate.


14/22

94 M.H. El-Haddad, O.M. Ramadan and A.R. BazaraaThe comp liance of an un cracked element , 2o is constant in the elas tic s tage whereas that of a cracked m embe r

is variable and increases as the crack s ize increases. The co mpliance of a cracked mem ber is expressed as the sumof the uncracked mem ber co mpliance, 2o and the increase in compliance, A2, due to crack presence. The increasein compliance in direction of the i th degreee of freedoms due to uni t appl ied load in direct ion of the j t h degreeof freedom is obtained by integrat ing ( l I-1) as fol lows:

A 2 u - 2( 1 - v 2 ) ( K I j ~ ( K I s ' ]"ff f A \ p i j \ p j j d A (II-2)

Through crack compliance relationsSolut ions for the s tress intens i ty factors for the rectang ular sect ion subjected to axial tens ion and bending m ome ntand containing through cracks , shown in Fig. 1-a are given below [2].

Kip = (P ~-~/Bd)(1.99 - 0.417 + 18.7072 - 38.48 73 + 53.8474) (II- 3)K IM = (6M xf~/Bd 2) (1.99 -- 2.477 + 12.9772 - 23.1773 + 24.8074) (I1-4)

where B and d are the width and depth of the membe r cross sect ion respectively, P and M are appl ied axial tens ionor bending moment respect ively, a is the crack s ize and 7 is the crack depth rat io.7 = (a/d). Subst i tut ing for K~eand Kil t given above into (II-2) , the increases in compliance due to axial tens ionand bending moment are obtained in the form:

2A2pp = ~ (1.9801q 2 -- 0.543973 + 18.648574 -- 33.696975 + 99.2611q 6 -- 211.90177 + 436.837678

- 460.47779 + 289.98237 l) (II-5)A2,,m 72EBd 2 (1"980172 - 3.276973 + 14.430474 - 31.257775 + 63.564176 - 103.363177 + 147.5201r/8

- 127.692479 + 61.5047 I) (II-6)

A/~pm = A .~mp = ; 2 d (1.9801q 2 -- 1.9173 + 16.00974 -- 34 .83875 + 83.93376 -- 153.649077 + 256.722078

-- 244.668q 9 + 133.5487 ~) (II-7)

Part through cracks compliance relationsThe assumption that cracks ini t ia te and propagate with ful l member width is convenient in relat ively deepmembers where depth is much larger than width but considering foundat ion elements with depth smaller thanwidth, the part - thro ugh cracks are considered. Mo st pract ical cracks take the shape of an el l ipse or semi-ell ipse.In this inves t igation, semi-el l ipt ical cracks are considered assuming the cracks ini t ia te and propag ate keeping thesame semi-axes ratio. The stress intensity factor of a semi-elliptical crack, s hown in Fig. II-a and Fig. II-b variesalong i ts c i rcumference according to

KO = [sin 2 0 + (ale) 2" cos 2 0] 1/4 gm a x (11-8)where KO is the s t ress intens ity factor at a locat ion m aking an angle, 0, with the major axis as shown in Fig. II-a ,Km,x is the maxim um value of the s tress intens ity factor that occurs at 0 = ~/2, a and c are the minor a nd m ajor


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d

Analysis o fra me s containing cracks and resting on elastic Jbundations8~ . . . . . . . . . . . ~k,,//////////./ ,. ,, ,:/..:.; ..,/,/,,///.2,1 ]"

i c ~ c ~~&c H-~. Part through ~emi-elliptical crack,~:,- , ,8 , , , , , , I

~ * d ! ' . I I ~- . I ~ IIIl'~J~'~ IIII I

~ / ~ ....~ ~F ig . I I - b . Element of area, dA.

95

semi-axes of the crack. Rice and Levy [8] have related this max imu m factor , K~,~, to the s t ress intens i ty factor fo rthrough cracks us ing sui table mult ipl iers that depend on the rat ios (a/d) and (2c/d) given below:

K ~ e = e~K p, K ~ = e~ t K M (II-9)where K,. .~ , K~..~ are the max imu m stress intensi ty factors for a semi-el lipt ical crack at a certain crack s ize inaxial tens ion and bending moment respect ively, Kp and KM are the s t ress intens i ty factors for a through crackat the sam e crac k size in axial tension and bend ing m om en t respectively and c~p and ~,. are mu ltipliers given byRice. Several values for the semi-axes rat io, a = c/ a were assumed an d both mult ipl iers ap and a~ were obtainedat some values for the crack depth rat io. To obtain a con t inuous funct ion for K ~ , values of~p and a . , were plot tedagains t q and leas t square l ines were obtained.The se lines ~vere con struc ted at differen t values of(c/a). Based on these l ines , the fol lowing solut ions are at ta ined[41:

K m ~ ' p = (A1 + Blr l ) (P~ '~/Bd)(1 .99 - - 0.41t/ + 18.70t/2 -- 38.48~/~ + 53.84~74) 0~4o)Km~~ - _~1 = (A 2 + B2t l ) (6Mx/~/Ba2)(1 .99 - 2.47~/ + 12.97,12 - 23.17tl 3 + 24.80t/~) (~-11)

Wher e A 1, B1, A2 and B2 are no n-dimen sional constants given in Table II-1.Substitu ting in to (II-8), using the so lution f or the stress intensity facto rs given by (II-10) and (II-11 ), increasesin compliance can be obtained via (II-2) as fol lows:

2 A-- E~2d f" (sinz 0 + ~ cosz O)~/2 (A , + Bltl)2rlY~~ dA2~ (II-12)~0


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96 M.H. El-Haddad, O.M. Ramadan and A.R. BazaraaTable I I -1 . Cons t an t s A 1 , B1 , A2 an d B2

C o n s t a n t A 1 B1 A2 B2c/a1 1 - - 1 . 3 1 - - 1 . 4 32 1 - 1.3 1 -- 1.403 1 - 1.25 1 -- 1.304 1 -- 1.10 1 - 1.205 1 --0 .98 1 - 1.106 1 0.0 1 0.00

A ~rnm 72 f A (si n 2 0 + ~2 co s 2 0)112 (A 2 ~_ Bzr/)2r/Ym dAE B Z d 3 do

12 f] (sin 2 0 + c~ cos 2 O) I/2 (A 1 + Blr/) (A 2 + B2r /)r/ YpYm dA.A2pm = A2mp - E B 2 d 3

where

Ym=dA =

(11-13)

( I1-14)

1.99 - 0 . 4 1 1 / + 1 8 . 7 0 r / 2 - 3 8 . 4 8 r / 3 + 53.85/'/4

1.99 - 2.47r/ + 12.97r/2 - 23.17r/3 + 24.80 q 4

(II-15)(II-16)

r dr dO

and r , q~ a r e t he po l a r coo rd i na t e sys t em. S i nce t he semi -axes r a t i o i s a s sumed cons t a n t d ur i ng t he c r ackpropag a t i on , t he above i n t egra t i ons r educe t o:

A2pp - 2 F dEB 2 f2 r/Z(A t + Biq)2Yp2" dr/ (II-17)

72FA~mm EB 2 d Jo r/2(A2 + B2r/)2 Ymz"dr/ (I1-18)_ 12FfA~Lpm = A/ ~mp E B 2Jo r/2(Al + BIr /)(A 2 + B2r/) YpY,,, dr/ (II-19)

where the m odif ica t ion factor F dep ends only on the semi-axes ra t io, e . Subst i tu t ing for Yp and Y,~ f rom (I I -15),( I I-16) i n t o ( II -17) , ( I I -18) and ( I I-19) t he above i n t egra ls can be ca r r i ed ou t and hence t he i nc r ease i n bo t h d i r ec tand i nd i r ec t compl i ances fo r s emi -e l li p t ica l c r acks can be ob t a i ned . Va l ues fo r t he modi f i ca t i on f ac t o r F a r e g i veni n T ab l e I I -2 a t va r i ous e/ a values .

F l ex i b il i t y m a t r i x f or t he c racked m em ber

Based on f r ac t u r e m echan i cs t echn i que , t he c r acked segment i s r ep l aced by an i m agi na ry spr i ng [2] w i t h sp r i ngcons t an t s s i mul a t i ng t he c r acked segment . T wo degrees o f f r eedom cor r espon di ng t o ax i a l f o r ce and b end i ng

Table 11-2. M odi f i ca t i on f ac t o r Fc/a 1 2 3 4 5 6F 3.14159 4.31303 5.05725 5.60241 6.03222 6.38679


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A n a l y s i s o f f r a m e s c o n t a i n in g c r a c k s a n d r e s t i n g o n e la s t ic f o u n d a t i o n s 9 7mom ent are assum ed a t both ends of the combin ed spring as shown in Fig. I I -c. I f the forces corresponding tothe degrees of f reedom at end I are taken as the redu ndan t forces as shown in Fig. I I -d, the assembled f lexibi li tymatr ix wi l l be in the form:

r 1 [ 1n f~2 A2.,,, A2,,,,, (II-20)gxx = L S~, s~2 = Axo. A.~o.The termf~l i s by defini t ion the increase in axia l deformat ion a t end I due to a uni t axia l force a t the same end,i .e., fH = A2pp. Similarly, othe r terms are obta ined as given in (II-20).Stiffness ma trix for the cracked sectionBy definition the stiffness is the inverse of the flexibili ty. Therefore, the inverse of the above flexibili ty matrix willyie ld the st i f fness matr ix corresponding to end I in the form:

S . = S,.p S,..,

where ,

g e m = -A2.,. ID&, , = - A . ~ . . , / D

D = A2 pp 'A 2,, m -- A.~.pm" A}.mp @Fig. II-c. Degrees of f reedom at the combine d spring ends.@

,4- ( " Fig. II-d. The redundant forces for the spr ing.

Fig. H. Represe nta t ion o f cracked sect ion.


18/22

9 8 M . H . E l - H a d d a d , O . M . Ramadan and A . R . BazaraaU s i n g e q u i l i b r i u m a n d s y m m e t r y c o n d i t i o n s o f t h e c r a c k e d s e g m e n t t h e f i n al s ti f fn e s s m a t r i x f o r t h e c r a c k e dsegment i s g i ven by :

s~m - s . -S.,~Smm -S ~ --Smm

--S~m S.~ S~m--Smo Sm~ Smm

Sin.[ /~] =- s~- s ~

C r a c k e d e l e m e n t s t i f f n e s s m a t r i c e s

a) S u b s t r u c t ur e e l e m e n tM a t r i x [A]

[A] = A ( I , J )

I = 1 , 6 & J = 1 , 6 , w = 1 - U

Al l e l ement s i n ma t r i x [A] a r e equa l t o ze ro excep t t he fo l l owi ng :

A(1, 1)A(2, 2)A(2, 3)A(3, 2)A(3, 3)A(4, 4)A(5, 5)

= E A / U L

= (SQO ~= (SQ~) l

= A(2, 3)

= (SQ3) 1

= E A / w L

= ( S Q 1 ) r

A ( 5 , 6 ) = - ( S Q 2 ) r

A ( 6 , 5 ) = - ( S Q ~ ) r

A(6, 6) = ( S Q 3 ) r

M a t r i x [B]

[B] = B(I , J)

I = 1 , 6 & J = 1 , 5


19/22

Analysis of fra me s containing cracks and resting on elastic foundation sA l l e l e m e n t s i n m a t r i x [ B] a r e e q u a l t o z e r o e x c e p t t h e f o l l o w i n g :

B(1, 1) = - E A / U LB( 2 , 2 ) = ( SQ 4 ) '

B(2 , 3 ) = (SQ5B( 3 , 2 ) = - ( SQ s ) zB(3 , 3 ) = (SQ6) ~

B(4 , 4 ) = - E A / w LB(5, 2) = (SQ4) ~B (5 , 5) = - ( so s ) ~B(6 , 2 ) = (SQs)rB(6 , 5) = (SQ6) r

Matrix [C ][c ] = c (L J)

I = 1 ,5 & J = 1 , 56 (1 , 1 ) = EA/uL + SppC( 1 ,3 ) = C(3 , 1 ) = Sp..C( 1 , 4 ) = C( 4 , 1 ) = - S : ;C( 1 , 5 ) = C( 5 , 1 ) = -Sp~C ( 2 , 2 ) = so~ + SQ~C ( 2 , 3 ) = C ( 3 , 2 ) = - S Q ~C ( 2 , 5 ) = C(5 ,2) = SQ~6(3, 3) = SQ~ + &~C( 3 , 4 ) = C( 4 , 3 ) = - Se ~

6(3 , 5) = 6(5 , 3) = --atom

C(4 , 4 ) = EA/wL + Spp

9 9


20/22


21/22

Analysis o f fram es containing crack,; and resting on elastic foundationsB(2, 3) = 6EI/(uL) 2B(3, 2) = - 6EI/(uL) 2B(5, 2) = -- 12EI/(wL) ~B(6, 2) = 6EI/(wL) ~B(6, 5) = 2EI /w L

Matr ix [C ][c] = c(~, J)I = 1, 5 & J = 1, 5

Al l e lements in matr ix [C] are equal to zero except the fol lowing:

C(1, 1) = EA/uL + SppC(2, 2) = 12EI/(uL) ~ + 12EI/(wL) 3C(3, 3) = 4E I/uL + S,,, ,C(4, 4) = 4EA/wL + Spp

C(5, 5) = 4EI/wL + SmmC(1 ,3) = C(3 ,1 ) = S p , ,

C( 1,4 ) = C(4, 1) = - S p ,c(1 , s ) = c(5 , 1) = -s ~

C(2, 3) = C(3, 2) = -6 EI / ( uL ) zC(2, 5) = C(5, 2) = 6EI/(wL) ~C(3, 4) = C(4, 3) = -Sp,~

c (3 , 5 ) = c (5 , 3 ) = -& . ,

C ( 4 , 5 ) = C ( 5 , 4 ) = Se,~

101

R e f e r e n c e s1. W . At t i a , "T hre e Di m ens i ona l Ana l ys i s f o r C racked S t ruc t u res" , M .S c . t hesi s , Ca i ro Uni ve r s i t y (1986) .2 . M .H. E1 Had dad , M .M . E1 Bahey and S . S aman , Res Meehanica 25 (1988) 371-386.3 . S .A . Hosn i , M .H. E1 Ha dd ad and A .R . Baza raa , Scientific Engineering Bulletin, F acu l t y o f E ngi nee r i ng ,

Ca i ro Uni ve r s i t y 4 (1988) 143-162 .


22/22

102 M.H. El-Haddad, O.M. Ramadan and A.R. Bazaraa4. O.M. Ramada n, "Analys is of Cracked Structures Rest ing on Elas tic Found at ion ", M.Sc. thes is, Cairo

Univ ersity (1987).5. C.K. Wang, Intermediate Structural Analysis, McGraw-Hi l l Book Company, New York (1983) .6. B.Y. Ting and E.F. Mockry, Journal of Structural Division, ASCE 110 (1984) 2324-2339, Paper 19229.7. R.J . Roark and W.C. Young, Formulas fo r Stress an d Strain, McGraw-Hi l l In te rna t iona l Book Company,London (1975).8. J .R. Rice and N. Levy, Journal of Applied Mechanics 39 (1972) 185-194.

R6sum6. On a introdui t dans la m6thode des matrices de r igidi t6 un mod61e de m6canique de rupture en vued'analyser des charpentes reposant sur des fondations 61astiques et comportant des fissures dans les 61~ments desupers tructure et de substructure.

Des m atrices de rigidit6 relatives ~i des 616ments de su bstructu re et de su perstruc ture fissur6s ont 6t6 d6velopp6esen se basant sur les techniques de la m6canique de la rupture et sur la m6thodologie de la matrice de r igidi t&

Le mod 6le actuel a 6t6 appliqu6 fi l '6tude des effets, su r le com po rtem ent de charp entes fissur~es, de la taille etde la posi t ion d 'u ne f issure, du type de sol l ic i ta tion, du m odule caract6ris tique d u sol, de la r igidi t6 de la fondat ionet de la g~ombtrie des charpentes , e t ceen tenant compte de l ' interact ion entre le sol e t la charpente.

analysis of frames containing cracks and resting

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