Rilem TC 162-TDF PDF

 

Materials and Structures/Mat6riaux et Constructions Vol. 35 June 200 2 pp 2 6 2 2 7 8

R I L E M T C 1 6 2 -T -T D F : T E S T A N D D E S I G N M E T H O D S F O R S TE T E EL E L F IB IB R E R E IN IN F O R C E D C O N C R E T E

De s i gn of s te e l fi bre re inforc inforc e d c onc re te u s i ng the ~-w me thod: pri nc ipl ipl e s a nd a pp l ic ic a ti ons

The text presented presented hereaf hereafter ter s a drafi or general general cons conside iderat ration ion.. Comm ents should be sent sent to the T C Chairlady: Prof. dr. dr. ir. Lu cie Vandewalle, K. U . Leuven, D e p a r t m e n t B u r g e r l ijij k e B o u w k u n d e , K a s t e e lp lp a r k A r e n b e r g 4 0 , 3 0 0 1 H e v e r le le e , B e l g i u m . F a x : 3 2 1 6 3 2 1 9 7 6 ; e - m a i l: l: lucie.vandewalle@bwk, uleuven.ac,be, uleuven.ac, be, by 31 October October22 0 0 2 .

). N em egeer, Belgium; Me mbe rs: L. Balazs, Hung Hungar ary*; y*; ~dt Poland;M. Poland; M. Cfiswe ll USA ; F. Denafi6, Suiss Suisse; e; M. Di P. Gro th, Sweden; Sweden;V. V. Haiisler Germ any; any;F. F. Katsaragakis, Finland; B. Massicotte, Canada; S. Mindess, Canada; Schumac her the Netherlands; B. S chniitgen, Germ any; a the Netherlands R Sw am y UK P ~Fat ~Fatnal nalll USA ; lds

T A B LE LE O F C O N T E N T S 2

3

4

2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3

4.4

Scope Background M e c h a n i c s o f c ra ra c k f o r m a t i o n a n d p r o p a g a t i o n Fracture mechanics for concrete Fracture mech anics for SFR C S t res res s-cr s-crack ack op eni n g rel at i ons hi ps for for des i gn General Mul t i -l i near rel at i ons hi p B i -l i nea r relat relat i ons hi p D r o p - c o n s t a n t r e l a titi o n s h ip ip F r e e - f o r m r e l at at io io n s h i p Ch aracte ristic values and safet safetyy factors factors E x p e r i m e n t a l d e t e r m i n a t i o n a n d v e r if i f i c a ti ti o n Cro ss section al analysis analysis for flexu re an d axial forces General - non-linear hinge G e o m e t r i c a l a s s u m p t io io n s A nal ysi ysi s for rect angul ar s ect ions ions S i m p l i f ie ie d a p p r o a c h b y P e d e r s e n S i m p l i f ie ie d a p p r o a c h b y C a s a n o v a E x p l i c it it f o r m u l a t i o n b y O l e s e n A p p l i c a t i o n o f t h e n o n - l i n e a r h i n g e i n s t ru ru c t u r a l analysis C r o s s s e c t io io n w i t h c o n v e n t i o n a l r e i n f o r c e m e n t C rac k s paci ng R e i n f o r c e m e n t s t ra ra in in s i n t h e c r a c k e d r e g i o n

A verage curvat ure 4.5 Limit states Service ability lim it state U l t i m at e l i m i t st st ate ate 1359 13 59-5 -599 997/ 7/02 02 9 RILEM

5 6 7

B e a m a n al al ys ys is is : l o a d - d e f l e c t i o n b e h a v i o r She ar capacit capacity: y: ultim ate lim it stat statee C r a c k w i d t h s i n s l a b s o n g r a d e u n d e r r e s t ra ra i n e d s h r in in k a g e a n d t e m p e r a t u re re m o v e m e n t 8 C o n c l u s i o n s a n d d i r e c t io io n s f o r f u t u r e w o r k Bibliography Appendix

1. SCOP E T h e d e s i g n p r in i n c i p le l e s d e s c r i b e d i n t h e f o l l o w i n g ar ar e b a s e d o n t h e f r a c tu tu r e m e c h a n i c a l a p p ro ro a c h k n o w n a s th th e fictitious crack model, which relies on the so-called s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p O w W as t he bas i c mat eri al i nput . The design method is applicable for Steel Fibre R e i n f o r c e d C o n c r e t e s ( S F R C ) t h a t e x h i b i t t e n s i o n s o f te te n ing behaviour. The method can also be used for other Fibre Reinforced Cementitious Composites (FRCC) t hat exhi bi t t ens i on s oft eni ng behavi our as w el l as pl ai n c o n c r e t e ( w h i c h i s a ss s s u m e d a lw l w a y s to to e x h i b i t t e n s i o n s o f t e n in in g b e h a v i o u r ) . T h e f o l l o w i n g c as as e s a re re c o v e r e d i n t h is is d o c u m e n t : 9 C r o s s s e c t i o n s u b j e c t t o c o m b i n a t i o n o f ax ax ia ia l f o r c e , b e n d i n g m o m e n t a n d s h e a r f o rc rc e . R e i n f o r c e m e n t c an an b e a c o m b i n a t i o n o f c o n v e n t i o n a l r ee- b a r s a n d f ib ib r es es . 9 S l ab on g rade s ubj ect ed t o s hri nkage.

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TC 162 TDF

mechan ics of metals. The first significant significant attempts to develop a non-linear frac fracture ture mechanics framework for concrete were taken in the nineteen sixties after it had been realized that linear fracture mechanics could not be applied. Excellent text books are now available on the subject [13, 31, 8]. Co n cep tu ally , a cr ack p r o p ag atin g in co n cr ete is m o d e l l e d b y a z o n e o f d if if f u se se m i c r o c r a c k i n g - t h e process zone, and a localized crack. The localized crack can be divided into a part where aggregate interlock is present, and a true traction free crack. A remarkably simplee description of crack formation in plain concrete simpl was suggested by [12]. [12]. H illerborg suggested the so-called F ictitio u s Cr ack M o d el ( F CM ) w h ich o r ig in ally w as intend ed for use in comb ination with FEM . How ever, as it will be show n here the approach can easil easilyy be adopte d in oth er num erical and analytical analytical models. The ficti fictitious tious crack crack model relies relies on a num ber of simple assumptions: 1 . O n ly b r id g in g tr actio n n o r m al to th e f r actu r e plane is considered. 2 . P r o ces s zo n e 1, lo calized cr ack w ith ag g r eg ate interlock a nd localized stre stress ss free free crack can be mod elle d by a single crack plane. 3. T he proce process ss zone, together with the part o f the local local--

2. BACKGROUND 2.1 Mechanics of crack formation and propa gation Wh en unreinforced concrete fails fails in uniaxial tension the failu failure re is governed by the forma tion o f a single single crack. crack. Wh en a crack crack is formed in fibre reinforced concrete, concrete, the fibres will typically typically st stay ay unbrok en. Th e fibres crosscrossing a crack will resist further crack opening and impose what is called crack closing or crack bridging effect on the crack surfaces. Different failure modes can result, depend ing on the eff effect ectiven iveness ess of the fib fibres res in providing crack bridging, see Fig. 1. If the fibres break or are pull ed out during crack initiation, initiation, or i f the fibres fibres cannot carry more load after after the formation of the first first throu gh crack, then the first cracking strength is the ultimate strength and further deform ation is governed by the o pening of a single crack and fibres pullin g out an d/or breaki ng along the edges of the crack, see Fig. Fig. l(a). Th is behav iour is alss o k n o w as ten s io n s o f ten ing al ing b eh av io u r. r. I f - o n th e othe r hand - the fibres are are able to sustain mor e load after th e f o r m atio n o f th e f irs irs t cr ack , m o r e cr ack ack s w ill ill b e f o r m ed an d w h at is k n o w n as m u ltip le cr ack in g , s ee Fig. l(b). This behaviour is also known as strain (or pseudo-strain) hardening behaviour. In the present text only materials exhibiting the first type of behaviour will be dealt with.

ized crack whe re aggr aggregat egatee in terlock is present is referred referred to as the fictitious crack. crack. Th e m echanical behaviour of the fictitious crack is characterized characterized by the stre stress-c ss-crack rack open ing relation rela tionship ship,, %(w), whe re o w is the traction applied to th e crack surface surface as a fun ction o f crack openin g w. 4. Stress-singularities2 Stress-singularities 2 can be disregarded. As soon as the largest principal stress reaches the tensile strength a fictitious crack is formed. 5. The length of the fictitious crack crack cannot be assumed to b e s m all co m p ar ed w ith a typical structural dimensio n. In Fig. 2, a real crack in plain concrete is outlined to g eth er w ith th e F CM m o d el of the same crack. crack. F r o m a m o d elling elling p o in t o f view the second assumption m ak es th e ap p licatio n o f th e Fig.. 1 - The principle o f sing Fig single le and multiple cracking. T he specimens are loaded in uniaxial tenFCM in FEM formulations sion and the schematic load load versus deforma tion, P 8), relationship is shown to gether w ith the particularly simple since the cracking pattern. T he left hand side a) shows sin single gle cracking or tension softening) while the crack can be modelled with right hand side shows shows multiple cracking or strain hardening). s o - called in ter f ace elem en ts

2.2 Fracture mechanics for concrete Fracture mecha nics deals deals wit h th e mec hanics of crack formation and propagation in mate materia rials. ls. When struct structures ures are made from softening materials materials,, crack formation often governs the structural behaviour. Fracture mechanics of concrete is a relatively young scientif scie ntific ic and engineering field compa red e g to fracture

i) In the original original ormulation of the FC M, the proces processs zone and the ictitiou s crack are syno nyms ; here however two different mechanisms are associated associated with the ficti tiou s crack: micro-cracking in the process zon e and aggregate interlock along with fibre brid bridging ging - in the case of tension softening softening SF R C ). 2) According to linear elastic analysis of cracks infinitely large stresses - stress singularit ies - alway s exi st at sharp crack crack tips. I n line ar elasti elasticc fract ure mechanics these singularities are characterized through so-called stress intensity factors which depend only on geometry and loading. loading. Crack propagati propagation on is assumed to take place when the stress intensit y a ctor reaches a critical value.

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Materials Materi als and Structures/M at~riaux et Constructions Constructions Vol. 35 June 2002

2 3 Fracture mechanics for SFRC

Th e stres stress-cra s-crack ck ope ning relationship lends itself in a very natural way to the description of fractu fracture re o f short fibre reinforced materials with SFRC behaviour. In fact the fictitious crack approach was suggested by Hille rborg for use in the description of formation of cracks cracks in fibre reinforced reinforc ed concrete [11 [11]] introducing an approac approachh where the stress-crack opening relationship now describes the

F ig . 2 - O u tlin e o f a c onc r e t e c r ac k and the essential features: the pr oc e s s z one aggr e gat e in te r lo c k a n d tr a c tio n f r e e c r a ck ck to g e th e r wit h the FCM . After [13] [13]). ).

containing information about the stress-crack opening relationship Ow(W . The third assumption means that the basic fracture property is the stress-crack opening relation Ow(W . It is usually assumed that this function is a monotonically decreasing function, indic ating softening behaviour. It is furthermo re assumed that: that: wheref is the tensile strength; Additionally, a characteristi is ticc crack open ing wc w c is defin ed by:

Ow W )--0

2)

see Fig. Fig. 2. Assum ing that the shape o f the stress stress-crack -crack opening relationship is more or less independent of material type, the stress-crack opening relation can be defined in terms of the area under the curve and since no energy dissipation is assumed to take place in the crack tip, this area can be equated with the fracture energy, GF:

GF=[.oOOw(w)dw

stresses carried across of a tensile crack in the com pos ite m aterby ial fibres as f unction the cr ack opening. Later this this approach has been taken by nu mero us authors in attem pts to describe the crack bridging ability of fibres fibres - the so-called fibre bridging - in different brittle matrix composite systems, [4, 15-18]. The fictitious crack in SFRC materials now represents the process zone, aggregate interlo ck as well as fibre bridging, see Fig. 3. Th oug h similar similar in in many aspec aspects ts,, in others the FC M approachh for crack initiation, approac initiation, propagation and opening in SFRC differs significantly from the FCM approach to concrete fracture. Since the fibre bridging is closely r elated to the f ibr es debonding and pulling out, and since the fibres often have a length that is not small co mpared to crack openings accepted in real structures, from a practical practical poi nt o f view, the para meter w c is not relevant. Thi s impli es that in analysis of SF RC materials, G F defined by Equation (3) loses its significance from a pr actical actical,, s tr uctur al poin t of view . F ur ther m or e, the stress-crack opening curve shape depends on the type and am ount of fibr fibr e us ed, ed, and the s hape of the the cur ve inf luences the s tr uctur al behaviour . T hus , the actual variation varia tion of the func tion ( w ) in the range o f acceptable crack openin gs - e.g. 0- 1. 1. 5 m m - becom es m or e im por tant than GF. Finally, it is very important to realize that in many practical applications the fibres are are not rand omly distrib-

(3)

F r om a m odel ling point of view , any any cons titutive continuum modelling can be chosen in the bulk material. Often a simple linear elastic modelling is chosen. Introdu cing the Young s modu lus E, the characteristi characteristicc length Ich can be defined: lch lc h

=

GF

f2

(4)

The characteristic length can be interpreted as the ratio between fracture energy per crack area and elastic energy per volu me for a given material. material. T he characteristic length can be used to characterize the brittleness of a given material but can also be used to define a dimensionless structural brittleness number B by taking the ratio of a representati representative ve structural dim ensio n L to the characteristic characteri stic length of the m aterial:

B-

Lf/

EG F

5)

Fig. 3 - A crack in SFR C and the essential features: zone with fibre bridging, the process zone and aggregate aggregate interloc k together with the FCM.

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TC 162 TDF

Stress MPa) 4.00

Stress (MPa) 4 00

l

3 00

3 00

Total response 2 00

1 QO

Low strength concrete, ~.~gh fibb e r c o n te fi te n t (1 v o l. ) ~ H ig h

0 00 0 0

s trtr en en gt gt h concrete,

ber content 0.3 vol.

I

I 0.2

01

e s s - c ra ra c k o p e n F ig . 4 - T y p ic a l s t r es i n g relat relationshi ionships ps (a) obta ined from experimental m e a s u r e m e n t s o n s t e el el f i b r e r e in in f o r c e d c o n c r e t e

2 00

)

w ram)

c o n t a i n i n g 0 .3 a n d I v o l. o f h o o k e d e n d ste e l fib re s in h ig h

1 00

0 3

0 00 0 01

0 10

a)

1 00

10 00

w mm)

b)

uted. As a consequence, the stress-crack opening relat i o n s h i p o f S F R C m a t e r i al al s c a n n o t b e c o n s i d e r e d a n i s ot ropi c propert y, i . e . t h e r e l a t i o n s h i p d e p e n d s o n t h e d i r e c t i o n o f c r a c k i n g re re l a t iv iv e t o t h e f i b r e o r i e n t a t i o n . O ft en t he fi bres are ori ent ed duri ng t he cas t i ng proces s . In s urface l ayers fi bres are t o a cert ai n ext ent ori ent ed

and low strength concrete, respect i v e l y . In (b ) is sh o w n a c o n c e p t u a l t h e o r e ti ti c a l m o d e l l i n g o f t h e r e l a t i o n s h i p , fo llo win g [1 8 ], ], s h o w i n g t h e c o n c r e te te a n d t h e f i b re re c o n t r i b u t i o n s , N o t e t h e l o g a rith mic l e n g t h s c a l e ) .

3.2 Multi-linear relationship V ery rea reall is is titi c rep res ent at i ons of m eas ured s tres tresss -crac -crackk o p e n i n g r e l a titi o n s h ip ip s c a n b e o b t a i n e d w i t h a m u l t i - l in in ear funct i on:

parallel to the extern al surface parallel surface the so-called wa ll effect effect)) a n d s o m e t i m e s t h e t h i c k n e ss ss o f t h e s t r u c tu tu r a l m e m b e r i s of the same ord er of magn itude as the fibre length e.g. p a v e m e n t s ) , w h i c h c a u se se s fi fi b re re s t o b e o r i e n t e d t h r o u g h o u t t h e t h i c k n e s s o f t h e s t r u c tu tu r a l m e m b e r .

( : ~w ~w :

(~i -- (~i w

W 0 =0 ,

0 11 ft

O'i+1 -- (~i C ~i ~ i+ 1 - - O ~ i '

W i - 1 < ~ 1 42 42 ~ W i

(6)

O ~1 ~1 > 0

The mul t i -l i near s t res s -crack openi ng rel at i ons hi p i s illustrated in Fig. 5. As ind icated in Fig. 5 th e slopes a i o f t h e d i f f e re re n t l in in e a r s e c t io io n s o f t h e m u l t i - li li n e a r f u n c t i o n can be pos i t i ve and negat i ve. It is is as s umed, how ever, t hat t he t ens i le le s t rengt h i s nev er reached agai n aft er s oft eni ng has been i ni t iat iat ed.

3. STRESS-CRACK OPENING RELATIONSHIPS FOR DESIGN 3.1 General E x p e r i m e n t a l d e t e r m i n a t i o n o f t h e s tr tr es es ss- cr cr ac ac k o p e n i n g r e la l a t io io n s h i p c a n b e d o n e i n a f u n d a m e n t a l w a y u s i n g t he uni axi al t ens i on tes tes t [3, 29, 30]. A s men t i on ed i n t he p r e v i o u s s e c t i o n , t h e s h a p e o f th th e s t re re s s -c -c r a c k o p e n i n g r e l at at io i o n s h i p d e p e n d s h e a v i ly ly u p o n t h e t y p e a n d a m o u n t o f f i b re re u s e d . T h e r e la l a t io io n s h i p c a n b e d i v i d e d i n t o a c o n c r e t e c o n t r i b u t i o n a n d a f i b re re c o n t r i b u t i o n . T h e c o n c r e t e c o n t r i b u t i o n i s t h e s o f t e n i n g s t r e ss s s - cr cr a c k o p e n i n g r e l a tionship for the unreinforced concrete, while the fibre con t ri but i on cons is is ts ts o f a st st eepl eepl y as cendi ng part fol l ow ed by a s l ow ly ly des cend i ng or s oft eni ng part . Th e fi rs t part o f t he res ul t i ng rel at ions ions hi p - up t o crack openi ngs of about 0 . 1 - 0 .2 .2 m m - i s a r e su s u l t o f th th e c o m p e t i n g c o n c r e te te a n d fi bre cont ri b ut i on, w h i l e t he rel at i ons hi p for for larger larger crack openings is due mainly to the fibre contribution. The res ul t i ng t ot al res pons e cons i st st s o f firs firs t a d es cendi n g part , t hen a s low low l y as cendi ng and fi nal l y a des cendi ng or s oft en ing part, see Fig. 4. F or des i gn purpos es , s i mpl i fi ed vers ions ions o f t he s tres tresss c r a c k o p e n i n g r e l a titi o n s n e e d t o b e d e f i n e d .

3 . 3 B i l in e a r r e l a t i o n s h i p

F o r m a n y S F P , .C .C m a t e r i a l s a b i - l i n e a r r e l a t i o n s h i p p r o v i d e s a re r e a s o n a b le le r e p r e s e n t a t i o n o f t h e m e a s u r e d b e h a v i o u r , c o m p a r e w i t h F ig ig . 4 :

ow

f,

1

WI

W2

W3

W

F ig . 5 - Illu stra tio n o f th e mu lti-lin e a r stre stre ss-c ss-c rac rac k o p e n in g r e l a t i o n s h i p . I n t h e fig u re a 1 a n d a 2 a r e p o s i t i v e , is n e g a t i v e .

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  ateri ateri als and

Structures/M at6riaux et C onstruct onstructions ions Vol. 35 Jun June e 2002

c r l - - C ~ ' IW IW c r w (W ) =

or

c r2 r2 - - c r l

Ou a0w ill alw alw ays r es ult in the m or e cr itical s ituation. ituation.

0

3 . 7 E x p e r ime ime n ta l de te r m i na ti on a nd v e r ifi ificc a tion tion F o r e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e s t re r e s s -c - c r a ck ck o p e n i n g r e l a t i o n sh s h i p i t i s u s e fu f u l t o d i s t in in g u i s h b e t w e e n t e st s t s a i m i n g f o r d i re re c t d e t e r m i n a t i o n o f t h e r e l a t i o n s h i p a n d m e t h o d s a i m i n g f o r v e r i fi f i c a t io io n o f a s i m p l i f i ed ed v e r s i o n o f th th e r e l a t i o n s h i p u s e d i n d e s i g n . I n t h i s c o n t e x t i t s h o u l d b e n o t e d t h a t n o g e n e r a llll y a c c e p t e d s ta ta n d a rd rd s f o r e i t h e r d i r e c t d e t e r m i n a t i o n o r v e r i f i c a t i o n o f t h e s tr tr es es ssc r a c k o p e n i n g r e l a t i o n s h i p e x i s t a t th th i s p o i n t i n t i m e . M e t h o d s f o r d i r e c t d e t e r m i n a t i o n o f th t h e s tr t r e ss ss -c - c r ac ac k

w

F i g . 9 - I l l u s t r a t i o n o f a v e r a g e , f f w ( w ), ), c h a r a c t e r i s t i c , o w k (w (w ) , a n d design, o w d(w), stress-crack openin g relation ships. T he ~verage and the cfiaracteristic curves are based on testing experience. The d e s i g n c u r v e i s r e l a t e d t o t h e c h a r a c t e r i st st i c c u r v e t h r o u g h a s a f e t y f a c t o r . T h e c h a r a c t e r is is t i c t e n s ilil e s t r e n g t h f t k a n d t h e d e s i g n v a l u e ft dare code related properties or pro perties obtaine d from inde13 endent t e s t i n g .

results with pre-calculated, expected responses. The exp ected response can be calculated for any of the sugg e s t e d m a t e r i a l m o d e l s , E q u a t i o n s ( 6 ) to t o ( 9 ). ). T h i s

o p e n i n g r e l a t i o n s h i p a r e t y p i c a l ly l y b a se se d o n a u n i a x i a l t e n s i o n t e s titi n g c o n f i g u r a t i o n . A s t a n d a r d f o r u n i ax a x i a l te te n s i o n t e s t i n g m e t h o d f o r S F R C (a s w e l l a s o t h e r t y p e s o f F R C C ) h a s b e e n p r o p o s e d i n [ 3 ]. ]. H e r e a m e t h o d f o r t h e d i r e c t d e t e r m i n a t i o n o f t h e c h a r a c t e r is is t i c s t re r e s s - c ra ra c k o p e n i n g c u r v e O w , w f r o m t h e a v e ra ra g e c u r v e ~ w W ) i s d e s c r i b e d . C a r e m u s t b e t a k e n t o o b t a i n a f ib ib r e o r i e n t a t i o n i n t h e t e s t s p e c i m e n t h a t is is re re p r e s e n t a t i v e o f t h e o r i e n t a t i o n i n t h e s t r u c t u r a l application, see Section 2.3. The uniaxial tension test m e t h o d i s n o t s u i ta ta b l e fo f o r d e t e r m i n a t i o n o f t h e t e n s i le le strength f. It is assumed that information about the chara c t e r is is t i c a n d d e s i g n t e n s i l e s t r e n g t h s , f t k a n d f t d i s o b t a i n e d i n d e p e n d e n t l y , t h r o u g h c o d e s o r s t a n d a rd rd t e s t i n g m e t h o d s i n t e n d e d f o r p l a in in c o n c r e t e . T h e p r i n c i p le le

m e t h o d i s s titi llll u n d e r i n v e s t i g a t i o n . I t s h o u l d al a l so so b e n o t e d t h a t m u c h w o r k h a s b e e n c a r r i e d o u t i n o r d e r t o s e t u p m e t h o d s o f i n v e r se se a na na ly ly s is is t o d e r iv i v e t h e s tr t r e ss ss -c - c r ac a c k o p e n i n g r e la l a t i o n s h ip ip f r o m b e n d i n g t e s t s , [1 [1 4 , 2 3 ] , S o m e d i f f i c u l t i e s w i t h t h i s t y p e o f a p p r o a c h w e r e p o i n t e d o u t i n [ 33 33 ]. ].

is il illus lusstrtrated in ack F ig.opening 9. A es s - cr cur ve f or us e in s er viceability lim it s tate tate analysis analysis m ay be ob taine d f r om the un iaxial tes ting result and information about tensile strength using c u r v e f i t t i n g o f a n y o f t h e s u g g e s t e d m a t e r ia ia l m o d e l s , Equations (6) to (9) to the characteristic stress-crack openi n g r e l a titi o n s h i p . It I t i s r e c o m m e n d e d t h a t t h e f i t t e d a n a l y t iical relatio nsh ip always lies be low th e characteristi characteristicc curv e. F u r t h e r m o r e , t h e f i t t e d s t re re ss s s -c -c ra ra c k o p e n i n g r e l a t i o n s h i p s ho uld alw ays be over all all des cend ing, i . e . the characte characterist ristic ic t e n s ilil e s t r e n g t h s h o u l d n e v e r b e r e a c h e d a g ai ai n. n. T h e d e s i g n s t re r e s s -c -c r a c k o p e n i n g r e l a t i o n s h i p d (w (w ) f o r u s e i n u l t i m a t e l i m i t s t a t e a n al al ys ys is is i s o b t a i n e d i ~r ~r om om t h e c h a r a c te te r i st s t i c r e l a titi o n s h i p b a s e d o n t h e l o w e r c h a r a c ter is tic tic tens ile s tr en gth by divis io n w ith a s af af ety ety f actor . E x p e r i m e n t a l v e r i fi f i c a t i o n o f a ss s s u m e d s t re r e s s -c - c r a ck ck

d e s cTrhi bei ni gd et ah eo fc rt ahcek ne od ns-elci nt ieoanr ahsi an gneo nm- loi nd ee al ri sh ito t no gaen. a ly ly ze ze

o p e n i n g c u r v e s c a n b e c a r ri ri e d o u t t h r o u g h a d i r e c t c o m parison with uniaxial test data. Stang and Olesen [34] suggested to carry out bending tests with a notched b e a m , e.g. accor ding to [ 1] , and to compar e the test

and as socia sociated ted load car r ying capacity capacity is car r ied ou t on ly f or the non- linear hinge loaded w ith the gener alized s tr es s es , i n t h i s c as as e a a x ia ia l f o rc rc e N a n d a m o m e n t M .

267

4. CROSS SECTIONAL ANALYSIS FOR F L EX EX U R E A N D A X I A L F O R C E S 4 .1 .1 G e n e r a l

n o n l in in e a r h i n g e

A c r o s s - s e c t i o n a l a n a ly ly s i s o f t h e c r a c k e d s e c t i o n o f la b c a n b e c a r r i e d o u t b y e . g . a b e a m , a p i p e o r a s la

s e p a ra r a t e ly ly t h e s e c t i o n o f t h e s t r u c t u r a l e l e m e n t w h e r e t h e c r a c k is is f o r m e d a n d a s s u m e t h a t t h e r e s t o f t h e s t r u c tur e behaves in a linear elas tic f as hion. I n or der f or the n o n - l i n e a r h i n g e t o c o n n e c t t o t h e r e s t o f t h e s t r u c tu tu r e , t h e e n d f a ce ce s o f t h e n o n - l i n e a r h i n g e a r e a s s u m e d t o r e m a i n p l a n e a n d t o b e l o a d e d w i t h t h e g e n e r a li li z e d s tr es s es in the elem ent. T h e p r i n c i p l e o f t h e a n al al ys ys is is i n t h e c a s e o f a s t ru ru c t u r a l e l e m e n t s u b j e c t ed e d t o a c o m b i n a t i o n o f f le l e x u r e a n d a xi x i al al f or or ce ce i s s h o w n i n F i g. g. 1 0 . T h e s t ru r u c t u r a l e l e m e n t is is d M d e d i n t o a n o n - l i n e a r h i n g e o f l e n g th th s w h e r e t h e m a i n n o n l i n e ar ar b e h a v i o u r d u e t o c r a c k in in g is is c o n c e n t r a t e d a n d i n t o oth er par ts that ar e con s ider ed to behav e elasti elastical cally. ly. T h e non - linear analysis analysis f or f ictitious ictitious crack crack pr opaga tion

 

M a t e r ia ia ls ls a n d S t r u c t u r e s M a t 6 r i a u x e t C o n s t r u c t i o n s Vol . 35 June 2002

i

the de form a tion is gov e rne d by the st stre re ssssc r a c k o p e n i n g r e l a t i o n s h i p , t h e c ra ra c k le ng th a nd the ove ra ra ll a ngula r de form a tio n of the n on-lin e a r hinge , se e [35] for line a r stree ss-c str ss-c ra ra c k op e nin g re la tionship tionship a nd [26] for a bi-linear stress-crack opening relat i o n s h i p a n d t h e d r o p - l i n e a r r e l at at i o n s h i p , (b) in F ig. 11. In a ll c a se se s the a ve ra ge ge c urva ture of the

~

~Xc!ac

Non-linear hinge

non -line a r hing hingee ,K ,K m s give n by:

nci ple ple o f th e no n-l i ne a r h i n g e a n a l y s i s i n the case o f a structural structural e l e F i g . 1 0 - T h e pr i nci m e n t subjected t o a c o m b i n a t i o n o f f l ex e x u r e a n d a x i a l f o r c e l e f t) t) . A s s h o w n i n t h e r i g h t h a n d s i de the e l e m e nt is is d i v i d e d i n t o a m i d d l e s e c t io io n o f w i d t h s c o n t a i n i n g t h e c r a c k - t h e s o - c a llll e d n o n - l i n e a r h i n g e - a n d t h e r e s t o f t h e e l e m e n t w h i c h i s a s s u m e d t o b e h a v e e l a s t ic pag a tio tio n a nd ic a l l y . T h e n o n - l i n e a r a n a l y s i s fo r c r a c k pr o pag associated associa ted load carrying capacity is carried carried out on ly for th e n o n - l i n e a r h i n g e .

A nu m be r of diffe re nt solutions solutions for rec rec ta ngula r c rossrossse c tion c a n be found in the lite ra ture ba se d on d iffe re nt a ssum ssum ptions ptions re ga rding kine m a tic a nd c on stituti stitutive ve c ond itions. ti ons. A n ove rv ie w of diffe re nt c onstitutive rela rela tions in te rm s of the st stre re ss-c ss-cra ra ck ck ope ning re la tion gove rn ing the tra c tion on the fic titious c rac rac k a s func tion o f the c rac rac k o p e n i n g c a n b e f o u n d a b ov ov e . Different kinem atic assumptions applied in various m o de ls a re show n sc he m a tic a lly in Fig. 11 a nd c a n be de sc ribe d in th e following wa y: 9 Th e fic titious titious c ra c k surfac surfac e s re m a in p la ne a n d th e c rac rac k o pe n ing a ngle e quate quate s the ove ra ra llll a ngula r de form a tion of the n on-lin e a r hinge , [28] [28],, (a) (a) in Fig. 11. 9 Th e fic titious titious c ra c k surfac surfac e s re m a in p la ne a n d the c rac rac k o pe n ing a ngle e qua te s the ove rall rall a ngula ngularr d e form a t i o n o f t h e n o n - l i n e a r h in in g e . F u r t h e r m o r e , t h e o v e r a llll c u r v a tu tu r e o f th th e n o n - l i n e a r h i n g e , t h e c u r v a tu tu r e o f t h e c r a c k e d p a r t a n d t h e c u r v a t u r e o f t h e e l a s titi c p a r t a re re linke d ba se d on a n a ssum p tion of pa raboli rabolicc va ria tion of the c urva ture [6, 7], (a) in F ig. 11. 9 Th e fic titious titious c ra c k surfa surfa ce ce s do no t re m a in pla ne ,

r

]

q,/2

a)

W cmod

q *a

(11)

w h i l e t h e c r a ck c k m o u t h o p e n i n g d i s p l ac a c e m e n t in in t h e third a pproa pproa c h is de te rm ine d from the str stree ss-c ss-c rac rac k o pe n ing relation ship, the overal overalll curvatu re ~cm a n d t h e l e n g t h of the fictitious crack.

4 3 A n a l y s i s f o r r e c t a n g u l a r s e c t io io n s T h e a n al al ys ys is is a c c o r d i n g t o t h e t h r e e d i f f e r e n t k i n e m a tic a ssum ptions is pre se nte d be low. In a ll c a se s the s o l u t io io n s c a n b e g e n e r a l i z ed ed t o c o v e r T - s e c t i o n s , n o n line a r m a te ria l be ha v iour in c om p re ssion e tc. tc.

Simplifiedapproach Simplified approach by ed edersen ersen

Th e follow ing a nalys nalysis is follows the a na lysi lysiss in [28] [28].. T he analysi analysiss allows for the ap plication o f any stress-crack stress-crack o p e n i n g r e l a titi o n u s i n g n u m e r i c a l i n t e g r a t i o n t o o b t a i n the solution. The a na lysis ta ke s into a c c ount c om bine d a xial xial forc e a nd m om e n t on the c ross se c tion. tion. C o n s i d e r a n o n - l i n e a r h i n g e w i t h r e c ta ta n g u l ar a r c r o ss ss -

section on lan pthmh. cross section subje an esecti xternal xterna bednddeing o mTehe nt M pe rsecti wid on th aisndsubjected the cted a xia xia l to forc for ce N pe r wid th o f the the e le m e nt. As long a s the the te nsil nsilee strength strength is no t reached th e e lem ent is assumed to behave linear elaselastically tica lly accordi according ng to cla classi ssicc Bem oui lli beam theory. W h e n t h e t e n s ilil e s t re re n g t h i s re re a c h e d , i t is is a s s u m e d that a single crack is formed with a maximum tensile stress f at the crack crack tip. Fig. 12 shows the assum ed distributio n of stre stre sse sse s w he re the post-pe a k te nsile stre stre ss is a f u n c t i o n o f t h e c r a c k w i d t h g i v e n b y t h e s t re r e s s - cr cr a c k opening relationship, Ow(W. Ow(W . No te tha t it is is a ssum e d tha t the compressive zo ne rem ains in the lin ear elasti elasticc range. range. As indic a te d in Fig. 12 it is a ssum e d tha t the c ra c k has a linea r profile (see also [ 1 9 ] , t h u s t h e c r ac ac k o p e n i n g angle ~0" ~0" is related to the crack m ou th op en ing Wcm Wcmoa nd the le ngth of the fic titious titious c ra c k a a s follows: follows:

I

r

(10)

In the first two a pproa c he s, the c ra c k m o u t h o p e n i n g d i s p la la c e m e n t W c m o d ( t h e c ra r a c k o p e n i n g at a t t h e b o t t o m o f th th e n o n linear hinge) follows directly from the crack opening angle ~0" ~0" and the len gth o f the fictitious crack, a:

4 2 Geometrical assumptions

I

= q s

w~,,oa (b)

F ig. 11 - D i f f e r e n t k i n e m a t i c a ss ss u m p t i o n s a p p l i e d i n v a r i ou ou s n o n l i ne a r h i n g e m o d e l s . I n a ) the fi c ti ti o us c r a c k s ur fac fac e s r e m a i n plane and the crack opening angle equates the overall angular d e f o r m a t i o n . I n b ) the fi c ti ti o us c r a c k s ur fa fa c e s do no t r e m a i n p l a n e t h e d e f o r m a t i o n i s g o v e r n e d b y t h e s t r e ss ss - c r a ck ck o p e n i n g r e l a ti o ns hi p.

tp* - Wcm ~ a

268

(12)

 

TC 162 TDF

order

N,

N,M)

N,(~) +

t-}

I

9

a~

is i

f

d e t e r m i n e t h e p o s i t io i o n o f t h e n e u t r a l ax ax is is :

T h e c ra ra c k m o u t h o p e n i n g f r o m E q u a t i o n ( 1 5) 5) :

i

hl

t

cmod

X o ) - N c(IP ,x0) = N

(19)) (19

G i v e n I p a n d N t h e p o s i t i o n o f th t h e n e u t r a l a x is is c a n b e d e t e r m i n e d f r o m E q u a t i o n ( 1 9) 9) . N o w t h e m o m e n t M r el el at at iv iv e to t o t h e c e n t e r l in i n e o f th th e crosss sect cros sectii on can be det erm i ned. Th e m om en t i s gi ven by:

r,,(w)

(h

stress distribution

F i g . 12 12 T h e a n a l y s i s a c c o r d i n g t o t h e s i m p l i f i e d a p p r o a c h b y Pedersen. The non linear hinge subjected to axial force and m o m e n t i s s h o w n a l o n g w i t h t h e a s s u m e d s t r es es s d i st st r i b u t io io n the resu lting forces in the different regions.

Ns(

Wcmoa a n b e s u b s t i t u t e d

x o l N + ( h + x o 2 a l N + ( h _ a l N f +M +M y 2 0 )

v=tT-T)

< tg T - T )

t,2

)

w h e r e a i s d e t e r m i n e d b y E q u a t i o n s ( 1 2 ) a n d ( 15 15 ) . T h u s t h e r e s u lt l t o f t h e c a l c u l a ti t i o n is is c o r r e s p o n d i n g v a l u e s o f a n g u l a r d e f o r m a t i o n , a xi xi al al fo fo r c e , a n d b e n d i n g m o m e n t :

nd

M = M O p ,N ,N )

(21)

N o w , t h e d i s t r i b u t i o n o f n o r m a l s t re re ss s s es es w i t h i n t h e c r a c k e d z o n e i s g i v e n b y t h e s t re re s s -c -c r a c k o p e n i n g r e l a t i ons hi p, Ow(W a n d t h e r e s u l titi n g fo fo r c e p e r w i d t h o f t h e c r a ck ck e d zo z o n e a n d t h e b e n d i n g m o m e n t p e r w i d t h o f th th e cracked zone taken at the crack tip, N f and M y respec-

a n d t h e s o l u t i o n c o n s t i t u t e s t h e c h a r a c t e r is i s t i c s o f th th e n o n - l i n e a r h in in g e . F urt hermore, res ul t s are obt ai ned for fi ct i t i ous crack l e n g t h a, a, c r ac a c k m o u t h o p e n i n g Wcmod nd stresses. T h e n o n - l i n e a r E q u a t i o n ( 1 9 ) c a n b e s o lv lv e d f o r x 0

t ivel ively, y, can be o bt ai ned by i nt egrat i ng:

u s i n g a s i m p l e n u m e r i c a l i te t e r a t io i o n t e c h n i q u e , e.g. bi s ect io io n . F u r t h e r m o r e , t h e i n v o l v e d i n t e g r a t i o n s i n Equations (13) and (14) can be carried out using a n u m e r i c a l i n t e g r a ti t i o n s c h e m e a l lo lo w i n g f o r th th e u s e o f a n y s tres tresss -crac -crackk ope ni ng rel at ions ions hi p Ow(W Ow(W..

U l = 4 f . o ~ 1 76 ~o

-

1

jo ,Oo ,.wt.juau

(13)

14)

S i mplif mplifii edapp apprr oach by asa asanov novaa

It is is now requi red t hat t he crack openi ng angl anglee corres ponds t o t he t otal otal angul ar deform at i on of t he non -l i near hi nge. W i t h t he gi ven as sumpt sumpt ions ions t he dept h of t he t ens ilil e zone, h - x0, (see (see F i g. 12) 12) i n t he non -l i near hi nge m ay now be rel at ed t o t he crack mo ut h open i ng by:

h-Xo=l(ft S+Wcmod)

(15)

q)\E w h e r e s is is t h e l e n g t h o f t h e n o n - l i n e a r h i n g e . T h e r e s u l titi n g f o rc rc e p e r w i d t h o f t h e e l as as titi c c o m p r e s s i o n z o n e i s d e n o t e d N < a n d t h e r e s u l t in in g f o r c e p e r w i d t h of t he el as as titi c tens tens i l e zone i s den ot ed Nt:

Uc = ~pEx~

(16)

N ,=

(17)

2s

(f )2s 2~pE

Casanova and Rossi, [7], proposed a model which adopt s t he as s um pt i ons s how n on F i g. 11 (a) (a) w i t h res pect t o t h e a n g l e at at t h e f a ce ce s o f th th e n o n - l i n e a r h i n g e a n d t h e c r a c k o p e n i n g a n g l e . I n t h e m o d e l i t is is a s su su m e d t h a t t h e angl e formed by t he crack vari es accordi ng t o t he crack m o u t h o p e n i n g a n d t h e d e p t h o f th t h e c r ac ac k :

q~= W c'~ a

T h e l e n g t h s o f th t h e n o n - l i n e a r h i n g e i n C a sa s a n ov ov a s m od el vari vari es w i t h t he dep t h o f t he crack s uch t hat: hat: s = 2a

Wcmod:NN f p , W c m o d.d. and Wcmod:

D e n o t i n g t h e r e s u l t in i n g a x ia ia l fo fo r c e p e r w i d t h o f t h e e l e m e n t N , f o r a g i v e n a n g u l a r d e f o r m a t i o n ~ p, p, t h e e q u i l i b r i u m o f t h e s e c ti t i o n is is w r i t t e n i n t h e f o l l o w i n g w a y i n 269

(23)

T h e f i n a l as as s u m p t i o n i s r e l a te te d t o t h e i n t e r n a l k i n e m a t i c s o f th th e h i n g e . T w o c u r v a t u r e s ar ar e c o n s i d e r e d , th th e elas el as titi c curvat ure o f t he u n-cr ack ed part o f t he h i nge, K 1, and t h e cu rvat u re i n t he crack ed zone , ~c ~c . Th e el elas as titi c curva t ure i s gi ven by: 12M 1~ 1

Eq uat i on (16) des cribes cribes ho w N < i s relat relat ed t o Ip and x0: N< ({P,Xo).E q u a t i o n ( 1 7 ) d e s c r ib ib e s h o w N t is related to qJ:N (q~ q~)). Eq uat i on (13) des cri bes h ow N f i s relat relat ed t o ~p

(22)

Eh 3

(24)

w h e r e M i s t h e m o m e n t p e r u n i t w i d t h i n t h e b ea ea m . Th e curvature in the cracked zone is given by: le22 = -~ -c le

xo

( 25 25 )

w h e r e e c i s th t h e s t ra ra i n a t t h e e x t r e m e f i b r e i n c o m p r e s s i on, and x 0 i s t he dep t h o f t he ne ut ral axi s at t he crack.

 

M a t e r i a ls ls a n d S t r u c t u r e s /M a t ~ r i au au x e t C o n s t r u c titi o n s V o l . 3 5 J u n e 2 0 0 2

T h e d e p t h o f t h e n e u t r a l a xi xi s i n t h e u n - c r a c k e d ( el el as as tic) ti c) zo ne in Casanova s m od el is based o n linear elastici elasticity. ty. Th us , al t hou gh q) -- ~ p*, p*, t he neu t ral axi s , and t herefo re t he cu rvat ure, are di fferent at t he faces faces of t he no n-l i n ear hi nge and at t he cra crack. ck. Th e curvature in the un-crack ed and the cracked part o f t h e h i n g e i s l i n k e d t o t h e a v e r ag ag e c u r v a t u r e Km K m (see E q u a t i o n ( 1 0) 0) ) o f t h e h i n g e b y a s s u m i n g a p a r a b o li li c v a r i a t i o n o f th th e c u r v a t u r e a l o n g t h e n o n - l i n e a r h i n g e :

S

X

X

~

~

a

2~:11 + K2 2~: (26) 3 C o m bi n i ng E quat i ons (22) to to (26) l eads eads to to t he bas i c c o n s t it i t u t iv iv e r e la l a t io i o n s h i p o f t h e m o d e l p r o p o s e d i n [ 7 ]: ]:

aw w)

l

// i i

]~m

W c m o a= a= 2 1 c m a e = 2 2 K 1 ; K 2 ) a 2

,

41

=I

cmod

u x)/2

(27)

stress

u x)/2

am W) dx

w

distribution

aw w) dx

F i g . 1 3 T h e m o d e l l i n g o f t h e p r o p a g a t i o n o f t h e f i c t itit i o u s crack t h r o u g h t h e n o n l i n e a r h in in g e u s e d i n t h e e x p l i c i t f o r m u l a t i o n . B e l o w t h e n o n l i n e a r h i n g e : i l l u s t ra ra t i o n o f a n i n c r e m e n t a l h o r i z o n associated stress stress d i s t r i b u t i o n . t a l l a y e r o f the band To the right: the associated

Equ at i ons (13) t o (14 ) and (16) t o (2 0) pres en t ed for P eders en s mo del are di rect l y appli appli cabl cablee w i t h C as anova anova s m o d e l w h e n q)/s s replaced replaced with Km Km..Again, the equ ilibrium equa tion (19) has to b e solved by n um erical iterati iteration. on. In the above a linear stress-strain relationship for the con crete in com pression or in tension ou tside the c r a c k e d z o n e i s a d o p t e d . I n t h e g e n e r a l c a se se w h e r e a no n-l i ne ar s t res res s-s s-stt rai rai n rel at ions ions hi p for concret e i n co m p r e s si s i o n i s c h o s e n , a n u m e r i c a l i n te te g r a t i o n o v e r d e p t h o f t he cros s s ect i on can be carri ed out . Thi s approach has b e e n u s e d b y [ 2 0 ] t o p r e d i c t t h e r e s p o n s e o f f ib ib r e r e i n forced s l ab el emen t s .

= m X- *0)= u *)

2S/

S

xplicit ormulat ormulation ion by by O lese lesenn

It is is pos pos s ibl ibl e t o obt ai n a cl os ed form s ol ut i on for t he non-linear hinge when using a multi-linear or bi-linear stress crack-opening relationship in combination with t h e k i n e m a t i c a s su s u m p t i o n t h a t th th e b o u n d a r i e s o f t h e n o n - l i n e a r h i n g e r e m a i n p l a n e w h i l e t h e f i c t i titi o u s c r a c k pl ane deformat i on i s governed by t he s t res s -crack openi ng rel at i ons hi p as w el l as t he overal l angul ar deformat i o n o f t h e n o n - l i n e a r h i n g e a n d t h e l e n g t h o f th th e f i c t i t i ous I n crack. p a r t ic ic u l a r a s o l u t io i o n f o r t h e m o m e n t - r o t a t i o n r e la la t i ons hi p i n t he cas casee of zero axi al force and a bi -l i nea r s t res s -crack openi ng rel at i ons hi p w as pres ent ed i n [33]. The complete solution for the bi-linear stress-crack o p e n i n g r e l a t io io n s h i p i n c l u d i n g a n o n - z e r o a x ia ia l fo fo r c e c a n be fou nd i n [26] [26].. Th e d eri vat i on i n t he fol l ow i ng fol fol low low s [33] closely. T h e n o n - l i n e a r h i n g e i s m o d e l l e d a s i n c r e m e n t a l la la y ers of s pri pri ngs t hat act w i t ho ut t rans ferri ng s hear be t w e en e a c h o t h e r , s e e F ig ig . 1 3 . T h e v e r t ic ic a l b o u n d a r i e s o f t h e h i n g e a r e a s su s u m e d t o r e m a i n s tr tr ai ai gh gh t d u r i n g d e f o r m a t i o n a n d t h e t o t a l a n g u la la r d e f o r m a t i o n o f t h e n o n - l i n e a r h i n g e i s ag a g a i n d e n o t e d ~ p .T . T h e a s s o ci c i a te te d l o n g i t u d i n a l d e f o r m a t i o n o f t h e s p ri ri ng ng s i s d e n o t e d u x ) w h e r e x i s a vert i cal co-ordi nat e, s ee F i g. 13. The average curvat ure o f t h e n o n - l i n e a r h i n g e , ~ :m :m , i s g i v e n b y E q u a t i o n ( 1 0) 0) , w h i l e t he m ean l o ngi t ud i nal s trai train, n, g*(x) g*(x),, is is gi ven by:

270

w h e r e x 0 i s t h e c o - o r d i n a t e o f th t h e n e u t r a l a x is is i n t h e hi nge. It i s as s umed t hat t he hi nge l ayers behave l i near elastical elast ically ly as lon g as the ten sile streng thft is no t reach ed. W h en t he s t res res s reaches ft a fict fict i t ious ious crack i s as s umed t o form w i t h a s tres tresss -crac -crackk o pen i ng rel at i ons hi p ~ w hi ch is a function of the crack opening w wh ich in tu rn is a f u n c t i o n o f x . T h e d e f o r m a t i o n u o f a la la y e r m a y t h e n b e obtained from:

u x)=

~

E

s + w x)

(29)

C ons i deri ng t hat t he axi al force i s zero and t hat t he s tres tresss -cra -crack ck ope ni ng rel at i on O w( w(W i s t he bi -l i near fu nctdevel i o n , op E qiuna ttihe o n cros ( 7 )s, sfect o u ri ond i as f f etrhe e n tfi ct s tirtei sous s dcrack i s t r i b grow u t i o nss. cross The di fferent phas es are s how n i n F i g. 14. The phas es a r e g o v e r n e d b y t h e p a r a m e t e r s x 1 a n d x 2 g iv iv e n b y t h e general exp res s ion: ion: i

= x o + lq~L [ g i - w i ~- i - 1 ] ] -

i=

1,2

30)

where

~

gi = E

otis

~i = E

i = 1,2

(31)

T h e f o l l o w i n g n o r m a l i z a t io io n s a r e i n t r o d u c e d :

6

p= f,--~ M ,

hE

O=2~](p,

a

ot= ~

(3 2 )

w h e r e a i s t h e d e p t h o f th t h e f i c ti t i ti t i o u s cr cr ac ac k . T h e m o m e n t is normalized with the moment which causes the crack initiation while the angular deformation is normalized with the angular deformation at crack initiation. Given

 

TC 162 TDF

Phase 0

II

1

g = 4( 1 _ 3 0 t + 3 o r a

II1

1 _a___~3q t) 0

(1 _ b )(3 ~ 2

(41)

( c { , ~ ) j)2 )

-3+6ct For phase F i g . 1 4 - D e f i n i t i o n o f t h e f o u r d i f f e r e n t p h a s e s o f s tr tr e s s d i s t r i b u t i o n e x p e r i e n c e d d u r i n g c r a c k p r o p a g a t io io n .

I III I w i th th

ct = 1 -

t h e s e n o r m a l i z a t i o n s t h e p r e - c r a c k e la la st s t ic ic b e h a v i o u r o f t h e hing e is de s c ribe d b y the re la tion Ix = 0 with 0 < 0 < 1. The complete solution in the three crack development phases is given below. For con venienc e the fo llowing prope rtie s b a nd c a re re introduc e d: b = % f

(33)

C m ( 1 - b ) ( 1 - I I]I] 1 / (1 (1 4 / 2 - 1 4 /1

(34)

1+

mm _0 the s olution re a ds :

~ll/1

~-14/2

( 4 2) 2)

1412 . )

together with: = 4(1 - 3r + 3 a 2 - 0~3) 0~3)00 3 + 6~ - 3o 3o~~ +

1 (1

b + c/(l-

b/(1

ggV

-

c

(43)

)+ ( ~ 0/2

l-v1

A c l o s e d f o r m s o l u t i o n f o r Wo~od t h e c r a c k o p e n i n g a t th th e m o u t h o f t h e c ra ra c k - i s o b t a i n e d i n [ 2 6 ].]. I n a c o m pact form this can be writte n as:

T h e a n a ly l y s is i s o f t h e n o n - l i n e a r h i n g e i s c ar ar r i e d o u t w i t h 0 a s th th e c o n t r o l l i n g p a r a m e t e r . T h u s , t h e v a lu lu e s o f 0 corresponding to phase transitions need to be known. T h r e e d i f f e r e n t v a lu lu e s a r e r e le le v a n t , 0 0 4 , 0 M I a n d 0 H _ m : 0o , = 1

(35)

l- c +

1 ii ~

_l (b + 011-11l=2~1112

l

c2,

(36)

+ b2 ~

( 3 7) 7)

1 - c 2 + g tlt l _ 1 ~

/(l -b )2 ~ I 1 /1/ 1 - - 1 1 /2/ 2

s f , ( 1 - b i + 2 o tO tO W c m o d - - E ( I _ l ltl t /)/ )

( 44 44 )

w h e r e b i , ~ i ) = 1 , ~ 1 ) in pha se I , (b (b,Ur ,Ur''2) in p hase I I a n d (0,0) in pha s e I I L W i t h t h e s i m p l e d r o p - c o n s ta t a n t s tr t r e ss s s -c - c r ac ac k o p e n i n g relationship defined in Equation (8) involving the residu a l s t r e n g t h ~ . t h e I x (O (O ) r e l a t io io n s h i p r e d u c e s t o t h e s i m pie re la tions h ip, a ll in pha s e I, i.e. for 0 > 1: 9

.Y

I x( x( 0) 0) = ~ ' ( 3 - 2

( 45 45 )

~ /7

with: Fo r pha s e I w ith 00 00__ < 0 < 0i 0i__u the s olution re a ds : r ~ = 1 - q /1 /1 - i f ( 1 - q t l ) ( 1 - g t l )

(38)

(46)

F u r t h e r m o r e , i n t h i s c a se se t h e s o l u t i o n fo fo r be writte n as:

together with:

W cm od a n

W c m o d ( O ) = ( 1 - - 7 + 2Or0) sf'

bt= 41 -3 ~

--~-7110-3+ 6ct

(3 9 )

F o r p h a s e / / w i t h 0 i_ i_ ////_ < 0 _< _< 0m ~ 11 he s olution re a ds : a = l - q * : - 1 --~ -b 20

Of ' y

l-V 2

0

-V 2 + ? 14/2) 14/ 2)

E w i t h t h e n o r m a l iz i z e d c r a cckk l e n g t h g i v e n b y :

1

( 40 40 )

1- r 20

n /Y /Y _ _

(47)

(48)

V

A p p l ici c a t ioi o n o f th th e n o n li n e a r h i n g e i n s t r u c t u r a l analysis

Structural calculations o f e . g . be a ms , [6, 7, 26, 28, 3335], be a ms o n e la s tic tic foun da tion [26] a nd pipe s , [28], c a n

together with:

b e d o n e b y i n t r o d u c i n g a n o n - l i n e a r h i n g e i n t h e s t ru ru c t u r e , p r e s c r ib ib i n g t h e a n g u l a r d e f o r m a t i o n o n t h e n o n - l i n e a r h i n g e , u s i n g t h e n o n - l i n e a r h i n g e s o l u t i o n t o s o lv lv e fo fo r the ge ne ra liz e d s tre s s e s a t the non-line a r hinge , e s ta blis h271

 

Materials Materi als and

S tructure ructuress M aMria aMriaux ux t Constructions

Vol . 35 June 2002

i ng t he l i near el as t i c s ol ut i on for t he s t ruct ure gi ven t he g e n e r a l i z e d s tr tr es es se se s a t t h e n o n - l i n e a r h i n g e a n d f i n a l l y solving solvi ng for the applied load and the total deform ations. As an exam pl e, s ee s ect i on 5 and t he ap pend i x i n cas e of a s i m p l y s u p p o r t e d b e a m s u b j e c te te d t o 3 - p o i n t l o a d i n g . Th e leng th of the non-linear hinge s has to be cons i dered a fi t t i ng param et er for t he cal cul at ions ions . In gen eral t h e o p t i m a l l e n g t h w i l l d e p e n d o n t h e t y p e o f st st r u c tu tu r a l element. The hinge width s in a beam has previously bee asns es as esass ed edw el[28, i ng no n-l iconcret n ear hienge el s for npl ai l as 35]) fi breusrei nforced andmod i t has b e e n s h o w n t h a t s = h/2 i s a n a d e q u a t e c h o i c e . C o m p a r i s o n s b e t w e e n t h e e x p l ic ic i t m o d e l b y O l e s e n , t h e s i mpl i fi ed approach by [28] and t he approach by [6] for t he bi -l i near s t res res s-cr s-crack ack ope ni ng rel at i ons hi p s ho w very s i mi l ar res ul t s . F urt hermore, al l model s compare favora b l y w i t h a n o n - l i n e a r F E M a n al al ys ys is is u s i n g th th e s a m e b i l i near s t res s -crack openi ng rel at i ons hi p, s ee t he appendix. This seems to indicate that, for a large number of stress-crack op enin g relationships, the simplified as s umpt i ons t hat t he fi ct i t i ous crack faces remai n pl ane a n d t h a t t h e d e f o r m a t i o n o f th th e n o n - l i n e a r h i n g e i s p r i ma ri l y due t o t he crac k open i ng are reasonabl reasonable. e.

able. However, attention should be drawn to the recent f o r m u l a t i o n o f a s o - c a l l e d a d a p t i v e h i n g e m o d e l [ 2 5 ], ], w h i ch es s ent ent ial ialll y i s a no n-l i n ear hi nge s ect i on 4.3) t hat t ak ak e s i n t o a c c o u n t b o t h t h e p r o p a g a t i o n o f th th e f i c titi t io io u s crack and the de-bonding between re-bars and SFRC. T h e r e a s o n f o r c al al l in in g th th e m o d e l t h e a d a p t i v e h i n g e i s t h a t t h e w i d t h o f t h e h i n g e a d ap ap ts t s i t s el el f t o t h e d e - b o n d e d r e g i o n o n b o t h s id id es e s o f th th e b e n d i n g c r a ck ck . T h e m o d e l a ls ls o p ro ro v i d e s in in f o r m a t i o n a b o u t c r a c k o p e n i n g a n d a v e r a g e c u r v a t u r e o f th th e a d a p titi ve ve h i n g e . H o w e v e r , t h e m o d e l has not yet been ex perim entally verified. Therefo re, unt i l s uch i nformat i on i s avai l abl e, crack s paci ng has t o be eval uat ed bas ed on reas onabl e as sumpt sumpt ions ions . Th e form u l a t i o n p r o p o s e d i n E u r o c o d e 2 , [ 1 0 ] , f o r t h e a v er er ag ag e c r a c k sp s p a ci ci ng ng a n d d e s c r i b e d i n T e s t a n d D e s i g n M e t h o d s f o r S te te e l F i b re re R e i n f o r c e d C o n c r e t e . R e c o m m e n d a t i o n s for the r~-a design method [2] gives an upper limit. Experimental evidence indicates that crack spacing in S F R C m e m b e r s s m a l le le r t h a n t h e m e m b e r d e p t h i s o b s e r v e d , bot h i n f l e x u r e a n d i n d i r e c t t e n s i o n . I t i s therefore proposed to adopt the following assumptions for c rac k spacing, srm: srm: Srm = 50 + 0.2 5k l k 2 q~ 9r

49)

w h e r e ~ i s t h e b a r d i a m e t e r i n m m w h e r e a s 9 t he effect iv i v e r e i n f o r c e m e n t r a t i o , a n d p a r a m e t e r s k 1 a n d k 2 are defi n ed i n [10]. T h e r e la la t io io n s h i p s p r e s e n t e d i n S e c t i o n 4 .3 .3 f o r u n r e i nforced concret e are appl i cabl e t o members rei nforced w i t h c o n v e n t i o n a l r e i n f o r c e m e n t . I n t h is i s c a se se t h e l e n g t h o f th t h e n o n - l i n e a r h i n g e d e f i n e d i n S e c t i o n 4 . 3 s t ilil l a p p li l i e s to to d e t e r m i n e t h e r e s p o n s e o f t h e s e c t i o n . H o w e v e r i n c o m p u t i n g t h e d e f l ec e c t io io n , [ 2 0 ] s h o w e d t h a t m o r e t h a n o n e c r ac ac k m a y g o v e r n t h e m e m b e r b e h a v i o u r a n d t h a t a l e n g t h e q u a l to to t h e m e m b e r d e p t h b u t n o t l e ss ss t h a n s o r n o t l es es s t h a n t h e v a l u e g i v e n b y E q u a t i o n 4 9 ) should be used. They also showed that adopting the variabl vari abl e no n-l i n ear hi ng e l engt h pre s ent ed by [7 [7]] leads leads to to reas onabl e resul resul ts ts as com pared t o e xpe ri me nt al res ult ult s. s.

4 . 4 C r o s s s e c t io io n w i t h c o n v e n t i o n a l reinforcement

The models presented in Section 4.3 were aimed at p r e d i c t i n g t h e b e h a v i o u r o f c r o s s -s -s e c t io io n s r e i n f o r c e d o n l y w i t h f ib i b r e s . I n a p p l ic ic a t io io n s w h e r e c o n v e n t i o n a l reinforcement is used with fibre concrete, additional a s s u m p t i o n s r e l a te t e d t o t h e l e n g t h o f th th e n o n - l i n e a r hi nge, t he s t res s i n t he rei nforcement at t he crack, and t he average curvat ure mus t be adopt ed. I n p r e s en en c e o f r e i n f o r c e m e n t , S F R C m e m b e r s exhibit multiple cracking until reinforcement yielding. A t t h a t p o i n t , o n e c r a c k g e n e ra ra l ly ly g o v e r n s t h e m e m b e r b e h a v i o u r d u e t o t h e s o f t e n i n g n a t u r e o f f i b re re c o n c r e t e . T h e a s su s u m p t io i o n s m u s t e n a b l e m o d e l l in in g t h e m e m b e r res c r apons c k oep eatn i ns mal g u nl dand e r smoderat e r v i c e leo acrack d s , b oopeni t h t hng. e mAatx ismmal u ml crack w i dt h and t he s t res res s l evel l i mi t s i n t he rei n forcem en t can have a si si gni fi cant i mp ort an ce e.g e.g.. for durabi l i t y purpos es or for t he fat i gue of t he rei nfo rcem ent , [5]). [5]). In general, the ultimate load of a conven tionally reinf o r ce c e d S F R C m e m b e r c o r r e sp sp o n d s t o t h e s it i t u at a t io io n w h e n t he co nven t i onal rei nforce m ent yi yi elds elds . A t t hi s st st age age t he fibre contribution to the load carrying capacity can be significa signi ficant. nt. In b oth cases it is is therefo re essential essential for the as s umpt i ons adopt ed t o be real i s t i c and repres ent at i ve of t he act ual behavi our.

rack spacing spacing

In direct tension and in flexure, the observed crack s paci paci ng i n t he p res ence of fi bres is is les les s t han i n i den t i cal members without fibres [9, 22, 36]. Although formulat io i o n s h a v e b e e n p r o p o s e d t o d e t e r m i n e c r a c k s pa pa c in in g i n t he pres en ce of fibres fibres [22], furt he r res earch has has t o be carri ed o ut i n t hi s area before a genera l form ul at i on i s avai availl 272

R e i n f o rc r c e m e n t s t r a in in s n t h e c r a c k e d r eg eg i o n

A d o p t i n g E u r o c o d e r e p r e s e n t a t io io n o f t h e i n t e r a c t i o n b e t w e e n c o n c r e t e a n d r e i n f o r c e m e n t a l lo lo w s u s t o c o n s i d e r r e i n f o r c e d S F R C m e m b e r b e h a v i o u r as as a w e i g h t e d average between the cracked and uncracked regions. Therefore the strain in conventional reinforcement in t he crac ked zon e i s gi ven by:

w h e r e d is i s t h e p o s i t io io n o f th th e r e i n f o r c e m e n t w i t h r e s p e ct ct t o t he compres s i on face. Thi s as s umpt i on i s appl i cabl e w i t h t h e s i m p l i f ie ie d m e t h o d s p r e s e n t e d i n S e c t i o n 4 . 3 .

A v e r ag ag e c u r v a t u r e The average curvature can be determined according t o t h e r e c o m m e n d a t i o n [ 1 0 ] , a v e r ag ag i n g t h e c u r v a t u r e a t t he crack and b et w een cracks: cracks:

 

T

1 6 2 T DF

K m =(1- -4)1(

1 q-~K

c o n s t r u c t i o n . L i m i t i n g c r a c k o p e n i n g s a re re s u g g e s t e d i n [ 2] 2] . B o t h t h e c a se se o f s te te e l f i b r e r e i n f o r c e d c o n c r e t e w i t h and without conventional reinforcement are covered. C o d e v a l u e s m a y b e a d o p t e d . C r a c k o p e n i n g s fo fo l l o w dir ectly f or the cr os s - s e ctional analys analys is is des cr ibe d above.

51)

2

with:

-- 1 - [31~

52)

ltimate limit state

I n c a se s e o f c r os o s s s e c ti ti o n s w i t h o u t c o n v e n t i o n a l r e i n forcement the ultimate limit state in bending is deter-

w h e r e ~ ] a n d ~ 2 a r e f actor s d ef i ne d in [ 10], 10], ~ s r is the s tr tr es es s i n t h e r e i n f o r c e m e n t a t t h e c r a c k j u s t a f t e r c r a c k i n g whereas 2 i s t h e c u r r e n t r e i n f o r c e m e n t s tr tr es es s a t t h e c r a c k . C u r v a t u r e s ~ c1 c 1 a n d ~ :2 a re re c o m p u t e d i n t h e u n c r a c k e d a n d c r a c k e d r e g i o n s r e s p ec e c t iv i v e l y , u s in in g a n iter ative pr ocedur e, [ 20] . A l t e r n a t i v e l y , E q u a t i o n ( 2 6) 6 ) p r o p o s e d i n [ 7] 7 ] fo fo r SFRC members only, can also be adopted to represent the average curvature over the non-linear hinge for m e m b e r s c o n t a i n i n g c o n v e n t i o n a l r e in in f o r c e m e n t . T h i s approach was used by [2 ] and showed good agreement w i t h e x p e r i m e n t a l r es es u l ts ts .

m i n e d s i m p l y b y t h e u l t i m a t e l o a d c a r r y i n g c a p a ci c i ty ty c a l culated according to the abov e-men tioned methods. I n t h e c a se se o f u n d e r - r e i n f o r c e d c r o s s - se se c t i o n , t h e u l t i m a t e l o a d c a r r y i n g c a p a c i t y is is u s u a llll y re re a c h e d a t o n s e t o f r e i n f o r c e m e n t y i e ld ld i n g . B e y o n d t h a t p o i n t , t h e m e m b e r e x h i b i t s a s o f t e n i n g t h a t is is g o v e r n e d b y t h e a m o u n t o f reinforcem ent and the fibre concrete properties. At yieldi n g o f th th e r e i n f o r c e m e n t t h e c o n c r e t e i n c o m p r e s s io io n c a n be f ir s t as s um ed to beh ave linear elas elas tica ticall llyy w h ich enables t h e u s e o f th th e m e t h o d s p r e s e n t e d i n S e c t i o n 4 . 3 . T h e maximum stress level in the concrete can then be c h e c k e d . I f t h e a s s u m p t i o n i s s h o w n t o b e a p p r o p r i at at e , t h e u l t i m a t e s t r e n g t h o f t h e c r o ss s s s e c t io io n d e t e r m i n e d with the methods presented above is adequate. In cases w h e r e t h e a s s u m p t i o n o f a l in in e a r b e h a v i o u r i n c o m p r e s s i o n is is n o t v a li li d , t h e n o n - l i n e a r i t y o f t h e c o n c r e t e i n c o m p r e s s i o n s h o u l d b e t a k e n i n t o a c c o u n t a s s p ec ec i fi fi e d b y c o d e s f o r n o r m a l r e i n f o r c ed ed c o n c r e t e m e m b e r s .

4.5 Lim it states

e r v i c e a b i lil i t y l i m i t s a t e

The serviceability limit state defines a maximum c ra r a c k o p e n i n g d e p e n d i n g o n t h e e x p o s u r e c la la ss ss o f t h e

cra ck ; L -

7

J

-

5 . B E A M A N A L Y S I S: S: L O A D DEFLECTION BEHAVIOR

u N o n l i n ea ea r h i n g e

F i g . 15 15 - T h e p r i n c i p l e o f t h e n o n - l i n e a r h i n g e a n a l y s i s i n t h e c a s e o f a b e a m w i t h r e c t a n g u h r c ro ro s s= s= s ec ec titi o n n o c o n v e n t i o n a l r e i n f o r c e m e n t s u b j e c t e d t o t h r e e p o i n t b e n d i n g . T h e b e a m g e o m e t r y a n d l o a d i n g i s s h o w n o n t h e l e f t h a n d s i d e . O n t h e r ig ig h t h a n d s i d e i t is is sh s h o w n h o w t h e b e a m i s d i v i d e d i n to to a n f i d d l e s e ct ct i o n o f w i d t h s c o n t a i n i n g t h e n o n l i n e a r h i n g e a n d t h e r e st st o f t h e b e a m w h i c h i s a s s u m e d t o b e h a v e e l a s t i c a llll y .

L I

c r o s s - s e c ti ti o n w i t h d e p t h h , w i d t h t a n d span L. The deflection u is calculated as a s u m o f t w o t e rm rm s :

u = u e + uc

I

(53)

the elastic u e and the cracking uc related p art of the d e f l e c t i o n . T h i s s u p e r p o s i t i o n o f d e f o r m a t i o n s i s i ll ll u s tr ated in F ig. 16. Acc ording to chssical bea m th eory ur is related to the l o a d P a n d t h e b e a m s p a n L a s fo fo l l o w s :

total deformation ~176176176176176176176176176176176176176176176

e l a s ti ti c d e f o r m a t i o n .....

In this section the non-linear hinge is i n t r o d u c e d i n a s im im p l e b e a m w i t h o u t c o n ventional reinforcement, loaded in three point bending and experiencing failure in bending, s ee F ig. 15. C o n s i d e r a b e a m w i t h r e c ta ta n g u l a r

~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7. .6.1. 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 p 7 6L1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6

ue -

r i g id i d b o d y r o t a titi o n + -r

i

(54)

48EI

whe re E is Young 's mo dulu s of the SFR C-m aterial and 1 i s t h e m o m e n t o f i n e rt rt i a o f th t h e c ro r o s s s ec e c t io io n . T h e c r a c k uc

tecdr apcakr t aos f at hgee ndeerfal el icztei od np l a si tsi cc ahl icnugl aet eads m li inngg r et hl ae te s hoodwe lnabove. In the ca lculation of this part the two halves of the beam are both assumed to be rigid and rotate an

ic a n d c r a c k F i g . 16 16 - T h e p r i n c i p l e o f t h e s u p e r p o s i t i o n o f e l a s t ic i n g r e l a te te d d e f o r m a t i o n . T h e e l a st s t ic ic d e f o r m a t i o n i s c a l c u l a t e d a c c o r d i n g t o c l a s s i ca ca l b e a m t h e o r y w h i l e t h e c r a c k i n g r e l a t e d d e f o r m a t i o n i s c a l c u l a te te d f r o m t h e d e f o r m a t i o n o f t h e n o n l i n e a r h i n g e a s s u m i n g t h a t t h e r e st st o f t h e b e a m r o t a t e s a s s t i f f b o d i e s .

273

 

June2002 2002 M a t e r i a ls ls a n d S t r u c t u r e s /M /M a t 6 r i a u x e t C o n s t r u c titi o n s Vol. 35 June

angle ~: r

steel fibres I~d: ~o2

T h e c o n t r ib i b u t i o n s o f th t h e m e m b e r w i t h o u t s h ea ea r rei nfo rcem ent and t he s t i rrups rrups an d/ or i ncl i ned bars bars can be cal cul at ed accordi ng t o Eu roco de 2. S ee al so so [2].

wh ere is t h e a n g u l a r d e f o r m a t i o n o f t h e n o n - l i n e a r hinge at the o nset of cracking given by:

Pe-

12sM Eth 3

(56) --

w h i lhi e nge cp i spres th e cri th t o bed t a l a inng ut he ta l a r cros d e f os r-smect a t iioonal n o f anal t h e ys n oi sn. - The linear deflection due to the rigid-body rotation ~ may be written in the form: c

L

.,:

(57)

d

Wm

Fig. 17 - A s s u m e d c r a ck ck g e o m e t r y in a beam w it ithh c o n v e n t i o n a l longitudinal reinforcement longitudinalreinf orcement oaded o the ul ti m a te s he a r loading capacity,V R d3. 45~ a n d t h e d3. The crack s a s s u m e d t o e x t e n d under 45~ c r a c k o p e n i n g a t t h e re-bar is limited to w m.

C o n s i d e r i n g a r e c t a n g u l a r c r o ss s s - s ec ec t io io n w i t h w i d t h b , effect eff ectii ve dep t h d (di (di st st ance ance fro m t he t op o f t he bea m t o t he rei n forci n g bars bars)) an d i nn er l ever arm z = 0.9d, 0.9d, s ee F i g . 1 7 t he fi bre cont ri but i on V f,d f,d i s cal cul ated ated from t he de s i gn s t re s s -c ra c k ope ni ng re l a t i ons hi p Ow,d W) i n t he following way:

58)

In this case the elastic deflection can be obtained from E quat i ons (54) and (58). The entire load deflection diagram for a beam in t h r e e p o i n t b e n d i n g c a n b e d e t e r m i n e d b y p r e s c r ib ib i n g angul ar deform at i ons cp cp > rac ackk of t he no n-l i ne ar hi nge , c a l c u la la t in i n g th th e l o a d o n t h e b e a m f r o m E q u a t i o n ( 5 8 ) a n d the two parts of the deform ation from Equ ations (54) and (57) (57).. It foll foll ow s t hat t h e l oad w hi c h i ni t i at at es cracki ng c r a c k i s gi ven by: P ~ ~ c k f , 2h2 2h2----~ -~tt

z 45

In t he el as t i c cas e t he defl ect i on i s gi ven by u = u e. W h e n a c ra ra c k h as as d e v e l o p e d t h e d e f l e c t i o n is is g iv iv e n b y u to L and M by: = u e + u c. c. P i s rel at ed to p=4M(~) L

(60)

VRa3 = Vc,a + Vw,d + Vr,a

(55)

Vf, d

(61)

= bZ-~p,d Wm)

with: a w

(5 9 )

p, \

3L I n E q u a t i o n ( 5 8 ) a n y o f th t h e m o m e n t - r o t a t i o n r e la la t i ons hi ps des cri bed above can be appl ied. ied. F urt h erm ore , it was shown in [34] how the above analysis can be extended to cover the case of a beam with a notch. Alternative Alternat ive approaches based based on integration o f the cu r-

] ~

m]

fw m o w a U ) d u

Wm JO

,

62)

T h e q u a n t itit y - ~ p ,d called ed the m ean design resid, d W m ) is call ual s t res res s at t he crack w i d t h w m and repres ent s t he mean v a l u e o f t h e p o s t - c r a c k in i n g s t re re ss ss b e t w e e n z e r o a n d w m . A d e f i n i t i o n o f w m i s neces s ary t o quant i fy t he ul t i ma t e l oad -carry i ng capaci t y of t he b eam fai lili ng in in s hear.

vat ure along along t he l engt h o f t he bea m can be adopt ed, [20] [20]..

6 . S H E A R C A P A C I T Y : U L T I M A T E L I M I T S T AT AT E

T h e s h e a r c a p a c itit y o f F R C b e a m s w i t h c o n v e n t i o n a l longitudinal reinforcing bars has been analyzed extens i vel y i n t he l i t erat ure [7] by cons i deri ng t he fai l ure t o occur due to crack propagation along known planes. O nl y s uch cas es w i l l be cons i dered i n t hi s s ect i on s i nce t h e r e i s n o g e n e r a l ly ly a c c e p t e d m e t h o d f o r t h e d e t e r m i n a t i o n o f th t h e s h e a r c a p a c itit y o f F R C e l e m e n t s w i t h o u t c o n ventional reinforcement. The approach of [7] to calcul a te te t h e c o n t r i b u t i o n f r o m t h e f i b r e s is is d e s c ri ri b e d i n t h e fol l ow i ng. F o l l o w i n g E u r o c o d e 2 , [ 1 0 ] , th th e u l t i m a t e s h e a r lo lo a d c a r r y i n g c ap ap a c i ty ty V R d i s t ak ak e n t o b e t h e s u m o f th th e c o n t r ib ib u t i o n s o f th t h e m e m b e r w i t h o u t s h ea ea r r e i n f o r c e m e n t Vc, a, of t he s t i rrups and/ or i ncl i ned bars , Vw , a , a n d t h e 274

Et ri uCd ie i ebse acm a rsr,i erde i on uf ot r w rxi epse roi fmsete tne el et al l FsRt ud c eidt hwdi ti hf f ecroennvt egne toi omnea-l l o n g i t u d i n a l r e -b -b a r s , h a v e a n a l y z e d t h e o n s e t o f i n c l i n e d c ra ra c ks ks a n d t h e f o r m a t i o n o f c o n c r e t e s tr t r u ts ts i n c o m p r e s s i on [6]. A c cor di ng t o t he res ul ts ts , t he s paci ng of thes thes e c r a ck ck s is is r o u g h l y e q u a l t o t h e i n n e r l e v e r a rm rm o f t h e beam and the ultimate crack opening is proportional to the heigh t of the beam . Since the crack opening is con t r o l le le d b y t h e l o n g i t u d i n a l r e in i n f o r c e m e n t , i t is is p r o p o s e d t h a t t h e m a x i m u m c r a c k o p e n i n g b e t a k e n as as : w m = s,z

( 6 3) 3)

w h e r e 8 i s t h e s t ra ra in in o f t h e l o n g i t u d i n a l r e i n f o r c e m e n t . Since Vf,d Vf,d typical typically ly decreases wi th a n increase in the maximum ~rack opening, the fibre contribution should b e d e t e r m i n e d f o r a m a x i m u m a llll ow ow a b le le c r a c k w i d t h . I f the maximum strain in the longitudinal steel re-bar is t a k e n t o b e 1 , t h e n wm w m s houl d be t aken as 0.009d 0.009d.. In order to obtain an equivalence relationship between conventional transverse reinforcement and

 

TC 162-TD 162-TDF F

fibres, we eq uate the co ntrib ution of 17w d to V~, given in j~ tr Equation (61) to obtain an equivalent mean design

)

esidual stress, -S -Sp, p,aa w m

(64)

bz

This implies that the equivalent mean residual stress o f O--*p,d(Wm) would yield the same load-carrying capaccapacVw,d. d. ity as stir stirrups rups and/o r inclin ed bars bars giving rise to Vw, Furthermore, the equivalence in Equation (64) can be used to extend the definition of the the mi nim um shear stirrup stir rup reinforcem ent given given in the Euroco de 2 by: -~p,d + Ptfy,d >- 0-02/c

a

the shrinkage and the thermal strains are homogeneous over the thickness. The normal stresses are assumed to be con stant over the cross-sec tion of the slab. slab. If the impos ed strai strainn - shrinkage or thermal contra ction or a combin ation o f both - att attain ainss such a value that the normal stress equals the tensile strengthf of the slab material, then a crack is initiated. The objective of this example is is to dete rmine the crack opening as a function function of the the par ameter s in the d r op- consta nt str essess- crack crack opening relationship. Fig. 18 shows the slab together with the str stress ess distributio n in th e slab after it has has cracked. The shear reaction acting on the bo ttom fac facee of the slab agai against nst the long itudinal deformation of the slab, u(x), is modeled as a twoparameter linear function:

65)

where f, d is the design compressive compressive strength strength o f the concrete,f~,d crete, f~,d is the design y!eld streng th o f the. stirr stirrups ups an d 9t xs the area of stirrup stirrup reinfor cem ent per unit length.

7 CRACK WID THS IN SLA SLABS BS ON GRADE

9 (x) ='% + ~ u ( x )

where % and ~ are constants describing cohesion and hardening on the slab-sub base interface, respectively. The shear reaction always acts in the opposite direction ofu(x). The norm al stress stress o is given by:

UNDE R RESTRAI RESTRAINED NED SHRINKAGE AN D

(67)

o = ft + E du dx

TEMPERA TEMP ERATUR TURE E MOVE ME NT

Slabs on grade under restrained shrinkage and temperature movement is an important field of application for SFRC. The present section presents an analysis for the crack opening at crack formation in such structures. Simple design formulae based on the drop-constant stress-crack opening relationship Equation (8) are presented but o ther m ore complicated formulae formulae can readily readily be applied. The present example follows closely [27]. An infinitely long slab of thicknes thicknesss h and unit w idth, cast onto a sub-base of e.g. graded gravel, is considered. Th e in-plan e slab deformations are assumed to be governed only by shear stresses acting on the interface betw een the slab and the subsub-base base.. Since the slab is infinitely long it will be completely restricted against longitudinal deformations. Thus, shrinkage or thermal strains developing in the slab will give give rise rise to con stant no rma l stresses throughout the length of the slab, assuming that

(66)

wh ere E is the sti stiffn ffness ess of the slab material. The differential equation governing the behaviour after crack initiation is readily established and reads as follows for the part o f the slab to the left of the crack: d2u

~

dx 2-

68)

Eh

Since all deformations are negative, Equations (66) and (68) may be com bined to furnish: dZu

~

dx 2

~,, U-

t~ Eh

u