2001 Analysis and Design of FRP Externally-reinforced Concrete Beams Against Debonding Type Failures...

8/11/2019 2001 Analysis and Design of FRP Externally-reinforced Concrete Beams Against Debonding Type Failures by Maalej

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M a t e r i a ls a n d S t r u c t u r e s / M a t e r i au x e t C o n s t r u c t i o n s Vol. 34 August-September 001 pp 418 -425

A n a lys is an d d es ig n o f F R P ex te rn a l ly re in fo rced

c o n c r e te b e a m s a g a in s t d e b o n d i n g t y p e fa ilu r e s

M . M aa lej W . H . Goh and P . Pa ramasi vam

Departmen t of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore 117 576

Paper received:June 22, 200 0; Paperaccepted: M arch 1 , 2 00 1

A B S T R A C T R I~ S U M I~

Epoxy -bonding of FR P plates to the tens ile face of

R C beams has been shown to be an effective repair and

s t r eng then in g techn ique . H ow ever , loca l f a i lu r e by

d e b o n d i n g o r r i p p i n g o f c o n c r e t e c o v e r h as b e e n

reported in experiments to be a likely mode of failure

due to high interfacial shear and normal s tress concen-

trations, Predictive mod els for f inding the interracial

shear s t ress have been reviewed and evaluated us ing

experimental data reported in the li terature. Th e most

c r i t i c a l p a r a m e t e r s g o v e r n i n g t h e i n t e r r a c i a l s h e a r

s t rength and s t ress as determ ined by the models were

also examined. Th roug h unders tanding of the condi-

tions that result in debonding failure, a better approach

towards des igning FRP-plated RC beams agains t th is

m ode of failure might be achieved.

Le ren forcemen t externe d e lemen ts en b~ton arme a l ' a ide de

p laques syn thd t iques ret~rc s de f ib res (F RP ), s ' es t rdvd ld e

u n e t e c h n i q u e e f f ic a c e d e r a b i l i t a t i o n d e s s t r u c t u r e s .

Cepend an t , la rup ture loca le par d o l lemen t ou f i ssure du bd ton

a d td pr en t dan s des essa is comm e le mode de rup ture le p lus

f rd q u en t a ca u se d e l a f o r t e co n cen t ra t io n d es co n tra in te s d e

c i sa i l l em en t e t n o rm a les a u x ex t rd m i t d es p la q u e s . D e s

modules th riques visan t 21 trouve r les contraintes de cisaillement

on t d td exam in e t a tu en u t i t isan t des donn s exp imen-

tales rapport s clans la li tt a ture. Les param ~tres les plus cri-

tiques gou verna nt la contrainte de cisailleme nt et la rdsistance au

cisa i llemen t de l ' in ter face co l le-bd ton , comm e d d termind par les

modeles th r iques , on t auss i d td ex am in Pour comprendre les

( fre ts qu i r u l ten t de la rup ture par d o l lemen t , une m ei l leure

mdthode de concep t ion peu t ~ tre r l i s a f in d 'd l iminer ou de

retarder ce mo de de rupture.

1 . I N T R O D U C T I O N

Repair and s t rengthening of RC mem bers with EB -

FR P (Externally-Bonded Fibre Rein forced Polymer) has

evolved progressively over the past decade. For beam

mem bers, failure can occu r due to f lexural compression,

b e a m s h e a r , F R P r u p t u r e , o r F R P d e b o n d i n g .

D e bond ing - ty pe f a i lur es a re p reva len t in beam tes ts

reported in the li terature. Th e prevalence of & bo nd ing

failures amon g each of the oth er modes emphasizes the

need either for reliable means of preventing this type of

f a i lu r e o r f o r a p r a c t i c a l m e t h o d o f p r e d i c t i n g i t.

Attempts to address this need can be seen in recent pub -

lications whe re approximate analyses were used to com -

pute the shear and normal s tress concentrations in the

adhesive layer of FRP -plated RC beams. Th e recently

published wo rk was motivated b y observations that pre -

mature failures may occur because of shear and normal

stress concentrations at FR P cut -off points and at f lex-

ural cracks a long the beam , resul t ing in d ebond ing or

ripping of the c oncrete cover along the level of conv en-

t ional in ternal re inforceme nt . Th e recent ly proposed

methods to p red ic t and p reven t p r ematu re f a i lu r e o f

FRP-plated RC beams are timely; however, such meth-

ods need further testing and evaluations before the y can

be relied upon in practice.

For a given strengthe ning application, the prim ary

issue is to decide what type of FRP reinforcing system

and how muc h EB -FRP f lexural re inforcement should

be used. The ideal system would be one whe re the FRP

properties are fully utilized. Published data in the litera-

ture indicate that the efficienc y of FRP external rein-

f o r c e m e n t a n d t h e d u c t i l i t y o f F R P - p l a t e d b e a m s

decrease with increasing FR P axial r igidity (area times

elastic mo dulus) due to prem ature failure [1]. In this

paper , s ta te-of- the-ar t methods for the analys is and

design ofFR P-plated RC beams agains t &b ondin g type

failures are reviewed and evaluated using experimental

1359-5997/01 9 RILE M 41 8


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Maalej Goh Paramasivam

d a t a re p o r t e d i n t h e l i te r a tu r e . A n i m p o r t a n t g o a l o f t h is

s t u d y is t h e n t o u s e t h e p r o p o s e d m o d e l s t o d e r i v e r e la -

t io n s h i p s b e t w e e n F R P e f f i c ie n c y a n d F R P a x ia l r i g i d i ty

t h a t d e s i g n e r s c a n u s e t o p r e d i c t f a i l u r e m o d e a n d

a c h ie v e a n o p p o r t u n e b a l a n c e b e t w e e n s t r en g t h g a i n a n d

def l ec t ion capaci ty .

2 . R E V IE W O F P R E D I C T IN G M O D E L S

A n um be r o f publ i shed a r ti c le s dea l t w i th the top ic of

p r e d i c t in g t h e f ai lu r e m o d e o f c o n c r e t e b e a m s s t r e n g t h -

e n e d i n f l e x u r e w i t h e x t e r n a l l y - b o n d e d r e in f o r c e m e n t [ 2 -

8 ] . A m o n g t h e s tu d i e s t h a t f o c u s e d o n t h e d e b o n d i n g

m od e of fa i lu re , Robe r t s ' s tudy [2] was the f i rs t t o provide

specific analyt ical equat ions , w hic h m ay be used to pre dict

bonding fa i lu re or des ign aga ins t i t . Spec i f i ca l ly , t he

a b o v e - r e fe r e n c e d s t u d y le d t o t h e d e v e l o p m e n t o f a m o d e l

f o r p r e d i c t i n g t h e s h e a r a n d n o r m a l s tr e ss e s at t h e

FRP/concre te in t e r face . Thi s mode l i s cons ide red in t i f f s

p a p e r f o r t h e p u r p o s e o f r e v i e w a n d e v a l u a t io n u s i n g

exper im enta l da ta repo r t ed in the l i te rature .

R o b e r t s ' m o d e l [ 2 ] w a s o r i g i n a ll y p r o p o s e d f o r t h e

ana lys is o f s t ee l -p la t ed RC beams . Th e ana lys is was pre -

sen ted in 3 s tages . In the f i rs t s tage, s tresses w ere d eter -

m i n e d a s su m i n g f u ll y co m p o s i t e a c ti o n b e t w e e n t h e R C

b e a m a n d t h e a d h e s i v e - b o n d e d st ee l p la te . I n t h e s e c o n d

and th i rd s t ages , t he ana lys i s was modi f i ed to t ake in to

account the ac tua l boundary condi t ions a t t he s t ee l p l a t e

c u r t a i lm e n t . T h e c o m p l e t e so l u t i o n w a s t h e n o b t a i n e d

by superpos i t ion . In th i s mo de l , a c racked sec t ion t rans -

f o r m e d i n t o a s t e e l p l a t e e q u i v a l e n t w a s u s e d i n t h e

a n al ys is . T h e g o v e r n i n g e q u a t i o n s f o r th e s h e a r a n d n o r -

m al s t res s d i s tr ibu t ions a re g iven by:

F F x ) , , 1

____1- T - - b p d p ( h p - h )

( x ) = b a [ + ~ t { _ t l0 sin h ~ x x~ t lo c ~ 1 7 6 t la c o s h ctx ~

sinh~ta j]

1 )

2 )

w h e r e

] 0 . 5

K ~

0 t = E p b p d p

3 )

_ [

K n ] 0 . 2 5

(4)

t l o = _ M ~ b p d p ( h p - h )

5 )

t l a = ~~ - a b p d p ( h p - h )

(6)

b a

K s = O a d--

K n = E b a

a d a

( E p I p ~

m 2 0 = M ~ I p + E c I c )

(7)

8 )

(9)

( E p l p ] + ( . t l 0 + ~ 2 0 ) b a d p / 2

f2 0 = F 0 / E p Ip + E c I c

l O )

1 : 1 0

1 [ F o

= b p d d h p - h

z2~ = b-~ [ct{t 1~c ~ a a - t l a } ]

i n h c t a

a n d

a =

b c, bp, b a =

d c , d[ , da =

Eo f fp, Ea

F 0 F a =

f20 =

h =

h P

Io Ip

=

K s

M 0 , M a

m20 =

t l 0 , t l a =

X =

( ~ =

y =

o x )

=

x ) =

~ 1 0 , C 2 0 =

(11)

1 2 )

Len gth of s t eal p l a t e

W idth of concre te , s t eel p la t e , adhes ive

De pth of concre te , s t eel p la t e , adhes ive

E l a s ti c m o d u l u s o f c o n c r e t e , s t e e l p l a t e ,

adhesive

Glob al shear force a t x = 0, x = a

S h e a r f o rc e i n p l a t e a t s o l u t io n d e v e l o p m e n t

s tage 2 (x = 0)

S h e a r m o d u l u s o f a d h e si v e

D e p t h o f n e u t r a l ax is c o m p u t e d b a s e d o n

cracked sect ion analys is

Effect ive de pth o f steel pla te

S e c o n d m o m e n t o f a re a o f th e t r an s f o rm e d

e q u i v a l e n t s t e e l s e c t i o n a b o u t t h e n e u t r a l

ax is based o n c racked sec t ion ana lys is

S e c o n d m o m e n t o f ar ea a b o u t i n d i v id u a l

cent ro id for con cre te , s tee l p la t e

Adh es ive nor m al s t if fnes s pe r u n i t l eng th

Adh es ive shea r s ti ffness pe r u n i t l eng th

G l o b a l b e n d i n g m o m e n t a t x - - 0 , x = a

B e n d i n g m o m e n t i n s t e e l p l a t e a t s o l u t i o n

dev e lop m ent s tage 2 (x = 0)

Axia l force in s t ee l p l a t e a t so lu t ion deve l -

op m en t s t age 1 a t x = 0 , x =a

D i s t a n c e a l o n g s t e e l p l a t e m e a s u r e d f r o m

p l a te c u t - o f f

C o e f f i c i e n t u s e d i n ~c ( x ) a n d d e f i n e d b y

E q u a t i o n ( 3 )

C o e f f i c i e n t u s e d i n o ( x ) a n d d e f i n e d b y

Equa t ion (4)

N orm al s tre ss a t a d i s t ance x f rom pla t e cu t -

o f f

In te r fac ia l shea r s t re s s a t a d i s t ance x f rom

p l a te c u t o f f

Shear s t re s s in adhesive a t so lu t ion de ve lop -

m en t s tage 1, 2 (x = 0)

R o b e r t s [ 2 ] c o m p a r e d t h e s h e a r s t r e s s d i s t r i b u t i o n

o b t a i n e d f r o m E q u a t i o n ( 1 ) w i t h a m o r e r i g o r o u s s o l u -

4 1 9


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Materials and Structures/Mat~riaux et Constructions Vol. 34 August-September 2001

Fig. 1 - Rei nforced concre te beam wi th external ly-bonded F RP show ing impor tant parameters used in

Rober ts model .

Fig . 2 - Resul ts predic ted by Ro ber ts for beam C wi t h load P/2 =

100kN.

t ion based on partial interaction theory [8] as well as

with experimental results presented by Jones e t al [4]. It

w a s c o n c l u d e d t h a t t h e a b o v e s o l u t i o n ( g i v e n b y

Equation (1)) underestimated the m agnitude of the stress

conc en t r a t ion by up to 30 , due p r im ar i ly to the

approximations made during the first stage of the solu-

tion. Robe rts [2] prop osed a corre ction by replacing M 0

( the va lue o f the g loba l m om en t a t x = 0 us ed in

E qua t ions (5 ) and (9 ) ) by a m od i f i ed m om ent M*,

whi ch is the value of the global mom en t at x = (dc+dp)/2

from the end o f the steel plate (see Fig. 1). This co rrec-

t ion resul ted in sat is factory correlat ion between the

mode l predict ion and both the more r igorous solution

based on partial interaction theory [8] and the available

test data [4]. Th e resulting mode l is referred to in this

paper as Rob erts' revised model.

W h e n t h e d e p t h o f n e u t r a l a x is h, t h e s e c o n d

moment of area of the equivalent steel section I, and the

second mom en t of area of the concrete about its individ-

u a l c e n t r o i d I c a re c o m p u t e d o n t h e b a s is o f a n

uncracked concrete section, Equations (1)-(12) lead to

Roberts ' uncracked section model, the results of which

will briefly be touche d upo n in this paper.

Comparisons were

initially made between

the results predicted by

the above-referenced

models for a RC beam

with reported proper-

t i es o f the concre te ,

s te e l r e i n f o r c e m e n t ,

FR P an d adhesive used

[9]. All mod els pre-

dicted that stress con-

centrations are rapidly

reduced as the distance

from the plate c utoff is

increased. However,

the r es u l t s f rom the

models differed in the

following:

(1) As indicated in

Fig. 2, Roberts ' revised model predicted shear stress

conce ntrations at the plate cuto ff that were significantly

higher than those predicted by Roberts ' original model.

This is expected as the end moment used in Rober ts '

revised model was a corrected moment (M*) at a dis-

tance of approximately half-beam depth from the plate

cut off (see

Fig. 1 ,

instead of the global moment at the

plate cuto ff itself(M0). Th e level of significance of using

this corrected moment would depend on the depth of

the beam and plate used. Deep beams and plates would

require a larger correction, thus making the corrected

and u ncorrected mom ents significantly more different.

(2) When an uncracked section is transformed into a

plate equivalent, and a plate end m om en t M* was used in

Roberts' model, the predicted results were significantly

lower than those predicted by both Rober ts ' or ig inal

model and Roberts' revised model (where a cracked sec-

tion transform ed into a plate equivalent was assumed).

3 PARAMETERS AFFECTING INTERFACIAL

SHEAR STRESS

Experimental data on thirty FRP-plated RC beams

with reported failure modes and FRP strains at failure

were gathered from an experimental database compiled

recently by Bonacc i and Maalej [1]. Th e original data

for the beams were reporte d in References [9-18]. Th e

beams were analyzed using the Roberts' revised model.

Properties of the adhesives were assumed if they had not

been reported. In addition to using the mod el for pre-

dicting the interracial shear stress, flexural analysis was

performed at the same time to determine the FRP strain

at the critical beam section for bending and to mon itor

the strain in the concrete at the extreme compression

fibre. This enabled the determina tion of whe ther the

beams would fail by flexural compression of concrete or

tensile rupture o f FRP prior to debonding.

Rober ts ' revised model predicted high in ter facial

shear stresses (compared to Roberts' original model or

Roberts ' uncracked section model) due to its adoption of

4 2


4/8

Maa lej , Goh, Paramasivam

Table 1 - Propert ies of FRP and adhe sive provided b y manufacturers

Property ProprietarySystem 1 Proprietary ystem2

Elasticmodulusof adhesive MPa) 12800 1470

Shearmoduluso f dhes i v e M P a ) 2 0 0 0 5 6 5

Adhesive ayer hickness mm ) 2 0.636

Elasticmoduluso f FRP M P a ) 1 6 5 , 2 1 0 ,3 0 0 2 3 0

Tensilerupturestrengtho f F RP M P a ) 2800 , 2400 , 1300 3400

Tensile upturestrainof FRP mm /mm ) 0.017, 0 .012, 0 .0045 0.014

a c ra c k e d s e c ti o n a n d u s e o f a c o r r e c t e d e n d m o m e n t

M *. Thi s sugges ts tha t the m od e o f fa i lure i s un l ike ly to

b e f l e x u r a l c o m p r e s s i o n o r F R P r u p t u r e a s t h e h i g h

in te r fac ia l shea r s t res s is l i ke ly to re su l t i n deb on din g fa i l-

u r e . I t i s n o t e d t h a t w h e n R o b e r t s ' r e v is e d m o d e l p r e -

d i c t i o n o f d e b o n d i n g f a il u r e c o r r e s p o n d e d w e l l w i t h t h e

a c t u a l f a i l u r e m o d e , t h e f a i l u r e l o a d s p r e d i c t e d b y t h e

s a m e m o d e l w e r e s i g n i f i c a n t l y l o w e r t h a n t h o s e

r e p o r t e d . I n c as es w h e r e b o t h p r e d i c t e d f a il u re m o d e s

a n d a c t ua l fa i lu r e m o d e s w e r e b y c o n c r e t e c o m p r e s s io n

o r F R P r u p t u r e , p r e d i c t e d and ac tua l fa ilure loads we re

fou nd to be in c lose agreem ent . In these cases , p red ic t ed

i n t e r f a c i a l s h e a r s t r e s s e s w e r e n o t h i g h

e n o u g h t o c a u s e b o n d i n g - t y p e f ai lu r es .

As such , the fa i lure loads and modes f rom

flexural analys is are expected to be c lose to

t h o s e f r o m e x p e r i m e n t a l d a ta .

U s i n g R o b e r t s ' r e v i s e d m o d e l , a p a r a -

m e t r i c s t u d y w a s c o n d u c t e d t o d e t e r m i n e

t h e m o s t c r i t i c a l p a r a m e t e r s c o n t r o l l i n g

in te r rac ia l shea r s t re s ses wi th in the beam.

T h e f o l l o w i n g p a r a m e t e r s w e r e f o u n d t o

b e t h e m o s t i m p o r t a n t : ( 1 ) F R P p l a t e

th icknes s , (2) FR P m odu lus , (3) adhesive shea r m od ulus

a n d ( 4 ) a d h e s iv e th i c k n e ss . S t u d i es w e r e c o n d u c t e d t o

f i n d o u t h o w e a c h o f t h e se p a r a m e t e r s a ff ec ts t h e F R P

e f f i c i e n c y ( d e f i n e d as t h e F R P s t ra i n a t a c t u a l

f a i l u r e / F R P r u p t u r e s tr a in ) a n d t h e m o d e o f f a i lu r e .

Factors affect ing the interfacial shear s t rength were a lso

e x a m i n e d b a s e d o n a m o d e l p r o p o s e d b y C h a a l al

e t a l

{6

3 .1 E f f e c t o f F R P t h i c k n e s s o n F R P e f f i c ie n c y

a t d e b o n d i n g

1.0

0.8

0.6

0.4

0.2

0.0

0.0

4 ~ o Exp. Data No Anch. )

~i , x \ [] E x p. D a t a ( A n c h . )

i~ i ~ - * - S M 2 CFRP, Type 1 )

~o ~

~. ~ . ~ -~- B2 CFR P, Type 1)

i i

0.5 1.0 1.5 2.0

ApEp/A~Es

Fig. 3 - Com parison between experimental da ta an d r e su l t s pre-

d i c t e d b y R o b e r t s r e v i s ed m o d e l . T h e te r m s

C, P2, SM2 and B2

r e f er t o F R P - s t r e n g t h e n e d b e a m s t e s t e d i n s t u d i e s [9], [10 ], [11],

an d [ 12] , r e sp e c t i ve l y . GF R P r e fe r s t o gl as s f i b r e reinforced poly-

mer , a n d Type 1 r e fe rs t o t h e a d h e s i v e o f p r o p r i e ta r y s y s t e m 1

Table 1).

1.0

C

C

~.4

C.2

.0

o Exp . Da t a No Anc h . )

o Exp. Data Anch. )

-_ . SM2 CFR P, Typ e 2)

o , - SM2 Propr ie ty Sys tem 2)

~ , t~ - - S M 2 C F R P , T y p e 1 )

q

0.0 0.5 1.0 1.5 2.0

A p Ep / A sE~

Fig. 4 - Effec t of a d h e s i v e p r o p e r t i e s o n F R P e f f i c i e n c y a s p r e -

d i c t e d b y R o b e r t s r e v is e d m o d e l .

O n e o f th e m o s t c r i t ic a l p a r a m e t e r s a f f e c ti n g t h e

i n t e r f a c i a l s h e a r s t r e s s i s t h e F R P p l a t e t h i c k n e s s .

E x p e r i m e n t a l d a ta h a d b e e n g a t h e r e d a n d c o m p i l e d

which re f l ec t the FRP s t ra in a t t he beam c r i t i ca l s ec t ion

for bend ing a t fa i lure . For p rogres s ive ly inc reas ing loads ,

R o b e r t s ' r e v i s e d m o d e l w a s u s e d t o p r e d i c t t h e i n t e r f a -

c i al shea r s tre s s a t the p la t e cu tof f . Th e co r respo ndin g

FRP s t ra in a t t he c r i t i ca l beam sec t ion was a l so ca lcu-

l a t ed f rom sec t iona l ana lys is . In the absence o f repo r t ed

v a l u e s o f a d h e s i v e e l a s ti c a n d s h e a r m o d u l i , a d h e s i v e

proper t i e s f rom two prop r i e t a ry sys tems (adhesive type 1

and adhes ive type 2) were a s sumed for beams re inforced

w i t h C F R P ( C a r b o n F i b re R e i n f o r c e d P o l y m e r ) .

U n l e s s s t a te d o t h e r w i s e , t h e a d h e s i v e u s e d i n p r o p r i e t a r y

s y s te m 1 w a s a d o p t e d f o r C F R P - p l a t e d b e a m s b y

defaul t. Th e prop er t i e s of these two propr i e t a rY systems

and the i r cor respo ndin g adhes ives a re shown in Table 1 .

R e p o r t e d e x p e r i m e n t a l d a t a a n d r e s u l t s f r o m t h e

above analysis (solids l ines) are show n in Figs. 3-4. N ot e

t h a t t h e t h e o r e t ic a l l y - p r e d i c te d c u r v e s h a v e b e e n g e n e r -

a t e d a s s u m i n g t h a t b o n d i n g f a il ur es ta k e p l a c e w h e n

t h e m a x i m u m s h e ar st re ss at th e F R P c u t - o f f p o i n t

e x c e e d s t h e i n t e r f a c i a l s h e a r s t r e n g t h ( t o b e d i s c u s s e d

l at er ). T h e e x p e r i m e n t a l d a t a s h o w a g e n e r a l t r e n d o f

dec reas ing FRP e f f i c i ency wi th inc reas ing FRP re la t ive

axia l r ig id i ty (A E/AsE s ) , w he re A , A s , E and E~ a re

. P P . P .P

t h e c r o s s - s ec t io n a l a re a s a n d e l a st ic m o d u h o f t h e F R P

and lon gi tu din al s teel, respectively. Th is is exp ecte d as

i n t e r r a c i a l s h e a r s t r e s s c o n c e n t r a t i o n i n c r e a s e s w i t h

i n c r e a si n g t h i c k n e ss o f t h e F R P p l a t e . A t t h e o n s e t o f

d e b o n d i n g , t h e s t r e s s i n t h e F R P p l a t e a t t h e c r i t i c a l

b e a m - s e c t i o n f o r b e n d i n g w o u l d t h u s d e c r e a s e w i t h

i n c r ea s i n g F R P t h ic k n e s s. E x p e r i m e n t a l d a t a r e v i e w e d

in th i s s tudy showed tha t measured FRP s t ra ins a t c r i t i -

c a l b e a m s e c ti o n s f o r b e n d i n g w e r e g e n e r a l ly h i g h e r f o r

a n c h o r e d b e a m s t h a n f o r t h o s e n o n - a n c h o r e d . T h i s w a s

4 2 1


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Materials and Structures/M ateriaux et Constructions Vol 34 August-September 0 0 1

b e c a u s e s t re ss c o n c e n t r a t i o n w a s r e d u c e d a t t h e p o i n t s o f

a n c h o r a g e , t h e r e b y d e l a y i n g t he o n s e t o f d e b o n d i n g .

T h i s a l l o w e d t h e F R P p l a t e to d e v e l o p a s ig n i f i c a n t p a r t

o f i t s te n s i le r u p t u r e s t ra i n p r i o r t o d e b o n d i n g , l e a d i n g t o

h i g h e r F R P e f f i c ie n c y . F r o m t h e e x p e r i m e n t a l t r e n d , it

a p p e a r s t h a t l i m i t i n g t h e F R P t h i c k n e s s , a n d h e n c e t h e

r e la t iv e ax ia l r ig id i ty , w o u ld in c r ease th e FRP e f f ic ien cy

a t d e b o n d i n g . T h i s w o u l d , h o w e v e r , r e d u c e th e c r os s

s e c t io n a l a re a o f t h e F R P r e q u i r e d f o r f l e x u ra l s t r e n g t h -

e n i n g . T h e w i d t h o f th e F R P p l at e s h o u ld , t h er e fo r e , b e

a s w i d e a s p o s si b l e t o p r o v i d e t h e n e c e s s a r y F R P c r o s s -

sec t io n a l a r ea r eq u i r e d f o r f l ex u r a l s t r en g th .

F r o m F i g . 3 , R o b e r t s ' r e v i s e d m o d e l a p p e a r s t o p r e -

d i c t r e su l t s t h a t a re g e n e r a l l y i n c l o se a g r e e m e n t w i t h t h e

a c t u a l t r e n d s e t b y t h e e x p e r i m e n t a l d a t a . A t l o w r e l a ti v e

a x ia l r i g i d it y (A E / A s E s ) , F R P & b o n d i n g t a k e s p l a c e

p

a f te r t h e F R P ~ v e l o p s a s ig n i f ic a n t p a rt o f it s te n s i le

r u p t u r e s t r ai n , l e a d i n g t o a h i g h F R P e f f i c i e n c y .

H o w e v e r , t h e f a i l u r e l o a d i s g e n e r a l l y n o t h i g h e n o u g h

t o c a u s e t h e b e a m t o f a i l b y f l e x u r a l c o m p r e s s i o n f i r s t .

T h e b e a m s r e in f o r ce d w i t h C F R P a d o p t i n g t h e a d h e si v e

p r o p e r t i e s o f p r o p r i e t a r y s y s t e m 1 ( a d h e s iv e t y p e 1 )

r e s u lt e d i n l o w e r F R P e f fi c ie n c ie s a t & b o n d i n g f a il u re

c o m p a r e d t o th e a c t ua l e x p e r i m e n t a l d at a . T h i s c o u l d b e

d u e t o t h e u s e o f i n a p p r o p r i a t e v a l u e s f o r t h e e l a st ic a n d

s h e a r m o d u l i o f t h e a d h e s i v e , w h i c h a p p e a r e d t o b e v e r y

s t i f f( E a = 1 2 8 0 0 M P a , G a = 2 0 0 0 M P a ) .

A c o m p a r i s o n w a s m a d e f o r b e a m S M 2 t e s t e d b y

A r d u i n i a n d N a n n i [ 1 1 ] f o r w h i c h a l e s s- s ti f f a d h e s iv e

( a d h e s i v e t y p e 2 ) u s e d i n t h e p r o p r i e t a r y s y s t e m 2 w a s

ad o p te d (see F ig . 4 ) . I t w as f o u n d th a t th e in te r f ac ia l

s h e a r st re ss i n t h e b e a m w a s l o w e r e d f o r t h e s a m e l o a d .

T h i s a l l o w e d t h e F R P p l a t e a t t h e c r i t i c a l s e c t i o n f o r

b e n d i n g t o d e v e l o p h i g h e r s tr es se s a t d e b o n d i n g , l e a d i n g

t o a n i n c re a s e d F R P e f fi c ie n c y . T h e a c t u a l F R P e f f i-

c ien c y o f 0 .4 0 a t a r e la tiv e ax ia l r ig id i ty o f 0 .6 0 f o r b ea m

S M 2 w a s s ti ll h ig h e r t h a n t h e F R P e f fi c i en c y c o m p u t e d

w h e n R o b e r t s ' r e v is e d m o d e l w a s u s e d . T h i s s u g g e st s

th a t th e ac tu a l ad h es iv e u sed in SM2 i s p r o b ab ly le s s s t i f f

t h a n b o t h a d h e s iv e s u s e d i n p r o p r i e t a r y s y s t em s 1 a n d 2 .

I t c o u l d t h u s b e s e e n t h a t t h e p r o p e r t i e s o f t h e a d h e s i v e

p l a y a n i m p o r t a n t r o le i n d e t e r m i n i n g t h e i n te r fa c i a l

s h e a r s tr e ss e s, a n d u l t i m a t e l y , t h e F R P e f f i c i e n c y a n d

f a il u re lo a d . I n w h a t f o l lo w s , R o b e r t s ' r e v i se d m o d e l

w a s u s e d a g a i n t o e v a l u a te t h e e f f e c t o f o t h e r p a r a m e t e r s

o n th e in te r f ac ia l sh ea r s t re s s an d F R P e f f ic ien cy .

3 2 Ef fec t of adhes ive shear mod ulus on FRP

ef f ic iency a t debo nding

T w o a d h e s i v e p a r a m e t e r s , t h e s h e a r m o d u l u s a n d

t h i c k n e s s , w e r e f o u n d t o a f f e c t t h e F R P e f f i c i e n c y a t

d eb o n d in g f a ilu re s ign i fican d y . T h e p r o p er t ie s o f th e FR P

s y s te m u s e d i n b e a m C t e s te d b y S a a d a t m a n e s h a n d E h s a n i

[ 9] w er e ad o p ted f o r an a ly s is in th i s case . T h e sh ear m o d u -

lu s o f th e ad h es iv e w as f i r s t v a ried an d i ts e f f ec t o n FR P

e f f ic i e n c y a t d e b o n d i n g w a s s t ud i e d . A s c a n b e s e e n f r o m

F i g. 5 , t h e F R P e f f ic i e n c y w a s r e d u c e d f r o m a b o u t 8 5 % t o

l es s t h a n 7 0 % w h e n t h e s h ea r m o d u l u s o f t h e a d h es iv e w a s

1 .0

0 . 8

.~ 0 .6

o 0 . 4

r~

0 . 2

0 . 0

2 4 0 2 6 0 2 8 0 3 0 0 3 2 0 3 4 0

A d h e s i v e s he a r m o d u l u s G a M P a )

F i g. 5 V a r i a t i o n o f F R P e f f i c ie n c y w i t h a d h es i v e s h ea r m o d u l u s .

in c r eased f r o m 2 4 0 M Pa to 3 2 0 M Pa . Th is su gg es ts th a t a

less-s t i f f adhesive sh ould be u sed to redu ce th e in ter racial

sh ear s tr e sses th a t w o u ld lead to d eb o n d in g . H o w ev er , i t

sh o u ld b e r ea lized th a t i t m ay n o t b e p o ssib le to in d e p en -

d e n t l y c h a n g e t h e s h e a r m o d u l u s o f t h e a d h e s i ve w i t h o u t

c h a n g i n g o t h e r a d h e s i v e - r e l a t e d p r o p e r t ie s s u c h a s t h e

in te r fac ia l sh ea r s t r en g th . Fo r th e p u r p o se o f d e te r m in in g

t h e s h e a r m o d u l u s o f t h e a d h e si v e , t h e S t a n d a r d T e s t

M e t h o d f or S h e a r S tr e n g t h a n d S he a r M o d u l u s o f

S t r u ct u r al A d h e s i ve s d o c u m e n t e d i n A S T M E 2 2 9 - 9 7

[ 1 9] , co u ld b e p e r f o r m ed .

3 3 Ef fec t of a dhes ive th ickness on FRP

ef fic iency a t debon ding

T h e o t h e r a d h e s i v e p a r a m e t e r t h a t a f f e c t s t h e F R P

e f f i c ie n c y at d e b o n d i n g is t h e t h i c k n e s s . P a r a m e t e r s f o r

b e a m C w e r e o n c e a g a in a d o p t e d . A l l p a r a m e t e r s

r e m a i n e d u n c h a n g e d w h i l e t h e a d h e s i v e t h i c k n e s s w a s

v a r i e d. T h e F R P e f f i c i e n c y w a s f o u n d t o in c r e a se s ig -

n i f ican t ly w i th in c r eas in g ad h es iv e th ick n e ss ( see F ig . 6 ).

T h i s i s d u e t o a s i g n i f i c a n t r e d u c t i o n i n i n t e rf a c i al s h e a r

s tr es s c o m b i n e d w i t h a s m a l l i n c re a s e i n i n t e rf a c ia l s h e a r

s t r e n g t h ( a s p r e d i c t e d b y C h a a l a l

e t a l

[ 6 ] . S e e n e x t s e c -

t i o n ) . I n g e n e r a l , t h e r e q u i r e d t h i c k n e s s o f a d h e s i v e

d e p e n d s o n t h e t h i c k n e s s o f t h e F R P p l a te , w i t h a

t h i c k e r p l a t e r e q u i r i n g a t h i c k e r l a y e r o f a d h e s i v e t o b e

1.0

s

g

[..r.

0 . 8

0 .6

0 .4

0 . 2

0 . 0

0 1 2 3 4 5

A d h e s i v e l a y e r t h i c k n e s s d a m m )

F i g. 6 V a r i a t i o n o f F R P e f f i c i e nc y w i t h a d h es i v e l a y er t h i c k n e ss .

4


6/8

M a a l e j G o h P a r a m a s i v a m

a p p l i e d f o r f u l l c o n t a c t b e t w e e n t h e F R P p l a t e a n d t h e

con cre te sur face . Ex per im enta l da ta , how ever , sugges t

t h a t t h e b o n d s t r e n g t h d e c r e a s e s w i t h i n c r e a s i n g g l u e -

l ine th icknes s [20] . Ev en tho ugh a th i cke r adhes ive l aye r

a p p e a r e d t o b e c o n d u c i v e i n r e d u c i n g i n t e r f a c i a l s h e a r

s t r e s s c o n c e n t r a t i o n s a c c o r d i n g t o R o b e r t s ' r e v i s e d

m o d e l , t h e n e e d t o c o n t r o l t h e a d h e s iv e th i c k n e s s w o u l d

s t i l l e x i s t d u e t o p o t e n t i a l l y w e a k e r b o n d s t r e n g t h s f o r

thick er adhes ive layers .

F rom the pa ram et r i c s tudies , i t was obse rved tha t the

mos t c r i t i ca l pa ramete rs cont ro l l ing the in t e r fac ia l shea r

s tre sses a re the F R P plat e th i cknes s, FR P mo dulus , adh e-

s ive shear modu lus and the adhesive th i cknes s. By c om -

par ing the t rends s e t by the exper im enta l da ta and resu l ts

genera ted by the predic t ive mod e l , i t appears tha t Rob er t s '

r ev is ed m o d e l c o u l d i n d e e d b e u s e d t o p r e d i c t d e b o n d i n g

type fa ilures . An exper im enta l program , howev er , should

b e c o n d u c t e d t o d e t e r m i n e t h e e la st ic a n d s h e ar m o d u l i o f

the adhesive as these pa ramete rs p lay an im por tan t ro le in

det erm inin g the interfacial shear s t resses .

4 PARAMETERS AFFEC TING INTERFAC IAL

SHEAR STRENGTH

Th e adhes ive in t e r fac ia l shea r s t rength ('Cu) was co m -

p u t e d b a s e d o n a n e x p r e s s io n g i v e n b y C h a a l a l e t a l [6] .

T h e p r o p o s e d e x p r e ss i o n to o k i n t o a c c o u n t th e e f f e ct o f

n o r m a l s t r e s s c o n c e n t r a t i o n a t t h e p l a t e c u r t a i l m e n t a s

g i v e n b y R o b e r t s ' m o d e l :

5.4

Zu - 1+ ' r tan 33 ~ (13)

wh ere i s 3 ' g iven by E qua t ion (4).

For a g iven adhes ive , factors tha t in f lue nce "~u inc lud e

the adhes ive th i cknes s , t he FRP pla t e th i cknes s and the

F R P e la st ic m o d u l u s . D e b o n d i n g o c c u r s w h e n t h e

i n t e r f a c i a l s h e a r s t r e s s e x c e e d s t h e i n t e r f a c i a l s h e a r

s t re n g t h . H o w e v e r , w h e n t h e F R P p l a te t h ic k n e s s i s

small , o the r m ode s o f fa i lure ( such as con cre te f l exura l

compres s ion and FRP rupture ) a re a l so pos s ib le be fore

the in t e r fac ia l shea r s t re s s can excee d the in t e r fac ia l shea r

s t re n g t h . A s c a n b e s e e n f r o m F i g . 7 b a s e d o n b e a m P 2

tes t ed by Shar i f

e t a l

[10], t he in t e r fac ia l shea r s t rength

drops wi th inc reas ing FR P pla t e th icknes s . Th e in te r fa -

c i a l s h e a r s t r e s s , h o w e v e r , i n c r e a s e s a s t h e F R P p l a t e

t h i c k n e s s i s i n c r e as e d . T h e r a n g e o f F R P t h i c k n e s s a t

w h i c h & b o n d i n g m i g h t o c c u r w a s f o u n d t o v a ry w i d e ly

d e p e n d i n g o n t h e t h i c k n e ss a n d t h e m e c h a n i c a l p r o p e r -

t ie s of the F R P and adhes ive used .

F igs. 7 -8 show the e f fec t of us ing a s t if fer FR P pla t e

o n t h e F R P t h i c k n e s s a t d e b o n d i n g . T h e F R P p l a t e

t h i ck n e s s at d e b o n d i n g d e c r e a s ed f r o m 1 .8 m m t o a b o u t

0 . 3 7 m m w h e n t h e F R P e l a s t i c m o d u l u s w a s i n c r e a s e d

f r o m 1 4 .9 G P a t o 4 0 G P a . F o r an F R P p l a te o f 0 .5 m m

t h i c k n e s s , w h e n E _ i s i n c r e a s e d f r o m 1 4 . 9 G P a t o 4 0

9 . P .

GPa, the in t e r rac ia l shea r s tre s s inc reased f rom 2 .0 M Pa

to 5 .2 M Pa , w hi l e the in t e r rac ia l shea r s trength inc reased

f r o m 4 . 2 M P a t o 4 .5 M P a . T h i s s u g g e st s t h a t th e

i n c r e a s e i n i n t e r r a c i a l s h e a r s t r e n g t h d i d n o t o u t w e i g h

5

~ ' , 4

2

1

0

~ ~ ~ p ~ 0 0 k N

S h e ar s tr en gth / ~ - -

" S h e a r s ~ " ~ / E p = 4 - g G P a

/ ~ E a = 3 0 0 M P a

J G a = 1 2 0 M P a

i l l( d a = l m m

0 0 . 5 1 . 0 1 . 5 2 . 0

F l i p p l a t e t h i c k n e s s dp r a m )

2 . 5

F i g .

7 - Variation of

i n t e r f a c i a l s h e a r s t r e n g t h / s t r e s s

with FRP

p l a t e t h i c k n e s s f o r

beam P2.

5.5

5.0

4.5

4.0

3.5

0.25

- , t

K / / / - + - Sh ear s u v n ~

--~- ~ stres s

F _p= 4 C ~

0.45 0.65

p ~ te t h i d ~ s s ~, m )

F i g .

8 -

V a r i a t i o n o f i n t e r r a c i a l s h e a r s t r e n g t h / s t r e s s

with FRP

p l a t e t h i c k n e s s f o r b e a m P 2 E p

= 40 GPa) .

4.4

4 2 9

3 . 8 = 2 m m

3.6

"~ 3.4 -- ~ Shear s t rength

Shear s tress

3.2

3 .0

2.0

2.2 2.4 2.6 2.8 3.0 3.2

F R P p l a te t h i c k n e s s d p r a m )

Fig. 9 - Variation of

i n t e r f a c i a l s h e a r s t r e n g t h / s t r e s s

with FRP

p l a t e t h i c k n e s s f o r

beam P2 d a = 2mm) .

t he inc rease in in t e r rac ia l shea r s tre ss wh en a s t if fe r FR P

pla te was used . Th e s ti ffness of the FR P pla t e i s the re -

f o r e a n i m p o r t a n t p a r a m e t e r t o c o n t r o l w h e n d e s i g n i n g

F R P - s t r e n g t h e n e d R C b e a m s ag a in s t & b o n d i n g f a il ur e.

F igs. 7 and 9 show the e f fect of va ry ing the th ickness of

the adhes ive laye r on the FR P th ickness a t deb ond m g ( the

adhesive th i cknes s was inc reased f rom 1 m m to 2 ram) .

T he effect of this change was to increase the FR P thickness

a t & b o n d i n g f r o m a b o u t 1 .8 m m t o a b o u t 2 . 7 r a m . T h i s

is due to a s ignif icant red uct io n in inteffacial shear s t ress as

4 3


7/8

M a t e r i a l s a n d S t r u c t u r e s / M a t e r i a u x e t

Const ruc t ions Vol . 34 August -September 2001

Tab le 2 - E f fec t o f chang ing va r i ous pa r am ete r s

on the in ter rac ia l shear s t rength and st ress

Factor Inter racia l she ar stress Inter racia l sh ear strength

R o b e r t s r e v is e d m o d e l [ 2 ] C h a a l a l

e t a l

[ 6 ]

Ea

da ~

? ? Strong dependenceon the actor , ? Wea ledepende,,a o, the actor,

predic ted by Rob er t s rev ised mod e l and a smal l i ncrease in

interracial shear s t rength as pred icted b y Chaalal

e t a l .

[6].

For an F R P p late of 2 mm thickness , the interfacial shear

s tre ss accord ing to Ro ber t s rev ised mo de l dec reases f rom

4 .5 M P a t o 3 . 4 M P a w h e n d a is in c re a se d f r o m 1 m m t o

2 m m . T h e c o r r e s p o n d i n g i n c r e a se i n i n t er r a c ia l s h e a r

s t rength accord ing to C haalal

e t a l .

[6] is f rom 4 .0 M Pa to

4.1 MPa. As pointed o ut earl ier , the la t ter resul t is inc on -

s is ten t wi th ex per im enta l da ta sho wing dec reas ing bo nd

s t r e n g t h w i t h i n c r e a s i n g g l u e - l i n e t h i c k n e s s [ 2 0 ] .

W hen eve r pos s ib le, d i rec t measurem ent s of the in te r rac ia l

b o n d s t r e n g t h b e t w e e n t h e F R P p l a t e a n d t h e c o n c r e t e

s u r f a c e s h o u l d t h e r e f o r e b e u n d e r t a k e n f o r t h e t y p e o f

adhes ive to be used. Fro m a pract ical s tandpoint , how ever,

m o s t p r o p r i e ta r y s ys te m s h a v e r e c o m m e n d e d a d h e s iv e

t h i c k n e s s e s t h a t s h o u l d b e u s e d a n d f o r w h i c h d a t a o n

in te r rac ia l bond s trength m ay be obta ined f rom the m anu -

facturer.

T a b l e 2 s h o w s a s u m m a r y o f t h e e f fe c t o f c h a n g i n g

di f fe rent FRP and adhes ive pa ramete rs on the in t e r fac ia l

shea r s t re ss and s t rength . Th e t ab le shows tha t the fac -

tors tha t a f fec t the in t e r rac ia l shea r s t re s s would a l so

affect the interfacial shear s t rength, but to a lesser extent .

Th e e f fec t of these pa ramete rs , pa r t i cu la r ly on the in t e r -

facia l shea r s tre ss , should the re fore be t aken in to a cco unt

w h e n d e s i gn i n g a n F R P - s t r e n g t h e n e d R C b e a m .

Bonacc i and Maa le ] [1] s tudied the behaviora l t rends

o f R C b e a m s s t r e n g t h e n e d i n f le x u r e w i t h e x t e rn a l l y -

b o n d e d F R P b y c o m p i l i n g a n d a n a ly z in g a n e x p e r i m e n -

t al d a t a b a se . T h e d e f l e c t i o n r a ti o f o r s t r e n g t h e n e d

b e a m s ( d e f i n e d as th e m i d s p a n d e f l e c t i o n a t p e a k l o a d o f

a s t r e n g t h e n e d b e a m d i v i d e d b y t h e m i d s p a n d e f l e c t i o n

a t p e a k l o a d o f a c o n t r o l b e a m ) w a s f o u n d t o i n c r e a se

w i t h i n c r e a s i n g F R P e f f i c i e n c y r a t io . F i g . 3 su g g es ts

t h a t t h e l a t t e r d e c r e a s e s w i t h i n c r e a s i n g r e l a t i v e a x i a l

r i g i d it y ( A p E p/ A sE s ). T h e r e f o r e , o n e w o u l d e x p e c t t h e

def l ec t io n ra t io to dec rease w i th inc reas inr A E /A E

I D p S S

r a ti o . B y l im i t i n g A E / A s E , i t w o u l d b e p os s ib l e t o

p 13 s

preve nt o r de lay debo nd lng type fa ilures a s we l l a s ensure

adequa te de f l ec t ion capac ity .

F o r a g i v e n s t r e n g t h e n i n g a p p l i c a t i o n , a p r e - d e t e r -

m i n e d s t r e n g t h e n i n g r a ti o ( d e f i n e d as th e s t re n g t h o f th e

b e a m w i t h e x t e r n a l l y b o n d e d F R P d i v i d e d b y t h e

s t re n g t h o f t h e c o n v e n t i o n a l l y r e i n f o r c e d c o n t r o l b e a m )

w o u l d b e ta r g e te d . G i v e n th a t n u m e r o u s p r o p r i e t a r y

FRP s t rengthening sys tems a re cur rent ly ava i l ab le , t he

o p t i m u m s y s te m t o u s e w o u l d b e o n e t h a t m e e t s th e t a r -

g e t e d s t r e n g t h e n i n g r a t i o w h i l e l i m i t i n g t h e A E / A s E ~

9 . P

ra t io . In th i s case , i t w ou ld be pos s ib le to achieve an

o p p o r t u n e b a l a n c e b e t w e e n s t r e n g th g a i n a n d d e f l e c t i o n

capaci ty . For th i s purpo se , F RP e f f i c i ency- re l a t ive ax ia l

r i g i d i t y t r e n d s ( s u c h a s t h o s e s h o w n i n F i g . 3 ) c a n b e

e s t a b l i s h e d f r o m e x p e r i m e n t s a n d / o r a n a l y t i c a l m o d e l s

( such as Ro ber t s rev i sed mo de l ) and used to guide the

o p t i m u m d e s i g n o f F R P - s t r e n g t h e n e d R C b e a m s .

W h i l e t h e f o c u s o f t h e p r e s e n t p a p e r w a s o n o n e

a sp e ct o f t h e s h o r t - t e r m s t ru c t u r a l p e r f o r m a n c e o f F R P

s t r e n g t h e n e d b e a m s , t h e l o n g - t e r m p e r f o r m a n c e i s a l s o

very impo r tan t . Spec if ica lly , the durabi l i ty of an FR P

s t r e n g t h e n i n g s y s t e m u n d e r c y cl ic f r e e z i n g a n d t h a w i n g ,

aggres s ive subs tances, and fa t igue needs to be co ns ide red

in design . In addi t ion , the des ign er should be aware tha t

F R P p l a t e s g e n e r a l l y d o n o t h a v e s u f f i c i e n t f i r e r e s i s -

t a n c e f o r m a n y a p p l i c a t i o n s , a n d t h e r e f o r e o f t e n n e e d

addi t iona l pro tec t ion .

5 C O N C L U S I O N S

In th i s s tudy , predic t ive mode l s for de te rmining the

i n t e r f a c i a l s h e a r s t r e s s d i s t r i b u t i o n i n F R P - p l a t e d R C

b e a m s h a v e b e e n r e v i e w e d a n d e v a l u a t e d u s i n g e x p e r i -

m enta l da ta repo r t ed in the l it e ra ture . Th e m os t c r it i ca l

p a r a m e t e r s g o v e r n i n g t h e i n t e r f a c i a l s h e a r s t r e s s ( a n d

s t re n g t h ) a s d e t e r m i n e d b y t h e r e v i e w e d p r e d i ct i v e m o d -

e ls w e r e a ls o e x a m i n e d . R o b e r t s r e v is e d m o d e l , d e r i v e d

o n t h e b a s is o f a c r a c k e d b e a m s e c t io n a n d a m o d i f i e d

m o m e n t M * , p r e d i c t e d re su lt s t h a t w e r e i n c l os e a g r e e -

m e n t w i t h a c t u a l t r e n d s e t b y t h e e x p e r i m e n t a l d a t a .

Exper imenta l a s we l l a s mode l re su l t s revea led tha t the

F R P e f f i c i e n c y i n c re a s e s w i t h d e c r e a s i n g F R P r e la t iv e

axia l r ig id ity . In addi t ion , da ta repo r t ed in the l i t e ra ture

sugges ted a d i rec t re l a t ionship be tween beam def l ec t ion

c a p a c i t y a n d F R P e f f i c ie n c y . W i t h t h e a v a i l a b il it y o f

n u m e r o u s p r o p r i e t a r y F R P s t r e n g t h e n i n g s y s t e m s , i t

would be pos s ib le to s e l ec t an FRP s t rengthening sys tem

w h i c h o f f e r a n o p p o r t u n e b a l an c e b e t w e e n s t r e n g t h g a i n

and de f l e c t ion capac i ty for a g iven appl i ca tion .

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