University of Manitoba Department of Civil and Geological Engineering

Course 23.735

Design Methods for FRP-Strengthened Concrete Using Sheets or Near-Surface Mounted Bars

Prepared by: Kevin Amy (6730274)

Prepared for: Dr. Svecova

March 28, 2002

ABSTRACT

The design of concrete member reinforced with fibre reinforced polymer (FRP) has many

different design considerations than a steel reinforced concrete member.

In flexure a steel reinforced member is designed to ensure steel yielding before concrete

crushing due to the ductility of steel. FRP is a brittle material with a linear stress-strain

curve until failure. Thus, when using FRP as reinforcement a conscious decision must be

made as to the type of failure mode since either FRP rupture of concrete crushing are

brittle and sudden.

FRP stirrups have inherently low strength despite the very high strength of the FRP

material. During the manufacturing residual stresses are created in the bends. As a

result, the strength at the stirrups may be reduced to 30% percent of the ultimate tensile

strength of the FRP material. Design limitations for the radius of the bend, the end length

and so on have been recommended which should ensure a stirrups design strength of 50%

of the tensile strength.

The crack width and deformability of a concrete member reinforced with FRP behaves

similar to that of a steel reinforced member but the design constraints are quite different.

FRP has a relatively low modulus of elasticity compared to steel allowing for larger

deflections. The larger deflections also allow for larger crack widths. Since FRP

materials are not susceptible to environmental degradation larger crack widths are

perfectly acceptable to the point of aesthetic limitations.

LIST OF SYMBOLS Ab area of the bar Afrp area of FRP reinforcement Afrpv area of FRP shear reinforcement As area of tensile steel reinforcement As’ area of compression steel reinforcement a1,a2 shear spans b width of member bfrp width of FRP sheet bw web width C compressive force in the concrete Ct,steel dimensionless variable defined by equation 71 Cs compression force in the steel compression reinforcement Cu ultimate concrete creep coefficient c depth to the neutral axis of a flexural member from the extreme compressive fibre cb distance to from the extreme compressive fibres to the neutral during the balanced

reinforcement condition D deflection d depth from the extreme compressive fibre to the tensile reinforcement d’ depth from the extreme compressive fibre to the compression reinforcement de effective diameter dfrp FRP sheet height along the side of the beam web Ea elastic tensile modulus of the adhesive or epoxy Ec modulus of elasticity of concrete Efrp modulus of elasticity of the FRP reinforcement Efrpl longitudinal elastic modulus of FRP Efrpv elastic modulus of FRP shear reinforcement Es elastic modulus of steel fbend strength of the bend portion of the FRP stirrup fc stress in the concrete fc’ specified ultimate compressive stress in the concrete fct tensile strength of concrete ffrp stress in the FRP reinforcement ffrpu ultimate stress in the FRP reinforcement fr rupture stress in concrete fs’ stress in the compression steel reinforcement fy yield stress of steel ffrpe effective strength of FRP ffrpu ultimate strength of the FRP reinforcement ffrpuv ultimate capacity of FRP in shear h height of the member hflange height of the flange hs minimum thickness as per CSA A23.3-94 Ie effective moment of inertia

Ig gross moment of inertia I1 moment of inertia of a non-cracked member I2 moment of inertia of a cracked member l length of the member Ksh dimensionless multiplier which depends on the type of supports M moment Mr moment resistance Mu ultimate moment Mcr cracking moment Qcr correction factor for nonstandard conditions rb bend radius of FRP reinforcement s spacing of shear reinforcement sfrp FRP sheet bands spacing T tensile force ta thickness of the adhesive or epoxy tbfrp thickness of a FRP sheet at the balanced condition tfrp thickness of FRP sheet Vc concrete shear Vcf factored shear resistance of concrete Vcfrp factored shear resistance of FRP Vfrp FRP shear Vrf factored shear resistance Vser shear under service conditions weff effective FRP width yCT moment arm for a flexural member reinforced with a single layer of reinforcement ymax the distance from the extreme tension fibre to the neutral axis yt distance from the centroid of the transformed section the extreme tension fibre α stress block factor for concrete αb bond dependent variable taken as 0.5 until further data is collected αd dimensionless coefficient defined by equations 68 and 68 β stress block factor for concrete α1, β1 dimensionless factors defined by equations 19 and 20 εc strain in the concrete εcu ultimate strain in the concrete εeff effective FRP strain in the principal direction of the fibres εfrp strain the FRP reinforcement εfrpu ultimate strain in the FRP reinforcement εo concrete strain at peak concrete stress εs strain in steel εy yield strain of steel φc concrete material resistance factor φfrp FRP material resistance factor φs steel material resistance factor ∆cp creep due to sustained load

∆i immediate deflection due to sustained loading ∆sh shrinkage deflection ∆t total deflection η dimensionless coefficient defined by equation 63 ρfrp reinforcement ratio of the FRP reinforcement ρfrpb balanced reinforcement ratio ρfrp min minimum reinforcement ratio ρfrpv FRP shear reinforcement ratio ρfrpvmin minimum shear reinforcement ratio ρs reinforcement ratio of the tensile steel reinforcement ρs’ reinforcement ratio of the compression steel reinforcement σ1, max stress at the extreme tensile fibre (ignoring cracking) τpeeling stress causing the a layer of concrete bonded to the FRP plate to fail τult ultimate debonding shear stress ψm, end1 mean curvature at the end of the beam ψm, end2 mean curvature at the other end of the beam ψm, centre mean curvature at the mid-point of the beam ψm mean curvature ψsh curvature due to shrinkage ψu curvature when moment is Mu ζ dimensionless coefficient defined by equation 49

LIST OF FIGURES

Figure 1 – Stress strain curve for concrete………………………………………………...3

Figure 2 - Stress-strain distribution in flexure: a) failure by rupture of FRP; b) failure by

crushing of concrete……………………………………………………………………….4

Figure 3 – Stress-strain relationship in a strengthened beam with tension and compression

steel reinforcement………………………………………………………………………...8

Figure 4 – Different failure modes of a reinforced concrete beam a) debonding of FRP

plate b) peeling off of concrete layer…………………………………………………….10

Figure 5 – Cross section and strain distribution at a cracked section reinforced with steel

and FRP bars……………………………………………………………………………..18

INTRODUCTION

Fibre reinforced polymer (FRP) has been used as reinforcement for concrete structures

for a number of years now. As the use of FRP in structural design becomes more

accepted, the current design codes and practices will have to be adjusted to reflect the use

of FRP materials.

The design procedure and critical design criteria must change to reflect the material

properties of FRP. FRP materials generally have a lower modulus of elasticity and a

higher tensile strength, than steel. FRP is also very resistant to aggressive environments

and could increase the durability of concrete structures.

Design considerations when using FRP materials are the increased deflections (when

used as flexural reinforcement), large tensile strength, linear stress-strain curve with no

yielding, and the ability to orient the fibres in any direction to optimize the structural

response of the material.

Unlike when using steel as flexural reinforcement, a flexural member reinforced with

FRP cannot be designed to ensure a progressive ductile type of failure. As stated

previously, FRP has a linear stress strain relationship with no yielding, thus, the designer

must make a decision as to the type of failure mechanism desired, FRP rupture or

concrete crushing. The development of the equilibrium equations must also change to

reflect the actual strain level in the concrete. Ultimate strain in the concrete is assumed

when designing a flexural member with steel reinforcement. Since the concrete does not

have to fail (FRP rupture), the strain level in the concrete could be lower than ultimate.

The varying strain in the concrete changes the stress block parameters for concrete thus

changing the equations of equilibrium for the system.

FRP stirrups also have different design considerations than steel stirrups. The bend in

FRP stirrups could be a weak point causing premature failure. The weakness of the bend

in the FRP stirrups is a result of the manufacturing process and must be considered when

designing the shear reinforcement.

The deflection of a concrete member reinforced with FRP is a major design

consideration. Deflections will be larger than a conventional steel reinforced member,

causing larger crack widths. The size of the crack width is not as critical when using FRP

instead of steel reinforcement since FRP is not affected by environmental attacks as steel

is.

1.0 ULTIMATE STRENGTH DESIGN IN BENDING

Failure of a flexural member may occur by rupture of the tension reinforcement or

crushing of the concrete.

1.1 Tensile Failure

A member which fails due to the rupture of the tension reinforcement is said to be under-

reinforced. Since the FRP bars fail and not the concrete as in a traditional steel

reinforced member, the compressive stress in the concrete cannot be idealized by a

rectangular stress block in the same fashion as when steel reinforcement is used. The

strain in the concrete does reach ultimate and therefore the factors α1 and β1 cannot be

used. A concrete stress-strain curve must be chosen and used to calculate the stress block

parameters.

The following analysis uses the concrete stress distribution proposed by Todeschini et al.

(1964) and MacGregor (1997) to determine the parameters α and β. The equation of the

curve, seen in Figure 1, is given as follows,

Figure 1 – Stress strain curve for concrete (Ghali et al., 2002)

2

0

0'

1

8.1

+

=

εε

εε

c

c

cc ff ……………………………………………………………………...1

where,

c

c

Ef '

0 71.1=ε ……………………………………………………………………………..2

Ec for this theory is taken as,

'4750 cc fE = MPa……………………………………………………………………….3

The parameters α and β are determined by equating the area under the concrete stress

curve with that of the rectangular stress block, as well, the centroid of each area must be

the same. By completing these two tasks the parameters α and β may be determined.

As for steel reinforced members, the assumption, plane sections remain plane is made.

This assumption leads to the following relationship through similar triangles from Figure

2a.

fuc

c

dc

εεε+

= ………………………………………………………………………………4

Figure 2 - Stress-strain distribution in flexure: a) failure by rupture of FRP; b) failure by

crushing of concrete (Ghali et al., 2002)

For equilibrium the compressive force in the concrete must equal the tensile force in the

FRP bars. This leads to the following relationship (Ghali et al., 2002),

∫= daffA cfrpufrp ………………………………………………………………………….5

where da is an elemental area of the compressive zone.

Using equations 1 through 5 and numerical integration, εc at the extreme compressive

fibre, the stress distribution, and the location of the resultant compressive force may be

obtained.

The ultimate moment and curvature for a tensile failure is given by the following,

CTfrpufrpu yfAM = …………………………………………………………………………6

where yCT is the location of the resultant compressive force relative to the resultant

tensile force.

dfrpuc

u

εεψ

+= …………………………………………………………………………...7

ISIS Canada (2001) proposed the use of a different concrete stress distribution. The

distribution used was developed by Collins and Mitchell (1997). This distribution is used

to determine the parameters α and β as before. The concrete distribution takes the form,

nk

c

c

c

c

n

n

ff

+−

=

0

0'

1εε

εε

……………………………………………………………………….8

where,

178.0

'cfn += ……………………………………………………………………………….9

1

'

0 −=

nn

Ef

c

cε ……………………………………………………………………………..10

0.162

67.0'

>+= cfk ……………………………………………………………………...11

The modulus of elasticity of the concrete, for normal density concrete having a

compressive strength between 20 and 40 MPa for this stress distribution, was taken as

'4500 cc fE = …………………………………………………………………………...12

or

'5000 cc fE = …………………………………………………………………………...13

as per CSA A23.3-94 (1994) and CSA S6-88 (1993) respectively.

The proposed process of determining the resistance of a flexural member is as follows:

The compressive force in the concrete is given by,

cbfC cc βαφ '= ……………………………………………………………………………14

and the tensile force in the reinforcement is given by,

frpfrpfrpfrp EAT εφ= ……………………………………………………………………….15

For equilibrium, the compressive force in the concrete and the tensile force in the

reinforcement must be equal. By equating equations 14 and 15, c, the distance of the

neutral axis from the extreme compression fibre, can be determined. Once c is

determined the moment resistance of the member can be found by,

−=

2cdTMr

β ………………………………………………………………………...16

The ultimate curvature is found by using equation 7.

1.2 Compression Failure

When failure is due to crushing of the concrete the flexural member is said to be over-

reinforced. The rectangular representation of the compressive stress block, found in the

CSA Standard A23.3-94 (1994) can be used since the ultimate strain in the concrete will

be reached. The analysis process is very similar to that of a tensile failure except Figure

2b must be used. The moment is obtained by (Ghali et al., 2002),

CTfrpfrpu yfAM = ………………………………………………………..……………….17

where yCT is given by 21cd β

− .

The location of the neutral axis, c, is found by equating the compressive and tensile

forces as before. The parameters α1 and β1 are to be used since the strain in the concrete

has reached the ultimate strain.

The stress in the reinforcement at failure, which is smaller than ffrpu, is given by the

following equation (Ghali et al., 2002),

−

+= 1415.0

21

'1

1cufrpfrp

ccufrpfrp E

fEf

ερβ

αε ……………………………………….…...18

When εcu=0.0035, α1 and β1 are given by empirical formulas, in accordance with CSA

A23.3-94 (1994).

67.00015.085.0 '1 ≥−= cfα .……………………………………………………………19

67.00025.097.0 '1 ≥−= cfβ ……………………………………………………………..20

The curvature, at ultimate, when the failure mode is crushing of the concrete is given by

(ISIS Canada, 2001),

dEf

frp

frpc

u

+

=

εψ …………………………………………………………………….…….21

1.3 Balanced Condition

The reinforcement ratio which allows for the failure of the concrete and tensile

reinforcement at the same time creates a situation known as the balanced condition. The

reinforcement ratio at the balanced condition is given by (Ghali et al., 2002, ISIS Canada,

2001),

=

frpu

cbfrpb f

fdc '

11βαρ ……………………………………………………………..….22

where,

frpucufrp

cufrpb

fEE

dc

+=

εε

……………………………………………………………………...23

1.4 Minimum Reinforcement Ratio

To ensure a sudden failure does not occur after cracking of the section the ultimate

moment, from equation 6, must be greater than the cracking moment by an accepted

safety factor. The cracking moment is given by (Ghali et al., 2002),

trcr yIfM = ………………………………………………………………………………24

this leads to the following relationship,

( )factorsafetybdyy

Iff

bdA

tCTfrpu

rfrp

>=

1ρ ……………………………………...……..25

For a rectangular cross-section, and '6.0 cr ff = , d=0.9 (height of section), dyCT 9.0≈ ,

and a safety factor of 3 are substituted into equation 16, the same minimum

reinforcement ratio given by the American Concrete Committee 440 (ACI, 2001)

guidelines would be obtained. The American Concrete Committee 440 (ACI, 2001)

suggests the minimum reinforcement ratio be determined from,

frpu

cfrp f

f125 '

min =ρ ……………………..…………………………………………..………26

ISIS Canada (2001) proposed the same equation, as equation 26, for the minimum

amount of reinforcement and suggest the moment resistance must be at least 50% larger

than the cracking moment, if it is not, the moment resistance must be at least 50% larger

than the applied factored moment.

1.5 Multiple Layers of Reinforcement

In the case where there is more than on layer of FRP reinforcement, each layer must be

considered separately. The centroid of the reinforcement may not be used as with steel

but rather the tensile force equations already seen must be slightly modified. The

modifications for equation 16 are as follows,

∑=

=n

ifrpifrpifrpfrpi EAT

1εφ …………………………………………………………………..27

where, i is a FRP reinforcement layer

n is the number of layers

The moment is found in the same manner as done previously except, the moment arm of

each layer must be determined, multiplied by that layers tensile force and the sum of all

the moments for all the layers is to be completed to determine the moment resistance of

the member.

1.6 Externally Bonded Flexural Reinforcement

Chaallel et al. (1998) present a procedure to strengthen a doubly steel reinforced concrete beam. As seen in Figure 3, strain compatibility is used to determine the strain in the steel, concrete and FRP plate.

Figure 3 – Stress-strain relationship in a strengthened beam with tension and compression steel reinforcement (Chaallal et al, 1998)

When using the strain compatibility formulas the thickness of the FRP plate is neglected. The bottom steel reinforcement should yield before the FRP plate ruptures or the concrete crushes. From internal equilibrium, the following equations were developed,

[ ] [ ] 0'''21

'1 =+−+−+ cufrpfrpfrpcussscufrpfrpfrpsyscussscc hEAdEAcEAAfEAbcf εφεφεφφεφβφα

for ysys εεεε <≥ ' ……………………………………………………………………...28

( )[ ] 0'21

'1 =−+−+ cufrpfrpfrpcufrpfrpfrpssyscc hEAcEAAAfbcf εφεφφβφα

for ssys εεεε ≥≥ ' …………………………………………………………………….29

The strain in the steel, εs, should be checked to ensure it is larger than the steel yield

strain, εy. If the strain in the steel is not larger than the yield strain the area of FRP used

is too large and should be reduced.

The moment resistance is given by,

( ) ( ) ( )chTcdTccCdcCM frpssr −+−+

−+−=

2' 1β …………………………………..30

It follows, the balanced reinforcement ratio for the FRP can be found by,

−−

−+

+=

cdch

Ef

E

ffff

s

yfrpfrp

syssssy

cc

bfrp

φ

ρφρφβφα

ρ

'''

11

700700

……………………………………………..…31

hence the balanced thickness of the FRP plate is,

frp

bfrpbfrp b

bdt

ρ= ………………………………………………………………………….....32

Once the above process is completed the premature failure modes for this type of

reinforcement must also be checked to ensure they do not govern.

The first premature failure mode is debonding at the interface between the concrete and

the FRP and is depicted in Figure 4a. The ultimate debonding stress is given by,

xkult tan14.5

1+=τ …………………………………………………………………………33

where 25.0

313

=

afrpfrp

afrp ttE

Etk ……………………………………………………………………34

The second premature failure is ripping of the concrete as seen in Figure 4b. The stress to cause this form of failure is as follows,

=

frpc

tctpeeling b

bdLf

6τ …………………………………………………………………...35

where the tensile strength of the concrete is given by Hatzinikolas et al. (1979),

'' 53.0 ct ff = …………………………………………………………………………….36

Figure 4 – Different failure modes of a reinforced concrete beam a) debonding of FRP

plate b) peeling off of concrete layer (Chaallal et al., 1998)

2.0 SHEAR DESIGN

2.1 FRP Stirrups

Stirrups used as shear reinforcement are normally the first to encounter environmental

effects as a result of the minimum concrete cover over them. Thus, the use of FRP

stirrups, and their ability to resist severe environments, could increase the durability of a

structure.

The use of FRP stirrups has some added design considerations, such as, their strength may be as low as 35% of the strength parallel to the fibres (Morphy et al., 2000).

This strength reduction may be attributed to the residual stress concentrations found in

the bend zone from the manufacturing process. The Japanese Society of Civil Engineers

(JSCE, 1997) recommends the use of the following equation to evaluate the capacity of

bent FRP bars.

30.005.0 +=e

b

frpu

bend

dr

ff ……………………………………………………………………37

where, de is the effective bar diameter and is given by,

πb

eAd 4

= ……………………………………………………………………………….38

Equation 37 gives a conservative estimate of the strength of a bent bar, for most CFRP

and GFRP specimens. Morphy et al. (2000) found this equation overestimated the

strength of the CFRP Leadline stirrups, thus the use of minimum bend radius of not less

than four times the effective bar diameter or 50 mm was recommended. This would

allow a FRP stirrup capacity of 50% of the strength parallel to the fibres to be obtained

(Shehata, 1999). As well, Morphy et al. (2000) recommended the use of a ed6 or 70 mm

tail length, whichever is greater.

Morphy et al. (2000) proposed the following changes to the current CSA 23.3-94 (CSA,

1994) shear design equations.

The factored shear resistance is given by,

sfcfrf VVV += …………………………………………………………………………….39

where,

s

frplwcccf E

EdbfV '2.0 λφ= ……………………………..………………………………40

s

frplwcc

fvfrpuvfrpsf E

Edbf

sdA

fV '8.04.0 λφφ ≤= ………………………………………….41

It has been observed that the direct application of the current CSA design codes results in

an unsafe prediction of the shear strength of concrete beams reinforced with FRP stirrups

(Morphy et al., 2000).

Minimum shear reinforcement is required to ensure sudden shear failure, just after the

formation of the first diagonal crack, does not occur as well as to control the diagonal

cracking at service loads. A minimum amount of shear reinforcement is provided when

FRP flexural reinforcement is used. This minimum reinforcement is to ensure the shear

capacity is higher than the cracking load. As well, minimum shear reinforcement is

required when the factored shear force is greater than the concrete contribution by 50%.

The minimum amount of shear reinforcement is obtained by,

sfrplfrpuv

c

frpuv

w

s

frpl

c

frpv EEforff

f

dbEE

V

<≥

−

=4.006.0

4.0

1

'

minρ ……………………………………..42

or

sfrplfrpuv

cfrpv EEfor

ff

≥=4.006.0 '

minρ ………………………………………………………..43

where Vc is the factored shear resistance of the concrete according to CSA 23.3-94 (CSA,

1994).

2.2 Externally Bonded FRP Sheets

Further modifications to the current design code must be made when using externally

bonded FRP sheets for shear reinforcement. The term Vfrp must be added to the factored

shear resistance equation. Cheng and Deniaud (2001) proposed the following equation

for the shear resistance contribution of externally bonded FRP sheets.

( )ccfrp

frpfrpefrpfrpfrp sd

ftbV θααθα

tancossintansin

+= ……………..…………………………..44

where, α is the angle of the principal direction of the fibers to the longitudinal axis

θc is the angle of the concrete crack to the longitudinal axis

The effective stress in the FRP, ffrpe, has many different formulations, some of which are

given below.

Chaallal et al. (1998) developed the following formulation for FRP sheets in the form of

U-jackets.

frpufrp

frpultfrpe fAdb

f ≤= τ …………………………………………………...……………...45

where,

τult is given by equation 33.

Khalifa et al. (1998) developed the following expression,

frpufrpe Rff = ……………………………………………………………………………...46

where R is the lowest of,

R=0.5

or

( ) ( )frpfrpfrpfrp EER ρρ 5622.02188.1778.0 +−= ………………………………………….47

where,

=

frp

frp

w

frpfrp s

bbt2

ρ ……………………………………………………………………...48

or

( )( ) frpfrpufrpfrp

effc

dtEwf

Rε58.0

32

'0042.0= ………………………………………………………………….49

where, ( )( )frpfrp Et

frpeff edw ln58.0134.6 −−= …………………………………………………………….50

frp

frpufrpu E

f=ε ………………………………………………………………………………51

CSA-S806 (CSA, 2000) has prepared a design standard for construction with FRP

materials. The shear design criteria uses CSA-A23.3 (CSA, 1994) simplified approach

with θ=45 degrees and β=0.2. The effective stress in the FRP is given by,

efffrpfrpe Ef ε= …………………………………………………………………………….52

where,

004.0=effε

3.0 SERVICEABILITY

3.1 Crack Width

The factors affecting the crack width are the crack spacing and bond between the

reinforcing bar and concrete. As a result of having a lower modulus of elasticity, flexural

concrete members reinforced with FRP bars allow for larger crack widths.

CSA A23.3-94 (1994) limits the crack width to 0.40 and 0.33 mm for interior and

exterior exposure respectively when using steel reinforcement. Currently, the steel

reinforcement is allowed to be stressed to 0.6fy, under service loads, in an attempt to limit

cracking. To take advantage of the high strengths, and good resistance to environmental

effects, FRP reinforcement bars could be stressed to a higher level, thus creating larger

crack widths than when using steel reinforcement. The JSCE (1997) recommends a

maximum crack with of 0.50 mm, when using FRP reinforcement, for all exposures. The

Canadian Highway and Bridge Design Code (CHBDC) (CSA, 2000) recommends crack

widths of 0.71 and 0.50 mm for interior and exterior exposure. ACI 440 (ACI 2001)

adopted similar limits as the CHBDC.

At the service load condition, Morphy et al. (2000) recommend limiting the strain in the stirrups to 0.002, instead of 0.0012 as used for steel reinforcing. Increasing the allowable

strain creates larger crack widths and is similar to the crack width limits already mentioned. For shear reinforcement and shear cracking the strain in the FRP stirrups, based on the 45 degree truss model, can be found as follows,

( )dEAVVs

frpvfrpv

cfserfrpvser

−=ε ……………………………………………………………………...53

3.2 Deformability

All design codes require the design of a flexural member reinforced with steel to be

under-reinforced. Generally, since FRP reinforcement has a very brittle and sudden

failure, and has only recently been used as structural reinforcement, the design codes

have recommended flexural members, with FRP reinforcement, be designed as over-

reinforced; ensuring the failure mode will be crushing of the concrete (Ghali et al., 2002).

Some new design guidelines have allowed for the design of a tensile failure mode for a

flexural member reinforced with FRP reinforcement. When designing for a tensile

failure, the deflections obtained must be monitored closely to ensure they do not become

excessive.

Bobey et al. (2001) present the following deflection calculation methodology based on

the CEB-FIP Model Code (1990).

The general deflection equation for the calculation of mid-length deflection of a straight

member is given by,

( )2,,1,

2

96 endmcentremendmlD ψψψ ++= ………………………………………………………54

ψm is the mean curvature at the ends and centre of the straight member. ψm is exact when

it’s geometric relationship varies parabolically which is sufficiently accurate for practical

application (Hall, 2000). For a simple beam the mean curvature at the ends can be

neglected, thus the deflection equation reduces to,

2,48

5 lD centremψ= …………………………………………………………………………55

For the following, ψm, will be taken to mean, ψm,centre, because even when the curvature at

the ends are not negligible, the deflection is mainly dependent upon ψm,centre. ψm is

determined as follows,

( ) 211 ςψψςψ +−=m ……………………………………………………………………..56

where,

ctct ff

>

−= max,1

2

max,1

5.01 σσ

ς …………………………………………………………57

also where,

max1max,1 yI

M=σ …………………………………………………………………………...58

11 IE

M

c

=ψ ………………………………………………………………………………...59

22 IE

M

c

=ψ ………………………………………………………………………………..60

Equations 56 – 60 apply to a reinforced, non-prestressed, section reinforced with steel or

FRP bars, without the application of a normal force.

Equations 56 – 60 can be rewritten as,

ecm IE

M=ψ ………………………………………………………………………………..61

where Ie is the effective moment of inertia of the section given by ISIS (2001),

( )21

2

2

21

5.01 IIMMI

IIIcr

e

−

−+

= ……………………………………………………….62

also where,

ctcr fyIMmax

1= ……………………………………………………………………………63

ACI 318-99 (ACI, 1999) and CSA A23.3-94 (CSA, 1994) adopted equation 54 but used

the Ie developed by Branson (1977), which is as follows,

21, 1 IMMI

MMI

mcr

mcr

Bransone

−+

= …………………………………………………..64

with m=3. Equation 57 with m=3 gives a constant effective moment of inertia over the

full length of the cracked member.

Equation 62 was chosen by Bobey et al. (2001) because equation 64, used by the before

mentioned codes, does agree with experimental data (Hall, 2000).

The effective moment of inertia of a GFRP-reinforced beam is given by (Benmokrane et

al., 1996 and Gao et al., 1998),

gcrcr

gs

frpb

cre II

MMI

EE

MMI ≤

−+

+

=

33

11α ……………………………………...65

where αb is a bond dependent variable, but for now is taken as 0.5 until further testing

can be completed to refine it’s value. This effective moment of inertia could be used in

equation 61 to find the deflection.

Bobey et al. (2001) present a relationship between deflection and the strain in the

reinforcement. The following is based on Figure 5.

Figure 5 – Cross section and strain distribution at a cracked section reinforced with steel

and FRP bars (Bobey et al., 2001)

cdfrp

−=

εψ 2 ……………………………………………………………………………….66

The depth of the compression zone, c, is given by,

−+−= 2

211 45.0 aaac ……………………………………………………………….67

where,

( )w

frpw

w

flange

bnA

bbbh

a22

1 +−= …………………………………………………………….68

( )w

frpw

w

flange

bnA

bbbh

a22

2 −−−= ……………………………………………………………69

c

frp

EE

n = ………………………………………………………………………………….70

The mean curvature is expressed as follows,

2ηψψ =m ………………………………………………………………………………...71

where η derived from equations 56 and 57 and is given by,

−

−=

1

2

2

max,.1

15.01IIfct

ση ……………………………………………………………..72

Equations 55, 65, and 71 give the mid-span deflection of a cracked member as follows,

frp

hcdh

lD εη

−=

2

485 …………………………………………………………………...73

The length to deflection ratio of a FRP reinforced section is given by,

−

−

=

frp

s

frp

s

sfrp

hcd

hcd

hl

hl

εε

η

η

……………………………………………………….74

The length to height ratio for steel can be found in the design codes.

ISIS Canada (2001) present a similar equation for calculation the length to height ratio of

a member reinforced with FRP and is given as, d

frp

s

sfrp hl

hl

α

εε

=

…………………………………………………………………...75

where,

5.0=dα for rectangular sections……………………………………………………….76

or

swd h

bbb

803.05.0 −+=α for T-sections……………………………………….…………77

where hs is the minimum thickness required by CSA A23.3-94 (1994) if steel

reinforcement was used.

Long term deflection calculations of flexural concrete members reinforced with FRP bars

are over estimated when using equations developed for steel (Kage et al., 1995).

However, Arockiasamy et al. (1996) found the steel developed deflection equations were

suitable for use when CFRP was used as the reinforcement type.

ACI Committee 209 (1993) recommends the use of the following equation to calculate

the long term deflection of a steel reinforced beam.

shcpt ∆+∆=∆ ……………………………………………………………………………78

Brown (1997) recommends the following equation be used if FRP reinforcement is used.

isteeltcrcp CQt

t∆

+=∆ ,6.0

6.0

6.010

…………………………………………………………….79

where,

crusteelt QCt

tC 6.0

6.0

, 10 += …………………………………………………………………...80

Cu can vary from 1.0 – 4.15, but an average value of 2.35 is suggested if specific data is

not available.

The shrinkage deflection term of equation 78 remains the same and is as follows, 2lK shshsh ψ=∆ ……………………………………………………………………………81

Ksh is a multiplier which depends on the type of support. Ksh is equal to 0.125 for a

simply supported member.

The ACI Committee 209 recommendation overestimates the deflections of both a steel and GFRP reinforced member. The overestimation is more significant for a GFRP reinforced member (Ghali and Hall, 2000).

REFERENCES ACI Committee 209, 1993. Prediction of creep, shrinkage, and temperature effects in concrete structures. ACI 209R-92, American Concrete Institute, Detroit, Mich. ACI Committee 318, 1999. Building code requirements for structural concrete. ACI 318-99, American Concrete Institute, Farmington Hills, Mich. ACI Committee 440, 2001. Guideline for the design and construction of concrete reinforced with FRP bars. ACI 440.1R-01, American Concrete Institute, Farmington Hills, Mich. Arockiasamy, M., Amer, A., Chidambaram, S., and Shahawy, M., 1996. Long-term behaviour of concrete beams reinforced with CFRP bars under sustained loads. Proceedings of the 2nd International Conference on Advanced Composite Materials in Bridges and Structures (ACMBS-II), Montreal, Que., p.673-680 Benmokrane, B., Chaallal O., and Mamoudi, R., 1996. Flexural response of concrete beams reinforced with FRP reinforcing bar. ACI Structural Journal, 93(1):46-55 Bobey, W., Ghali, A., and Hall, T., 2001. Minimum thickness of concrete members reinforced with fibre reinforced polymer bars. Canadian Journal of Civil Engineering. 28:583-592 Branson, D.E., 1977. Deformation of concrete structures. McGraw-Hill, New York. CEB-FIP, 1990. Model for concrete structures (MC-90). Comite Euro-International du Beton-Federation International de la Precontrainte, Thomas Telford, London, England Chaallal, O., Nollet, M.-J., and Parraton, D., 1998. Strengthening of reinforced concrete beams with externally bonded fiber-reinforced-plastic plates: design guidelines for shear and flexure. Canadian Journal of Civil Engineering, 25:692-704 Cheng, J.J.R., and Deniaud, C., 2001. Review of shear design methods for reinforced concrete beams strengthened with fibre reinforced polymer sheets. Journal of Canadian Civil Engineering. 28:271-281 Collins, M.P., and Mitchell, D., 1997. Prestessed concrete structures. Response Publications, Canada. P.766 Collins, M.P., and Mitchell, D. 1987. Prestressed concrete basics. 1st ed. Canadian Prestressed Concrete Institute, Ottawa, Ont. CSA, 1998. Design of highway bridges. Standard CSA S6-88, Canadian Standards Association, Rexdale, Ont.

CSA, 1994. Design of concrete structures. Standard CSA-A23.3-94, Canadian Standards Association, Rexdale, Ont. CSA, 2000. Canadian highway bridge design code (CHBDC), Section 16, fibre reinforced structures. Standard CSA-S6-00, Canadian Standards Association, Rexdale, Ont. CSA, 2000. Design of construction of building components with fiber reinforced polymers. Draft, CSA Standard S806, Canadian Standards Association, Rexdale, Ont. Gao, D., Benmodrane, B., and Masmoudi, R., 1998. A calculating method of flexural properties of FRP reinforced concrete beams: Part 1: Crack width and deflection. Technical Report, Department of Civil Engineering, University of Sherbrooke, Sherbrooke, Quebec. p.24 Ghali, A., and Hall, T., 2000. Long-term deflection prediction of concrete members reinforced with glass fibre reinforced polymer bars. Canadian Journal of Civil Engineering. 27:890-898 Ghali, A., Newhook, J., and Tadros, G., 2002. Concrete flexural members reinforced with fiber reinforced polymer: design for cracking and deformability. Journal of Canadian Civil Engineering. 29:125-134 Hall, T.S., 2000. Deflections of concrete members reinforced with fibre reinforced polymer (FRP) bars. M.Sc. thesis, Department of Civil Engineering, The University of Calgary, Calgary, Alta. Hatzinikolas, M., MacGregor, J.G., and Mirza, S.A., 1979. Statistical descriptions of the strength of concrete. ASCE Journal of the Structural Division, 105(ST6): 1021-1037 ISIS Canada, 2001. Reinforced concrete structures with fibre reinforced polymers (FRPs). ISIS-M04-01, The University of Manitoba, Winnipeg, Man. JSCE, 1997. Recommendation for design and construction of concrete structures using continuous fibre reinforced materials. Concrete Engineering Series, No. 23, Japan Society of Civil Engineers, Tokyo, Japan. Kage, T., Masuda, Y., Tanano, Y., 1995. Long-term deflections of continuous fiber reinforced concrete beams. Proceedings of the 2nd International RILEM Symposium on Non-Metallic (FRP) Reinforcement for Concret3e Structures, Ghent, Belgium, p. 251-258 Khalifa, A., Gold, W.J., Nanni, A., and Aziz, M.I., 1998. Contribution of externally bonded FRP to hsear capacit of RC flexural members. ASCE Journal of Composites for Construction, 2:195-202

MacGregor, J.G., 1997. Reinforced concrete-mechanics and design. 3rd ed. Prentice-Hall, Upper Saddle River, N.J. Morphy, R., Rizkalla, S., and Shehata, E., 2000. Fibre reinforced polymer shear reinforcement for concrete members: behaviour and design guidelines. Journal of Canadian Civil Engineering. 27:859-872 Shehata, E, 1999. FRP for shear reinforcement in concrete structures. Ph.D. thesis, Department of Civil and Geological Engineering, The University of Manitoba, Winnipeg, Man. Todeschini, C.E., Bianchini, A.C., and Kesler, C.E., 1964. Behaviour of concrete column reinforced with high strength steels. ACI Journal, 61(6):701-716