Bond Slip Analysis of Fiber-Reinforced Polymer-Strengthened Beams

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Bond Slip Analysis of Fiber-ReinforcedPolymer-Strengthened BeamsHayder A. Rasheed, M. ASCE1, and Shariq Pervaiz2

Abstract: The second order differential equation of interface shear is formulated for fiber-reinforced polymer-strengthened beambeam theory with a shear deformable adhesive layer. The solution of the boundary value problem is obtained in closed form anto derive deflection expressions for different loading conditions. The solution is also extended to analyze partially plated bearesults converge to the extreme cases of very poorly and perfectly bonded plates and they help identify values of the adhesmodulus for effective stiffening. Furthermore, the solution of partially plated beams aids in defining anchorage lengths needed tothe full or the highest possible composite action at midspan.

DOI: 10.1061/~ASCE!0733-9399~2002!128:1~78!

CE Database keywords: Slip; Adhesive bonding; Beams; Fiber-reinforced materials.

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IntroductionThe importance of bond slip in studying the behavior of fibereinforced polymer~FRP!-strengthened beams is recently reconized ~Taljsten 1997; Lee et al. 1999!. Lap joint simple shearexperiments, on FRP strips bonded to concrete blocks, wereried out to develop a basic understanding~Taljsten 1997; Bizin-davyi and Neale 1999!. The results obtained, within the lineaelastic range, compared well with the Volkersen solution for ljoints ~Taljsten 1997; Bizindavyi and Neale 1999!. This solutionwas found for the interface shear based on the axial deformatonly, in the FRP strip and the concrete. Lee et al.~1999! haveconducted tests on concrete prisms reinforced with an embedsteel bar and external carbon FRP sheets to generate initialmodulus values. They have also developed similar bond slip alytical solutions strictly based on axial response.

Other investigators studied the effect of the plate end disctinuity on the premature plate separation. Roberts~1989! devel-oped approximate expressions to predict shear and normal sconcentrations based on partial interaction theory~Roberts andHaji-Kazemi 1989! assuming no bond slip deformations. Zirabet al. ~1994! and Quantrill et al.~1996! suggested modificationsto Roberts equations. Malek et al.~1998! also studied local failuredue to stress concentrations at plate end assuming linear elbehavior and ignoring bond slip. They proposed another seequations similar to those of Roberts. Mukhopadhyayat aSwamy ~2001! and EI-Mihilmy and Tedesco~2001! have inde-pendently applied all the above analytical models to show thatresults are not consistently related to this failure mode usin

1Assistant Professor, Dept. of Civil Engineering, Kansas State UnManhattan, KS 66506.

2Bridge Design Engineer, Biggs Cardosa Associates, FresCA 93711.

Note. Associate Editor Arup K. Maji Discussion open until June2002. Separate discussions must be submitted for individual papersextend the closing date by one month, a written request must be filedthe ASCE Managing Editor. The manuscript for this paper was submifor review and possible publication on January 5, 2001; approvedApril 9, 2001. This paper is part of theJournal of Engineering Mechan-ics, Vol. 128, No. 1, January 1, 2002. ©ASCE, ISSN 0733-9399/200278–86/$8.001$.50 per page.

78 / JOURNAL OF ENGINEERING MECHANICS / JANUARY 2002

J. Eng. Mech. 20

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wide range of experimental data. The prediction of this failumode is beyond the scope of the present work, which is focuon addressing the partial composite action and FRP plate deopment length due to bond slip in the adhesive layer.

In this study the mechanics of the shear deformable adheslayer, combined with simple beam theory, is invoked to develrigorous analytical solutions for the flexural response of beaconsidering bond slip. The kinematics of the epoxy interfaceseen to have similarities to those of the foam core in sandwbeams. Hartsock~1969! used a sandwich beam differential element with an incompressible core to develop a differential eqution in terms of the deflection, curvature, and the double integof moment. He introduced an indirect solution, in which thsecond-order differential equation was equated to a generalfourth degree polynomial. The deflection solution was then otained, for simple load cases, through the help of a compuprogram by following a few steps of differentiation and substittion involving 13 unknown constants. Ha~1992! differentiatedHartsock’s equation, twice, to get the general fourth-order diffeential equation of deflection. He then explained that difficultiarise from applying certain boundary conditions to the genesolution of such a differential equation yielding insufficient conditions to solve for the unknowns. Accordingly, he developedspecial linear elastic sandwich beam element to avoid such dculties. Other solutions considering the transverse flexibilitythe sandwich beam core exist in the literature~Frostig and Baruch1990!. However, the negligible compressibility of the thin adhesive layer in FRP-strengthened beams, does not justify the highorder solution for the present application.

In the present treatment, a different approach is deviseddevelop the second-order differential equation of interface sheOnce the interface shear function is determined, the deflectexpression is obtained by direct double integration. The pressolution is substantially easier to get than that using the fourorder differential equation of deflection~Ha 1992!. It offers directinterface shear results without the need to solve for deflection ayields closed-form expressions for bond shear stress distribuand deflections. In addition, it involves no solution singularity duto boundary condition difficulties for the case of span load dcontinuities~e.g., four-point bending! as is the case with the so-

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lution of the fourth-order differential equation, verified by theauthors.

Interface Shear Differential Equation

In modeling the bond line effects in FRP-strengthened beams,following assumptions are made:1. Linear elastic behavior of the beam section, FRP plate, a

interface layer;2. Unidirectional FRP plates with their fibers aligned with th

beam axis;3. Bond line can only deform in shear with zero bending stif

ness;4. The adhesive is homogenous and uniform along the bo

line;5. The adhesive material is incompressible~points in the cross

section of the beam and FRP plate have the same vertdeflection!; and

6. Linear strain distribution is assumed valid in the beam athe strengthening plate sections following the BernoulliNavier hypothesis of shallow beams.

Accordingly, the resulting analytical solution is only applicable to reinforced concrete beams prior to cracking as wellsteel beams before yielding and timber members before reachtheir ultimate loads. Solutions for reinforced concrete beamunder service loads may be obtained by assuming cracked ccrete sections. The shear behavior of the epoxy-bonding layeexpected to be nonlinear, in general. Thus the present solutshould be extended to include such effects if the full nonlineresponse is of interest.

Kinematics

It may be easily shown that the shear strain in the adhesive la~g! is directly related to the difference between the rotation of thbeam-FRP plate (n8) and the average cross-sectional rotation~f!when interface shear deformation is included, see Fig. 1

2gt i5 jd~n82f! (1)

Fig. 1. FRP strengthened beams considering interface slip:~a! beamlayout; ~b! cross section kinematics

J. Eng. Mech. 20

e

d

d

al

d

sngsn-isnr

ere

where t i5thickness of the interface, andjd5distance betweenthe centroid of the beam and the composite laminate. Note thboth (n8) and~f! are counterclockwise angles and~g! is a clock-wise angle according to the present sign convention, see Fig. 1~b!.The average cross-sectional rotation~f! may be defined in termsof the relative axial displacement of the centroid of the beam anthe FRP plate, see Fig. 1

f5u22u1

jd(2)

Constitutive Equations

The axial strain in the beam and the FRP plate can be written iterms of the axial force and the axial rigidity of each, see Fig. 2

u185C

EAB, u285

T

EAP, (3)

whereEAB andEAP5axial rigidity of the beam and FRP plate,respectively. Since the deflection is assumed to be the same forpoints in the section, the curvature (n9) may be directly used todefine the bending moments in the beam and the FRP plate, sFig. 2~a!

MB5EIBn9, M P5EIPn9 (4)

whereEIB andEIP5flexural rigidity of the beam and FRP plate,respectively,MB and M P5the corresponding moments. The in-terface shear stress~t! is directly related to the shear strain~g!through the shear modulus~G!

t5Gg (5)

Equilibrium Conditions

Considering the equilibrium of a differential element of the FRPplate @Fig. 2~b!#

t51

b

dT

dx(6)

whereb5width of the FRP plate. Taking the equilibrium of forcesand moments at any cross section

Fig. 2. Equilibrium of forces:~a! force/moment distribution in crosssection;~b! interface shear stress on differential plate element

JOURNAL OF ENGINEERING MECHANICS / JANUARY 2002 / 79

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d.

C52T (7)

M ~x!5MB1M P1T jd (8)

Differentiating Eq.~2! with respect tox and substituting Eqs.~3! and ~7! into it

f85T

jdEAs(9)

where 1/EAS51/EIB11/EIP . Substituting Eq.~4! into Eq. ~8!and rearranging

T5M ~x!2EISn9

jd(10)

whereEIS5EIB1EIP . Differentiating Eq.~10! with respect toxand substituting it into Eq.~6!

n-5M 8~x!2b jd t~x!

EIS(11)

Differentiating Eq.~1! twice with respect tox, substituting Eq.~9!and rearranging

g91jd

t iS n-2

T8jdEAx

D50 (12)

Substituting Eqs.~5!, ~6!, and ~11! into Eq. ~12!, the interfaceshear differential equation is obtained

t9~x!2a2t~x!52jdG

tiEISM 8~x! (13)

wherea25bG/t i@1/EAS1 jd2/EIS#

Solution of Boundary Value Problem

Fully Plated Beams

The above ordinary differential equation with constant coefcients has the well-known general solution

t~x!5A cosh~ax!1B sinh~ax!1tP~x! (14)

where tp(x)5particular solution. The latter is determined fothree different load cases prior to the application of the boundaconditions. These are three-point bending, uniform loading, afour-point bending applied to half the span due to symmetry, sFig. 3

tP~x!5jdG

a2t iEISM 8~x! (15)

where M 8(x)5P/2 for the three-point bending,M 8(x)5w(L/22x) for the uniform load andM 8(x)5P/2 in the shear span ofthe beam with four-point bending. In this section the FRP plateassumed to extend along the entire span of the beam while ifree to slip at its ends, see Fig. 3. Accordingly, the boundaconditions are

g850 at x50(16)

g50 at x5L

2

The first boundary condition may be easily confirmed by diffeentiating Eq.~1! with respect tox, substituting Eq.~9! into it, andknowing thatv95T50 at x50. The second boundary conditionis a direct result of symmetry about midspan. For the casefour-point bending, the continuity ing andg8 is maintained at the


J. Eng. Mech. 20

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load position to satisfy the additional conditions needed. Usithe above boundary conditions to solve for the coefficients of E~14!, the interface shear function is obtained for the differeloading cases considered:

Three-point bending

t~x!5jdG

a2t iEIS

P

2 S 12cosh~ax!

cosh~aL/2! D (17)

Uniform loading

t~x!5jdG

a2t iEISwF S L

22xD1

sinh~ax!2tanh~aL/2!cosh~ax!

a G~18!

Four-point bending

tL~x!5jdG

a2t iEIs

P

2$12@cosh~aa!

2tanh~aL/2!sinh~aa!#cosh~ax!% (19)

tR~x!5jdG

a2t iEIS

P

2sinh~aa!@ tanh~aL/2!cosh~ax!2sinh~ax!#

(20)

wheretL(x) andtR(x)5 interface shear functions to the left anright side of the load, anda5 length of the shear span, see Fig. 3It can be easily verified that Eq.~19! reduces to Eq.~17! if a5L/2 is substituted.

Partially Plated Beams

As noted earlier, the present formulation allows for the interfashear solution of the partially plated beams to be determinwithout the need to solve for the overall deflections. To do so, tfirst boundary condition in Eq.~16! is redefined. DifferentiatingEq. ~1! with respect tox and substituting Eqs.~9! and~10! into it

Fig. 3. Loading cases for fully plated beams:~a! three-point bending;~b! uniform loading;~c! four-point bending

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g85M ~x!2EIn9~x!

t i jdEAS(21)

whereEI5EIS1EASjd2. Knowing that the curvature at the plateend section is the same as that of the unplated beam just to theof the section (vo95Mo /EIB), the new boundary condition is defined, see Fig. 4

g85go85Mo

ti jdEASS 12

EI

EIBD at x5

L2LP

2or xP50

(22)

whereMo5the moment value at that section~i.e., plate end sec-tion!. Applying the condition in Eq.~22! along with the secondone in Eq.~16!, the interface shear stress function for the thrload cases identified above is obtained:

Three-point bending

t~x!5jdG

a2t iEIS

P

2 S 12cosh~axP!

cosh~aLP/2! D1G

2at i jdEAS

P

2~L2LP!

3S 12EI

EIBD @sinh~axP!2tanh~aLP/2!cosh~axP!# (23)

whereMo5P/2@(L2LP)/2#Uniform loading

t~x!5jdG

a2t iEISwF S L

22xPD

1sinh~axP!2tanh~aLP/2!cosh~axP!

a G2

jdG

2a2t iEISw~L2LP!

cosh~axP!

cosh~aLP/2!

1G

at i jdEAS

w

8~L22LP

2 !S 12EI

EIBD @~sinh~axP!

2tanh~aLP/2!cosh~axP!# (24)

whereMo5w/4@(L22L2P)/2#

Four-point bending

tL~xP!5jdG

a2t iEIS

P

2$12cosh~axP!@cosh~aaP!

2tanh~aLP/2!sinh~aaP!#%1G

2at i jdEAS

P

2~L2LP!

3S 12EI

EIBD @sinh~axP!2tanh~aLP/2!cosh~axP!#

~25!

Fig. 4. Layout and coordinate system for partially plated half-beawith three-point bending

J. Eng. Mech. 2

left

e

tR~xP!5F jdG

a2t iEIS

P

2sinh~aaP!

2G

2at i jdEAS

P

2~L2LP!S 12

EI

EIBD G

3@ tanh~aLP/2!cosh~axP!2sinh~axP!# (26)

where Mo in Eq. ~25! is the same as that in Eq.~23!, xP ismeasured from the plate end as shown in Fig. 4.

Beam Deflection Formulation

Fully Plated Beams

Once the interface shear function is obtained, Eq.~21! is used todefine the curvature and determine the slope and deflectiondirect integration

n9~x!5M ~x!

EI2

t i jdEAS

GEIt8 (27)

n~x!5E E M ~x!2t i jdEASt8/G

EIdx dx1C1x1C2 (28)

SubstitutingM (x) for the different loading cases and Eqs~17!–~20! into Eq. ~28!, integrating twice and applying theboundary conditions:

Three-point bending

n~x!5Px3

12EI2

P sinh~ax!

2a3 cosh~aL/2! F 1

EI2

1

EISG2

PL2x

16EI

1Px

2a2 F 1

EI2

1

EISG (29)

Uniform loading

n~x!5w

EI S Lx3

122

L3x

242

x4

24D2w

a4 F 1

EI2

1

EISG@12x2cosh~ax!

1tanh~aL/2!sinh~ax!# (30)

Four-point bending

nL~x!5Px3

12EI2

P sinh~ax!

2a3 F 1

EI2

1

EISG@cosh~aa!

2tanh~aL/2!sinh~aa!#2Pax

4EI~L2a!

1Px

2a2 F 1

EI2

1

EISG (31)

nR~x!5P sinh~aa!

2a3 F 1

EI2

1

EISG@ tanh~aL/2!sinh~ax!

2cosh~ax!#2Pa

4EI S Lx2x22a2

3 D1Pa

2a2 F 1

EI2

1

EISG

(32)

whereEI5EIS1EASjd2. The boundary conditions applied arewell known to be ofn50 at x50 andn850 at x5L/2. In thecase of four-point bending, the extra two boundary conditionneeded are satisfied by matching then and n8 equations fromboth sides at the load position.


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Partially Plated Beams

The deflection solution of the partially plated beam follows thsame steps adopted for the fully plated one, Eqs.~27! and ~28!,adding the deflection function of the unplated portion from standard beam theory. For the sake of brevity, the deflection of thpartially plated beam is developed for the three-point bendincase only. This could be similarly extended to the other loacases. It is important to mention that the curvature and deflectiequations below are written in terms of bothx andxP , which arelinearly dependent, since the curvature expressions for fulplated beams can be directly used here for the plated portion, F4. Therefore the integration could be carried out in terms of bovariables and the difference between them will be absorbed by tintegration constants

xP5x2L2LP

2⇒dxP5dx (33)

Unplated Beam Portion

nu95M

EIB⇒nu5

Px3

12EIB1C1x1C2 (34)

Plated Beam PortionUsing Eqs.~23! and ~27!

npl9 5M

EI2

t i jdEAS

GEIt8⇒npl5

Px3

12EI1

P

2a2 S 1

EI2

1

EISD Fa2xP

2

2

2cosh~axP!

cosh~aLP/2!G1

P

4a3 ~L2LP!S 1

EI2

1

EIBD @sinh~axP!

2tanh~aLP/2!cosh~axp!#1C3xP1C4 (35)

Applying the boundary conditionsnu50 at x50, nu5npl ,nu85npl8 at xP50, andnpl8 50 at xP5LP/2

C452P

96~L2LP!3S 1

EI2

1

EIBD1

P

2a2 S 1

EI2

1

EISD sech~aLP/2!

1P

4a3 ~L2LP!S 1

EI2

1

EIBD tanh~aLP/2! (36)

C352PL2

16EI2

P

2a S 1

EI2

1

EISD FaLP

22tanhS aLP

2 D G2

P

4a2 ~L2LP!S 1

EI2

1

EIBD sech~aLP/2! (37)

C15C31P

16~L2LP!S 1

EI2

1

EIBD FL2LP1

4

a2G , C250

(38)

FRP Tensile Force

The FRP tension force function can be easily obtained by intgrating Eq.~6! with respect tox and applying the boundary con-dition, T50 at xP50, in a partially plated beam. This yields, forthe case of three-point bending, the following expression:


J. Eng. Mech. 2

-egdn

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-

T~x!5jdEAS

EI

P

2 S xP2sinh~axP!

a cosh~aLP/2! D1

EIS

2 jd

P

2~L2LP!S 1

EI2

1

EIBD Fcosh~axP!

2tanhS aLP

2 D sinh~axP!21G (39)

The above equation reduces to that of the fully plated beamsimply dropping the second term sinceL5LP in this case.

Transverse Plate-End Anchorage

The solution for fully or partially plated beams may also be obtained for the case of anchoring the plate ends with transvesheets, as experimentally done by Arduini et al.~1997!, throughreplacing the first boundary condition in Eq.~16! or the boundarycondition in Eq.~22! with

g50 at x50 ~ for fully plated beam!

or (40)

g50 at xp50 ~ for partially plated beam!

Applications

Example 1

The reinforced concrete beam~beam A! designed, constructed,strengthened with glass FRP, and tested under four-point bendby Saadatmanesh and Ehsani~1991!, see Fig. 5, is used as aprototype to study bond-slip, interface shear, and deflectionthe three loading cases considered above, see Fig. 3. A load bethe cracking level~30 KN! is assumed, for which the linear elasticassumptions hold, and a convergence study is conducted to vethe results and evaluate the minimum levels of the adhesive shmodulus ~G! yielding effective beam stiffening for the case ofully plated beams.

It is evident from Fig. 6 that interface shear stress distributiofor three-point bending, changes from very low values, an almounstrengthened beam, to those of the perfect composite actioG varies from 0.005 to 750 MPa. It may also be seen that tinterface shear stress closely reproduces the corresponding dibution from perfect bond analysis withG5750 MPa. On theother hand, it is seen to converge to the perfect bond value atplate end with a shear modulus as low as 10 MPa while it faconsiderably below that for lowerG values indicating poor effec-tive composite action. This may justify using interface shear vaues from perfect bond analysis to study interface fracture acrack initiation at plate ends for adhesives with a minimum secaG value, at any load level, greater than 10 MPa.

Fig. 7 shows the variation of interface shear strain along tspan for the same load case andG values. The figure simplyconfirms that significant shear strains take place forG lower than10 MPa whereas it converges closely to the expected perfect bsolution forG510 MPa.

The midspan deflection results in Fig. 8 indicate convergentowards the limiting cases of an unstrengthened beam and a Fstrengthened beam with a perfect bond. The deflection valuappear to approach those of a perfect interface for a shear molus higher than 10 MPa as well.

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Fig. 5. Beam A specimen of Saadatmanesh and Ehsani~1991!: ~a! beam layout;~b! cross section details;~c! section and material properties

ateieldhat

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The beam is reanalyzed assuming the same concentrload, above, to be distributed along the span. Figs. 9–11 ysimilar trends for this case with the added observation tthe results converge faster to the perfect bond solution with low

Fig. 6. Interface shear stress distribution in beam A for three-pobending case

J. Eng. Mech. 200

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G values. This is attributed to the low shear forces towards tmidspan providing the chance for interface shear stresses tovelop for the same levels of slip~shear strains! with lower Gvalues.

Fig. 7. Interface shear strain distribution in beam A for three-poinbending case


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Fig. 8. Midspan deflection of beam A for three-point bending cas

Fig. 9. Interface shear stress distribution in beam A for uniformloading case

Fig. 10. Interface shear strain distribution in beam A for uniformloading case


J. Eng. Mech. 20

The same beam is also analyzed under four-point bendinwith the same total load~30 KN!. It is interesting to see that thereis interface shear variation within the constant moment region fG of 10 MPa and less although overall cross-sectional shear fois zero, see Fig. 12. However, the interface shear stress is seeconverge to that of perfect bond analysis forG of 20 MPa andgreater. The variation of interface shear strain and midspan dflection appears to be the same as that of three-point bendiFigs. 13 and 14.

Norris et al.~1997! reported epoxyE values as low as~2.9–4.5 MPa! corresponding to (G51.12– 1.73 MPa!. EI-Mihilmyand Tedesco~2001! indicated that initialG values of 100–2500MPa have been reported by others for the same adhesive thness~1.5 mm!. However, effectiveG values were seen to drasti-cally reduce after cracking~Lee et al. 1999!. This is attributed tothe nonlinear adhesive shear response as well as crackRasheed and Pervaiz~2000! demonstrated the significant bondslip effect on the response of beam A, solved here, after crackthrough load-deflection comparisons of the experimental resuand nonlinear analysis considering local bond slip-tension stiffeing due to cracking and a perfect adhesive bond.

Fig. 11. Midspan deflection of beam A for uniform loading case

Fig. 12. Interface shear stress distribution in beam A for four-poinbending case

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Example 2

The same beam, analyzed above, is assumed to be subjectedservice load of 200 KN in three-point bending, which is substatially higher than cracking level~around 35 KN!. Thus fullycracked beam section rigidity is used in the calculations, see F5~c!. In this example the length of the FRP plate is varied and thtensile FRP force at midspan is computed for the partially platbeam with variousG values using Eq.~39!. It can be seen fromFig. 15 that the full composite action can be developed using veshort plates for values ofG greater than 100 MPa~e.g., Lp/250.49 m forG5100 MPa!. However, the interface shear stress iexpected to be higher for shorter plates. This may require longplates to maintain the interface shear stress below a certain le~e.g.,Lp/250.59 m forG5100 MPa andtavg.int50.15 MPa!, seeFig. 16.

It can also be seen from Fig. 15 that longer plates are needto develop the maximum possible tension force forG valueslower than 100 MPa due to the higher slip involved. This maxmum possible tension force is shown to be less than that corsponding to full composite action. As mentioned above, Fig. 1presents the variation of the average interface shear stress afunction of the plate length for differentG values. It can be seenthat a half plate length of~LP/250.147 m and 0.425 m! yields

Fig. 13. Interface shear strain distribution in beam A for four-poinbending case

Fig. 14. Midspan deflection of beam A for four-point bending case

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tavg.int50.15 MPa for G510 MPa. On the other hand,LP/250.94 m is needed to develop the maximum possible tensforce in this case, see Fig. 15. This may be the oppositedifferentG and/ortallowablevalues. Therefore it is important to useboth graphs to determine the FRP plate development lengthdesign. It is worth noting that Fig. 16 could be plotted in termsthe maximum interface shear stress instead of the average vby simply maximizing Eq.~23! with respect toxp .

Like other polymers, epoxy is expected to have a nonlineresponse in shear. Accordingly, it is important to study the effeof the shear modulus variation along the bond line on the behior of such structural elements, which is currently under invesgation by the authors.

Conclusions

A closed-form analytical solution is developed for the bond slanalysis of beams strengthened with full or partial FRP platusing beam theory with a shear deformable adhesive layer. Tsolution yields the interface shear stress distribution along tbond line, which can be used to determine the beam elastic cuby double integration. Direct analytical expressions are presen

Fig. 15. Development of FRP midspan tensile force for partiallplated beam under three-point bending case

Fig. 16. Average interface shear stress distribution for partialplated beam under three-point bending case


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for interface shear stress, slip distribution~in terms of the inter-face shear strain!, and deflections for the cases of three- and foupoint bending as well as uniform loading. The results show ththe present solution converges to the limiting cases of perfebond and no bond when high and low interfaceG values are used.They further help in defining the epoxy interface shear modulvalues needed for effective stiffening. The secantG values of theepoxy are known to reduce for higher stress levels due to tnonlinear shear response of such material. Accordingly, tpresent solution provides a useful tool to define the acceptasecantG values in design under service loads. In addition, thpresent solution is shown to offer guidance in identifying thminimum FRP plate development length. Such is the lengneeded to fully develop the midspan tensile force while maintaiing the interface shear stress level below allowable limits.

Acknowledgment

This work was supported by the College of Engineering anTechnology at Bradley Univ. through the Heuser Research Awa

References

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Bond Slip Analysis of Fiber-Reinforced Polymer-Strengthened Beams

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