ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Title no. 94-S45

Parametric Study of Beams with Externally Bonded

FRP Reinforcement

by Marco Arduini and Antonio Nanni

FRP reinforcement may be externally bonded to the soffit of Independently of the repair strategy, it is essential to
existing flexural members in order to increase their strength and understand the consequences of the design choice in terms of
rigidity. A parametric analysis is conducted to investigate the crack propagation and failure mechanism. In fact, it has been
effects of FRP reinforcement on serviceability, strength, and failure observed in the literature that different failure mechanisms,
mechanisms of repaired RC beams. FRP reinforcement parameters
from ductile to very brittle, occur as externally bonded FRP
considered in the analysis are: stiffness, bonded length, thickness,
and the adhesive stiffness. The choice of the repair material reinforcement is added to a flexural member.1,2,3,4
parameters is important in the design phase in order to obtain the With reference to a simply supported, FRP-repaired, RC
desired results of strengthening or stiffening without other unfore- beam loaded at 4-point, four possible failure mechanisms4
seen effects. In this paper, three typical RC beam cross sections are have been summarized in the sketches of Figure 1 and are
considered with height-to-width ratios of 0.5, 1, and 4. Two char- listed below:
acteristic compressive strength levels (20 and 30 MPa), and two • FRP tensile rupture (R) when the FRP strain exceeds its
shear span-to-reinforcement depth ratios (4.5 and 7) are consid- ultimate value in the zone of maximum moment.
ered. All other parameters related to material and geometry of the • Concrete crushing (C) when the concrete compressive
beams are maintained constant. strain exceeds its ultimate value in the zone of
The results of the analysis are shown in terms of repaired-to-un- maximum moment.
repaired strength and deflection ratios. They indicate that brittle
• Debonding between FRP and concrete (D) due to
failure mechanisms can develop at loads much lower than expected
when considering only flexural performance controlled by failure at the concrete-adhesive interface. This failure
concrete crushing and FRP tensile rupture. The analytical model mechanism can initiate at any flexural crack and propa-
used for the parametrization accounts for brittle failure mecha- gates from there to the end of the FRP reinforcement.
nisms induced by debonding of the FRP reinforcement or shear- • Shear-tension failure (S) resulting from a combination
tension failure in concrete in the plane of the main longitudinal of shear and normal tensile stress in the concrete in the
steel reinforcing bars. Even when considering the limitation of the plane of the longitudinal steel bars. This failure mecha-
RC member due to its un-modifiable shear resistance, it is shown nism initiates at the ends of the FRP plate, results in the
that the application of FRP reinforcement can considerably propagation of a horizontal crack, and causes separa-
increase load resistance capacity and limit deflection at service. tion of the concrete cover.
The first two failure mechanisms occur after large deflec-
Keywords: analytical model; cracking; failure mechanism; FRP; numeri- tion of the member and are synonyms of better structural
cal simulation; repair; reinforced concrete; stiffening; strengthening.
performance. In the case of FRP rupture, the main steel rein-
forcement is past yielding. Moreover, from an economical
INTRODUCTION point of view, the rupture of the FRP plate seems to be desir-
The reinforcement of existing reinforced concrete (RC) able because it means that all the mechanical resources of
beams and slabs with fiber reinforced plastic (FRP) materials FRP (an expensive material) are utilized.
bonded to their soffit may be needed for different reasons: The third and fourth failure mechanisms are brittle and
reduce the vertical deflection at service (stiffening criterion), occur at values of the applied load lower than expected with
improve the maximum load capacity (strengthening criterion), conventional design equations. In both cases, the stiffening/
or limit the width and the distribution of cracks in concrete strengthening resources of the FRP plate are of little advan-
(durability criterion). The durability criterion is not addressed tage. Anchoring the FRP plate ends, not applicable to slabs,
in this paper. The designer has generally no control over the may attain a higher ultimate load and an increase in ductility.
existing structural element in need of repair. Geometry and However, the improvements may not be very significant.4
properties of existing steel reinforcement and concrete cannot
be modified. To satisfy stiffening and strengthening require-
ments, the designer may select the thickness (area) of the FRP ACI Structural Journal, V. 94, No. 5, September-October 1997.
Received November 5, 1995, and reviewed under Institute publication policies.
reinforcement and its stiffness. For the choice of the adhesive, Copyright © 1997, American Concrete Institute. All rights reserved, including the
the designer usually relies on the selection made by the manu- making of copies unless permission is obtained from the copyright proprietors.
Pertinent discussion will be published in the July-August 1998 ACI Structural
facturer of the FRP material system. Journal if received by March 1, 1998.

ACI Structural Journal/September-October 1997 493

Marco Arduini received his BSC at the University of Bologna in 1989. As a technical
officer at Testing and Modeling Laboratory of the University of Bologna, Italy, he has
conducted experimental and numerical projects in the areas of concrete and composites
for repair and strengthening of structures.

Antonio Nanni is a professor in the department of civil engineering at the University

of Missouri, Rolla. He is a member of ACI Committees 325, 440, 530, 544, and 549.

Fig. 1—Typical failure mechanisms of RC beams repaired

with FRP composites.
This paper presents an analytical study of non pre-cracked
RC beams of representative geometries and materials,
repaired with FRP plates of various thickness and mechan-
ical properties. The validity of the model used for the study
was previously verified.4 The model allows for the non-
linear behavior of the RC member due to the diffusion of
flexural cracks. This is essential for a correct interpretation
of experimental results. In fact, the elastic solution to the
problem of repaired RC with FRP composites cannot be
effectively applied as explained below.
Fig. 2—(a) Analytical discrete model; and (b) concrete and
Researchers5,6 have shown that in the elastic state, at the
steel constitutive laws.
interface of concrete and adhesive, shear and normal stress (τ
and σ) can be calculated using equations in hyperbolic func-
tions derived from seventh order (or higher) differential equa- the results reported in the paper are not absolutely general but
tions when applying boundary conditions that depend on relate to selected RC beam geometries and materials.
beam load and support configuration. The resulting stress
distribution shows significant stress concentration only at the MODEL
very ends of the FRP reinforcement. Experiments have shown With reference to Figure 2(a), a simply supported beam
that, during the phase of flexural cracking in concrete, the with a span equal to 2l is subdivided into n segments of length
distribution of τ and σ along the adhesive-concrete interface Dx. For each segment j, the external moments due to the
changes dramatically from that of the elastic phase. In the area applied load (4-point configuration) are computed and equi-
around each crack, high stress concentration originates due to librium conditions are imposed for the three subsystems of the
the presence of the FRP reinforcement that opposes the segment (i.e., concrete, adhesive, and FRP). Via equilibrium,
opening of the flexural crack. Generally, this occurs early the shear (τ) and normal (σ) stresses along the adhesive-
during the loading stage of the beam, since concrete has low concrete (τa1, σa1) and adhesive-FRP (τa2, σa2) interfaces for
tensile strength. Therefore, the brittle mechanism detected by each segment of the beam are calculated.4 Failure can origi-
the elastic solutions at the end of the plate is never activated, nate at the end of a segment where the combination of
nor does it control the true failure mechanism. Only in partic- maximum τa1 and σa1 crosses the Mohr-Coulomb failure
ular situations, such as with very thick FRP plates, could the domain of the interface adhesive-concrete (failure types S or
failure of the beam occur before flexural cracking of the D). It has been shown that the strength of the FRP-adhesive
interface is stronger than the concrete-adhesive interface.4 The
concrete and be successfully predicted by the elastic analysis.
other possible failure types occur when the maximum tensile
strain of the plate is reached (failure type R) or when the
RESEARCH SIGNIFICANCE maximum compressive strain in concrete is reached (failure
The paper intends to identify in a rational fashion the param- type C).
eters that affect performance of flexural members repaired The constitutive laws for the four constituent materials
with externally bonded FRP reinforcement. These parameters considered by the model are as follows. Compressive
must include pre-existing materials and geometry as well as concrete is non-linear and is influenced by the confinement
repair materials. The study of the inter-relationship among action due to closed stirrups (if provided) according to the
these parameters leads to the understanding of the limiting CEB-FIP Model Code 90.7 Tensile concrete is bilinear
factors and the possible modes of failure. Analytical studies elasto-softening as shown in Fig. 2(b). Steel is bilinear
such as this and their experimental verification are necessary elasto-hardening, as shown in Fig. 2(b). FRP and adhesive
for the development of sound design guidelines. It is noted that are perfectly linear elastic.

494 ACI Structural Journal/September-October 1997

 Some assumptions are made to simplify the problem: Table 1—Mechanical properties of concrete and
• Plane cross sections remain plane during loading. steel
• The Mohr-Coulomb failure surface for the adhesive- E, fck, fyk, ftk, τu, εcu, εu,
concrete interface does not change in the presence flex- Material GPa v MPa MPa MPa MPa percent percent
ural cracks. This means that no interaction exists Concrete C20 24 0.2 20 n/a 1.5 3.5 0.3 0.04
between horizontal normal stress, shear, and vertical Concrete C30 26 0.2 30 n/a 2.6 5.5 0.3 0.06
normal stress. Steel 200 0.3 n/a 440 600 n/a n/a 10.0
• The interface between adhesive and FRP is considered n/a = not applicable
stronger than the corresponding concrete-adhesive 1.0 ksi = 6.985 MPa; 1.0 Msi = 6.985 GPa
interface.
• The FRP plate and the adhesive layer are “thin,” there-
fore the stresses due to the bending moment acting on Table 2—Mechanical properties of FRP and
adhesive
them are disregarded.
In order to allow for the study of the effects of stiffness and Material Type E, GPa ft, MPa εu, percent
thickness of the adhesive layer, the model considers the E1 400 2000 0.5
normal (horizontal) force in the adhesive (Na) as part of the FRP E2 150 2250 1.5
equilibrium equation expressions (see sketch in Fig. 2[a]). E3 50 1500 3.0
During the concrete crack propagation, local tensile failure A1 3 60 2.0
in the adhesive may be recorded. Where this phenomenon Adhesive
A2 11 33 0.3
occurs, the stress transfer for both σ a1 and τ a1, applied to the Note: 1.0 ksi = 6.985 MPa; 1.0 Msi = 6.985 GPa
adhesive-concrete interface of that segment, is neglected.
The shear stress distribution at the interface adhesive-
concrete for each segment is considered triangular and the forcement ratio as for ACI 318-89.8 A minimum area of
maximum value is calculated, for a generic j segment, from: compressive reinforcement was taken into account. The
shear reinforcement ratio ρv = Av /bs was assumed constant
and equal to 0.003. Table 1 reports the characteristic
( τ al ,j ) max = ⎛ N j + 1 + N a ,j + 1 – N j – N aj ) --------------
2 (1)
⎝ b ⋅ Dx mechanical properties of concrete and steel as used in the
study adopting the symbols of the notation.
Since the parametrization is intended for mean values
Where the symbols nomenclature is given in the notation. rather than characteristic values, the mean strengths adopted
The normal stress distribution at the same interface is also in the model were selected to be 1.18 higher than the corre-
triangular and its maximum value in a generic j segment of sponding characteristic strengths and are reported with the
the beam is: same symbol without the subscript k.
The second group of parameters refers to the properties
⎛ ⎛ ⎛
( σ al ,j ) 6 -
= ⎜ N j + 1 – N j ) ⋅ ⎜ dv j + t a + t p ⁄ 2 ) + ⎜ N a ,j + 1 – ( N a ,j ) ⋅ ( dv j + t a ⁄ 2 ) ----------------- (2) and geometries of the repair materials (FRP composite and
max ⎝ ⎝ ⎝ b ⋅ Dx
2
adhesive). In this study, the thickness of the FRP (tp) was
varied in the range of 0 to 2 mm (0 to 0.08 in.). The bonded
Since no shear reinforcement can be added to the existing length of the FRP-to-shear span (p/a) ratio was varied
member, its nominal shear capacity (Vn) is also calculated in between 0.60 and 0.95. Three FRP stiffness values with
order to determine whether or not it becomes the controlling corresponding maximum strain at failure were considered.
factor after repair. Vn is computed according to ACI 318-89,8 Table 2 reports the mechanical characteristics of the FRP
equations 11-3 and 11-17 and can be expressed as (in SI units): materials. For the adhesive, two stiffness values with corre-
sponding maximum strain at failure were adopted. The adhesive
thickness was maintained equal to 1 mm for all cases. Mechan-
f ck bd A v f yk d
V n = ----------------
- + --------------- (3) ical properties of the adhesives are reported in Table 2.
6 s
Results for one beam type
PARAMETRIC STUDY Stiffness—If the scope of the design is the stiffening crite-
Materials and geometries rion, the expected outcome must be a reduction of the
The parameters that influence the behavior of an RC beam maximum deflection of the member under service loads. For
repaired with FRP can be further subdivided into two groups. this study, it was necessary to define the service load level as
The first group consists of properties and geometries of the a fraction of the ultimate flexural capacity. The ratio between
constituents of the existing RC member including support ultimate and service load was assumed to be 1.5, as it would
conditions and loading configuration. In this paper three result from a reasonable combination of dead and live loads,
doubly-reinforced rectangular cross-sections were consid- which have load factors of 1.4 and 1.7, respectively, according
ered with height-to-width (h/b) ratios of 0.5, 1, and 4. The to ACI 318-89. The deflection at service load (Fs – Fu /1.5)
first cross-section was intended to represent the case of a slab was computed for beams with FRP repair (dsr) and for iden-
and did not include any shear reinforcement. For all beam tical beams without repair (ds). The ratio dsr /ds is plotted as a
types, a simply-supported configuration was adopted (4- function of the FRP plate thickness tp in Fig. 3. This diagram
point loading) with shear span-to-reinforcement depth (a/d) represents the case of beams with the following characteris-
ratios of 4.5 and 7. Two characteristic compressive concrete tics: h/b = 1; a/d = 4.5; fck = 30 MPa (4.35 ksi) and the adhe-
strengths were adopted (20 and 30 MPa, 2.9 and 4.35 ksi). sive type is A1. Curves were obtained for p/a ratios varying
The longitudinal steel reinforcement ratio was taken as 0.5 between 0.6 and 0.95. Only the two limiting cases are shown
ρmax, with ρmax equal to 75 percent of the balanced rein- in the figure, since the effect of this ratio is almost insignifi-

ACI Structural Journal/September-October 1997 495

Fig. 3—Ratio between deflection of the FRP reinforced
beam (dsr) and the deflection of the RC on (ds) vs. FRP plate
thickness (tp) (identical service load, different type of FRP
materials); 25.4 mm = 1 in.

Table 3—Effect of adhesive

Adhesive A1 Adhesive A2
Failure Failure
FRP type p/a tp, mm mode Fur/Fu mode Fur/Fu
0.1 R 1.27 R 1.27
0.6 0.5 S * S *
1 S * S *
2 S * S *
0.1 R 1.27 R 1.27
0.5 S 2.00 S 2.00
E1 0.85 1 S 2.20 S 2.00
2 S 1.82 S 1.82
0.1 R 1.27 R 1.27
0.5 R 2.73 D 2.00
0.95 1 D 3.09 D 2.73
2 D 2.91 D 2.91
0.1 R 1.36 S 1.09
0.5 S 1.18 S 1.09
0.6 1 S * S *
2 S * S *
0.1 R 1.46 D 1.09
E2 0.85 0.5 S 2.55 D 1.64
1 S 2.00 D 1.82
2 S 2.00 S 2.00
0.1 R 1.46 D 1.09
0.95 0.5 D 2.73 D 1.64
1 D 3.09 D 1.82
2 D 3.27 D 2.55
0.1 C 1.18 D *
0.6 0.5 C 1.64 D 1.18
1 S 1.46 S 1.27
2 S 1.09 S * Fig. 4—(a) Ratio between ultimate load of the FRP rein-
C 1.18 D * forced beam (Fur) and the ultimate load of the RC one (Fu)
0.1
E3 0.85 0.5 C 1.64 D 1.27 vs. FRP plate thickness (tp) for FRP type E1; (b) Ratio
1.2 C 2.18 D 1.46 between ultimate load of the FRP reinforced beam (Fur) and
S 2.18 D 1.64
the ultimate load of the RC one (Fu) vs. FRP plate thickness
0.1 C 1.18 D *
0.5 C 1.64 D 1.27 (tp) for FRP type E2; (c) Ratio between ultimate load of the
0.95 1 C 2.18 D 1.46 FRP reinforced beam (Fur) and the ultimate load of the RC
2 D 2.91 D 1.64 one (Fu) vs. FRP plate thickness (tp) for FRP type E3. 25.4
*Shear failure or debonding are detected at load lower than the unrepaired beam mm = 1 in.
(25.4 mm = 1 in.)

FRP repaired beam (Fur) and the ultimate load of the un-
cant. Three families of curves can be observed depending on repaired beam (Fu). Figures 4(a), (b), and (c) are the summary
the FRP stiffness. As expected, the reduction of the deflection of the Fur/Fu ratio plotted as a function of the FRP plate thick-
ratio at service is strongly influenced by FRP stiffness and ness for the same beam material and geometry combinations
thickness. used in Fig. 3. Each portion of the figure represents a family
Strength—When one considers the strengthening criterion, of four curves obtained for a given FRP stiffness at the varia-
the outcome of interest is the ratio between ultimate load of the tion of the FRP bonded length-to-shear span (p/a) ratio. With

496 ACI Structural Journal/September-October 1997

reference to the family of curves obtained for the highest FRP
stiffness (i.e., E1 in Fig. 4[a]), it is observed that the ultimate
strength ratio is strongly affected by the p/a ratio. For values
of p/a less than 0.65, there is practically no benefit in repairing
the beam for strength. Moreover, points on the diagram at FRP
thickness values of 0.0, 0.1, 0.5, 1.0, and 2.0 mm are labeled
with a letter that indicates the type of failure as previously
discussed. When no FRP material is used (tp = 0), concrete
crushing is the dominant failure mode. When the thickness of
FRP is 0.1 mm, the dominant failure mode is rupture of the
FRP independently of the p/a ratio. When thickness of the
FRP is 0.5 mm, rupture of the FRP is only obtained for the
case of p/a equal to 0.95. In all other cases, shear-tension
failure is the dominant mode. This failure type is brittle and
therefore undesirable. In addition, the occurrence of shear-
tension or debonding failure indicates that it is no longer
possible to increase the flexural capacity of the member by
increasing the FRP thickness. This is clearly shown in the
diagram for the remaining parts of the four curves. A final
observation is related to the horizontal line indicating the
value of the ultimate load ratio as controlled by shear capacity
assuming that ρv is equal to 0.003. The line is horizontal
because the repair method considered in this paper does not
improve the shear strength of the existing RC member.
Considerations similar to the ones reported above can be
repeated for the remaining two family curves obtained for E2
and E3 in Fig. 4(b) and (c). In general, the lower the FRP
stiffness, the higher its thickness needs to be to obtain a
given strength improvement.
When one considers the strengthening criterion, it is also
mandatory to determine whether or not the deflection of the
repaired member under the new (and higher) service load is
acceptable. A possible way to address this issue is by consid-
ering the ratio between deflection at service of the repaired
system under the new load (dnsr) and the deflection at service
of the un-repaired system under the old load (ds). The cases
of new and old loads were assumed to be equal to Fur /1.5 and
Fu/1.5, respectively. Figures 5(a), (b), and (c) are a summary
of the variation of the dnsr /ds ratio as a function of the FRP
thickness for the same cases given in Fig. 4(a), (b), and (c).
The diagrams are constructed as previously described and
the data labels correspond to the ones given in Figures 4(a),
(b), and (c). The most important observation at this point is
that, if the deflection for the repaired system under the new
service load should not exceed that of the unrepaired system
under the old load, then only dnsr /ds values equal or less than
1.0 become acceptable. An example can be given using a
type E3 plate with thickness of 1 mm (0.04 in.) and p/a ratio Fig. 5—(a) Ratio between deflection at new service load of
of 0.95. From Fig. 4(c), it is seen that the ultimate flexural the FRP reinforced beam (dnsr) and the deflection a the old
capacity could be increased 2.2 times after repair. If the service load of the RC beam (ds) vs. FRP plate thickness (tp)
service load were increased of the same amount, it would for FRP type E1; (b) ratio between deflection at new service
result in a deflection nearly 2.0 times the existing deflection load of the FRP reinforced beam (dnsr) and the deflection at
(see Fig. 5[c]). It can be shown that, if the service load for the old service load of the RC beam (ds) vs. FRP plate thick-
this beam is increased only 1.5 times, then the new deflection ness (tp) for FRP type E2; and (c) ratio between deflection at
coincides with that of the original member. new service load of the FRP reinforced beam (dnsr) and the
To better understand the failure mechanisms of the deflection at the old service load of the RC beam (ds) vs. FRP
repaired beams presented in the previous diagrams, Fig. 6 plate thickness (tp) for FRP type E3. 25.4 mm = 1 in.
and 7 present the distribution of selected stresses as a func-
tion of the position along the beam axis (expressed as
distance from the support-to-half span ratio = x/l) under the stress in the FRP plate (σp). σ1 becomes 0 if the tensile
ultimate load. These stresses are: maximum shear stress at strength of concrete is overcome and this does represent a
the adhesive-concrete interface (τa1,max), maximum tensile flexural crack propagation. The case presented in Fig. 6 (i.e.,
stress at the same interface (σa1,max), horizontal tensile stress E1, tp = 0.5 mm, and p/a = 0.95) is that dominated by rupture
at the bottom concrete fiber (σ1), and longitudinal tensile of the FRP. Rupture of the FRP occurs in the constant

ACI Structural Journal/September-October 1997 497

 the thinner the adhesive layer, the lower the likelihood of
concrete-adhesive interface (or concrete) failure.
Table 3 presents a comparison of the results obtained with
the RC beam described above when A1 and A2 adhesives are
used. The first three columns in the table show the FRP
parameters. Columns 4 and 6 compare the failure mode for
adhesives A1 and A2, respectively, and Columns 5 and 7,
compare the Fur/Fu ratio. When A2 is used, the debonding
failure mechanism prevails with a significant decrease of the
ultimate load capacity. This failure mechanism is facilitated
by the low ultimate strain of the adhesive. As the adhesive
fails in uniaxial tension in a given zone, there is no shear and
normal stress transfer between FRP and concrete. At the ends
of this zone, a high shear and normal stress transfer is needed
to balance the normal (horizontal) force in the unbonded FRP
plate. This concentration of stress causes the concrete failure.
With regard to the stiffening criterion, the results (not shown
Fig. 6—Distribution of selected stresses vs. position along here) are similar for both adhesives because deflections are
the axis of FRP reinforced beam; 1 ksi = 6.985 MPa. comparatively small under service load.

Results for three beam types

In the second part of the parametrization analysis, the
effects of different h/b and a/d ratios and strength of concrete
are considered. The parameters maintained constant are ρ =
0.5 ρmax, p/a = 0.85, FRP type E2, and adhesive type A1.
Figures 8(a), (b), and (c) present the evolution of the dsr /ds,
Fur/Fu and dnsr /ds ratios as a function of a parameter k
defined as

Ep Ap h 3
k = -----------
- --- a (4)
EI b

In each figure three families of two curves are shown. Each

family was obtained for a given h/b and a/d ratio. The two
curves per family are a function of the concrete compressive
strength, 20 and 30 MPa (2.9 and 4.35 ksi), respectively.
Fig. 7—Distribution of selected stresses vs. position along Figure 8(a) seems to indicate that deep beams (h/b = 4) can
the axis of FRP reinforced beam; 1 ksi = 6.985 MPa. hardly be stiffened. Sizable results can be obtained with slab-
type sections (h/b = 0.5). The best results are obtained for a
square-shaped section at a/d = 4.5.
moment region (σp = 2000 MPa [290 ksi]) and is reached Similarly, Fig. 8(b) seems to indicate that strengthening is
when almost the entire beam has experienced flexural more suitable for slab-type sections and square-shaped
cracking (σ1 is indicated as 0 value). The shear stress has sections. It is worth noting that more FRP reinforcement could
maximum values at the end of the plate and near the constant be added in the slab-type section with 30 MPa (4.35 ksi)
moment region, but they are below the ultimate value of 5.5 concrete since the failure mode is not of the brittle type. The
MPa (0.8 ksi). The case presented in Figure 7 (i.e., E1, tp = three pairs of horizontal lines represent the Fur /Fu ratio based
2.0 mm, and p/a = 0.85) is that dominated by shear-tension on the shear strength of each beam type. Two lines are given
failure in the concrete. When the FRP thickness is high, the for each beam as the compressive strength of concrete varies
maximum shear stress is responsible for failure at the end of from 20 to 30 MPa (2.9 and 4.35 ksi). Three observations are
the FRP plate. In this case, only 60 percent of the beam has made:
experienced flexural cracking, and the stress in the FRP plate • For two out of three cases, the Fur /Fu ratio of the 20 MPa
is small (500 MPa = 72.5 ksi). (2.9 ksi) concrete is higher than that corresponding to
The FRP debonding mechanism (the other brittle failure 30 MPa (4.35 ksi) concrete. In fact, as fck increases,
mode) can be activated at the FRP plate end or in any zone more flexural steel reinforcement is added to maintain
where a flexural concrete crack is generated. The latter case ρ = 0.5 ρmax. Therefore, flexural capacity increases in a
is mainly noted with long and high strength FRP plates. way directly proportional to fck, whereas the concrete
Effect of adhesive—Resins used for adhering FRP to contribution to shear capacity only increases proportion-
concrete may have low modulus and high deformability as ally to f ck .
well as high modulus and low deformability. Two represen- • The slab-type beam is without stirrups and therefore it
tative types are shown in Table 2. Even though the thickness has a relatively low shear strength. The efficiency of the
of the adhesive layer was kept constant in this study, it is FRP repair may be low in this type of application.
noted from Equation 2 that the normal vertical stress (σa1) is • For the deep beam, shear strength seems not to be a
directly proportional to the adhesive thickness. Therefore, limiting factor.

498 ACI Structural Journal/September-October 1997

 Fig. 9—(a) ratio between deflection at service load for the
reinforced beam (dsr) and deflection at service load for the
RC one (ds) vs. FRP thickness (tp) for RC beams with distrib-
uted uniformly load; and (b) ratio between ultimate linear
load of the FRP reinforced beam (qur) and the ultimate linear
load of the RC beam (qu) vs. FRP thickness (tp) for RC beams
with distributed uniformly load. 25.4 mm = 1 in.

FRP strengthened beams may need to be reduced to less than

Fur /1.5 to prevent unacceptable deflections.
Results for a beam subjected to uniformly distributed load
The case presented in Fig. 9(a) and (b) is relative to a
simply supported beam with a total span-to-height (2l/h)
ratio of 6.7 and subjected to uniformly distributed load. The
h/b ratio is equal to 3, lp/l is equal 0.85, concrete has a
strength of 30 MPa (4.35 ksi), ρ equals 0.5 ρmax, ρv equals
Fig. 8—(a) Ratio between deflection at service load for the 0.003, and the adhesive is type A1.
FRP reinforced beam (dsr) and deflection at the old service For the stiffening criterion, Fig. 9(a) shows the ratio
load for the RC one (ds) vs. parameter k; (b) ratio between between the mid-span deflection of the repaired beams (dsr)
ultimate load of the ultimate load of the FRP reinforced and the mid-span deflection of the un-repaired beam (ds) as
beam (Fur) and the ultimate load of the RC beam (fu) vs. a function of the FRP plate thickness (tp), for three different
parameter k; and (c) ratio between deflection at new service FRP types. It is reminded that deflections are at the same
load of the FRP reinforced beam (dnsr) and the deflection at service load level for repaired and un-repaired beams. A
the service load of the RC beam (ds) vs. parameter k. significant reduction of vertical displacement is only attain-
able with a very stiff FRP plate.
The grouping of the three families of curves is not that For the strengthening criterion, Fig. 9(b) reports the ratio
evident in Figure 8(c). For both the slab-type section and the between ultimate linear load of the repaired beams (qur) and
deep section, deflection under the new service load is higher the ultimate linear load of the un-repaired one (qu) as a func-
than deflection under the old service load. As for the tion of the FRP thickness. The shear capacity due to concrete
previous case of Figures 5(a), (b), and (c), the service load and stirrups is only 40 percent higher of the flexural un-
levels are computed by dividing the ultimate loads by a repaired strength. Therefore the extent of FRP repair is
factor of 1.5. This diagram points out that service load for limited. FRP with low modulus and high thickness gives the

ACI Structural Journal/September-October 1997 499

best performance even in terms of failure mechanism provided Fu = the ultimate load of the un-repaired beam
that shear strength near the supports does not become the Fur = the ultimate load of the repaired beam
fck = the characteristic compressive strength of concrete
controlling factor. ftk = the characteristic concrete tensile strength
fyk = the characteristic tensile yielding of steel
CONCLUSIONS h = the beam height
In summary, FRP repair of existing RC flexural members I = the gross moment of inertia (bh3/12)
l = the half span of the beam
may be structurally necessary for one of two reasons: stiff- lp = the half span of the FRP reinforcement
ening or strengthening. Depending on the criterion and the Nj = the normal (horizontal) force in the FRP plate in the j segment
conditions (i.e., materials and geometries) of the existing Na,j = the normal (horizontal) force in the adhesive in the j segment
member, the repair method may be more or less effective. p = the FRP bonded length in the shear span zone
qu = the ultimate linear load of the un-repaired beam
If a designer is only concerned with stiffening, the repaired qur = the ultimate linear load of the repaired beam
element is not required to carry any additional service load. In s = the stirrups spacing
general, stiffening is always attainable. For the same FRP thick- ta = the adhesive thickness
ness, the higher the FRP stiffness, the better the results. The tp = the FRP thickness
failure mode of the repaired system may become brittle, Vn = the nominal shear capacity of reinforced concrete
εcu = the ultimate compressive strain, according to Ref. 7
depending on several parameters, which include existing εu = the ultimate tensile strain
member conditions as well as repair parameters (e.g., p/l ratio). v = the Poisson’s ratio
If a designer is concerned with strengthening an existing ρ = the reinforcement ratio equal to As/bd
structural member and improving a given amount its load ρmax = equal to 75 percent of the balanced reinforcement ratio as for
ACI 318-89
carrying capacity at service, the success of the repair and the ρv = the shear reinforcement ratio equal to Av/bs
selection of the FRP stiffness, thickness, and bonded length σa1,j = the normal (vertical) stress at the concrete-adhesive interface in
has to be based on the limitations imposed by: the j segment
• Shear strength of the existing member σa2,j = the normal (vertical) stress at the FRP-adhesive interface in the j
segment
• mode of failure of the repaired system σp = the longitudinal tensile stress in the FRP plate
• Deflection at new service load τa1,j = the shear stress at the concrete-adhesive interface in the j
segment
In general, the bonded length of FRP should be as long as τa2,j = the shear stress at the FRP-adhesive interface in the j segment
possible to have a better use of the FRP strength resources τu = the concrete shear strength
and to activate failures such as concrete crushing or FRP
rupture. The adhesive should have high ultimate elongation. REFERENCES
1. Saadatmanesh, H., and Ehsani, M., “RC Beams Strengthened with
GFRP Plates,” Part I and Part II, Journal of Structural Engineering, ASCE,
ACKNOWLEDGMENTS
Financial support was partially provided by the Italian Ministry of V. 117, No. 11, Nov. 1991, pp. 3417-3455.
University Research (MURST) and the National Science Foundation (NSF). 2. Chajes, M. J.; Thomson, T. A.; Januszka, T. F.; and Fin, W., “Flexural
Strengthening of Concrete Beams Using Externally Bonded Composite
Materials,” Construction and Building Materials, 1994, V. 8, No. 3, pp. 1212-
NOTATION 1225.
Ap = the area of the cross section of FRP (tp b) 3. Arduini, M.; D’Ambrisi, A.; and Di Tommaso, A., “Shear Failure of
Av = the stirrups area Concrete Beams Reinforced with FRP Plates,” Proceedings, New Materials
a = the shear span and Methods for Repair ASCE, San Diego, Nov. 13-16, 1994, pp. 415-423.
b = the beam width
4. Arduini, M.; Di Tommaso, A.; and Nanni, A., “Brittle Failure in FRP
Dx = the segment length
Plate and Sheet Bonded Beams,” ACI Structural Journal, V. 94, No. 4,
d = the effective depth of the steel reinforcement
July-Aug. 1997, pp. 363-370.
dnsr = the midspan deflection of the repaired beam under the new
service load 5. Frostig, Y. et al. “High-Order Theory for Sandwich-Beam Behavior
ds = the midspan deflection of the un-repaired beam under the old with Transversely Flexible Core,” Journal of the Engineering Mechanics
service load Division, ASCE, V. 118, No. 5, May 1992, pp. 1026-1043.
dsr = the midspan deflection of the repaired beam under the old 6. Arduini, M., and Di Leo, A., “Analisi elastica lineare di travi,”
service load INARCOS, V. 522, No. 9, Sept. 1991, pp. 450-453.
dvj = the increment of vertical displacement of the j+1 segment with 7. Model Code 90, CEB-FIP Committee, Lausanne, 1993.
respect to j 8. ACI Committee 318, “Building Code Requirements for Reinforced
E = the elastic modulus Concrete (ACI 318-89) and Commentary (318R-89), American Concrete
Ep = the longitudinal elastic modulus of FRP Institute, Detroit, 1992, 338 pp.

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