Mechanics of Composite Materials, Vol. 36, No.

5, 2000

CRACKING ANALYSIS OF FRP-REINFORCED CONCRETE

FLEXURAL MEMBERS1

M. A. Aiello and L. Ombres

Keywords: reinforced concrete, fiber-reinforced polymer rebars, bond, slip, bending, cracks, prediction

The paper is dedicated to the cracking analysis of FRP (Fiber-Reinforced Polymer)-reinforced concrete elements.
A general nonlinear calculation procedure, based on the slip and bond stresses, is described and adopted for the
prediction of the crack width and crack spacing in FRP-reinforced concrete beams. An analytical expression of the
bond-slip law is estimated using the corresponding experimental results available in the literature. A numerical in-
vestigation is carried out and the influence of the mechanical and geometrical parameters of the material
(bond-slip law, reinforcement ratio, concrete strength, diameter of rebars, etc.) on the crack formation is investi-
gated. Referring to glass-FRP-reinforced concrete beams, a comparison between the theoretical predictions and
experimental results is made. The results obtained are presented and discussed.

1. Introduction

The use of fiber-reinforced polymers (FRP) as a reinforcement for concrete structures is becoming widespread in retrofit-
ting works and new constructions thanks to their numerous advantageous properties such as light weight, high strength, and
non-corrosive nature. However, because of the different behavior of FRP reinforcement with respect to the steel reinforcement, some
drawbacks arise and further studies are needed in this field. In particular, the low modulus of elasticity and the linear-elastic behavior of
FRP reinforcement involve high deformability, lack of ductility, and high crack width of FRP-reinforced flexural concrete members.
The structural behavior of FRP-reinforced concrete members, as is well known, is strongly conditioned by their serviceabil-
ity behavior; an analysis of cracking and deflection is, thus, essential for a correct design of such structures.
The physical and mechanical properties of FRPs are very different from those of steel. Such differences arise both from the
material properties and the interaction mechanisms between the FRP reinforcement and the concrete matrix. Consequently, the
analysis of FRP-reinforced concrete structures have to be based on procedures and design methodologies which, taking into account
the real behavior of the materials, allows one to discern the advantages and effectiveness of the use of FRPs [1].
The paper suggests a general approach to the analysis of the cracking behavior of FRP-reinforced concrete structures. An
analytical procedure, based on the slip and bond stresses, is described and adopted for the prediction of crack width and crack spac-
ing in FRP-reinforced concrete beams. The interaction between the FRP rebars and the concrete matrix is discussed and an analyti-
cal expression of the bond-slip law is estimated using the experimental results of bond-slip tests available in the literature.
Some numerical investigations are carried out and the influence of mechanical and geometrical parameters (bond-slip law,
reinforcement ratio, concrete strength, diameter of rebars, etc.) is investigated. Furthermore, referring to glass-FRP-reinforced con-
crete beams, a comparison between the theoretical predictions and experimental results is made and the results obtained are pre-
sented and discussed.

2. Problem Formulation

A mathematical model for cracking analysis is found based on the equilibrium and compatibility conditions for a cracked
element between two contiguous cracks. Referring to a flexural member the static problem is solved using the following equations:
1
Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11-15, 2000).

Department of Engineering Innovation, University of Lecce, Via per Arnesano, 73100 Lecce, Italy. Published in
Mekhanika Kompozitnykh Materialov, Vol. 36, No. 5, pp. 645-654, September-October, 2000. Original article submitted February
11, 2000.

0191-5665/00/3605-0389$25.00 ã 2000 Kluwer Academic/Plenum Publishers 389

 · compatibility condition for strains between two points, initially fully bonded, belonging to a FRP rebar and the concrete,
du
u( z ) = = e r ( z ) - e ct ( z ),
dz (1)
where u( z )is the slip between the concrete and the reinforcement and e r ( z ) and e ct ( z )are respectively the strains in the reinforce-
ment and the concrete at the reinforcement level, respectively,
· axial equilibrium condition for the reinforcement,
4
sr = t( z ),
db (2)
where t( z ) is the bond stress and d b is the diameter of rebars, and
· equilibrium conditions for sections,
ò s c dW c + i=1å, nwri s ri = 0,
Wc
(3)
ò s c ydW c + iå
=1, n
s ri y i wri = M ,
W2

where W c is the area of concrete, wri is the area of the ith rebar, and y i is the distance between the neutral axis of the member and the
ith rebar.
Equations (1), (2), and (3) constitute a system of differential equations, which can be solved if the boundary conditions and
the bond-slip law in an explicit form are given. The cracking analysis of reinforced concrete members must consider two different
situations, the first corresponding to the crack formation phase and the second one to the crack stabilization phase. At the crack for-
mation phase, in a section where the bond stress starts to develop, both the slip and its first derivative are equal to zero [u( 0) = 0,
u¢ ( 0) = 0]; the reinforcement stress is minimum at the points where the slip is zero and the bond stress changes its sign.
Considering the crack formation phase, the boundary conditions are homogeneous and the system can be solved in a closed
form. The boundary conditions for the crack stabilization phase are inhomogeneous and, consequently, it is impossible to solve the
system in a closed form; in this case, numerical solutions are nedeed.
2.1. A general numerical procedure for the cracking analysis. With reference to the crack stabilization phase in flexural
reinforced concrete members, we will present a general numerical procedure for the solution of the static problem. The analysis re-
fers to a beam element between two contiguous cracks in the region of a constant bending moment.
Basic assumptions. The analytical relationships are defined on the basis of the following underlaying assumptions:
1. Except at the cracks, the concrete participates in resisting the load (i.e., the bond stresses are produced).
2. The rebars and concrete follow the Hooke’s law.
3. Due to the differences in strains, slipping arise between the rebars and concrete. The bond-slip law is an interfacial prop-
erty, which is supposed to be valid along the rebar.
4. The bond stress reaches zero at a point halfway between the cracks.
5. There is no splitting cracks along the bars.
6. The crack width at the surface of a rebar is equal to the difference between the elongations of the rebar and concrete (i.e.,
to the sum of slips from both sides).
7. An effective area of the concrete in tension is considered; it is the area of the concrete surrounding the main reinforce-
ment and having the same centroid as the rebars.
8. The bond-slip law is nonlinear, namely the Bertero—Popov—Eligehausen law [1] is assumed.
The numerical procedure is based on the finite-difference method, by dividing the space between two cracks in n -1
subintervals of a small length Dz. An iterative procedure is adopted, which transforms the problem of limit conditions into the itera-
tive solution of an initial-value problem.
In the block between two cracks, the initial values of the stress and strain state can be found both at the halfway and the
cracked sections; these values are determined imposing the limit conditions on the problem. At the halfway point, the slip value is
zero, while at the cracked section, the tensile strain in the concrete at the reinforcement level is equal to zero.
The procedure starts from the halfway section, where the stress and strain states are determined by means of the compati-
bility and equilibrium conditions (initial values). Using these values, it is possible to obtain from Eqs (1), (2), and (3) the stresses
and strains at the end of the first subinterval of length Dz measured from the halfway section. The same procedure is also adopted for
the subsequent subintervals.
The procedure is stopped when the reinforcement strain reaches the value corresponding to the cracked section (initial
value), where the tensile strain in concrete at the reinforcement level is zero.

390
 The crack width at the reinforcement level, w r , is twice the sum of the slips occurring in each subinterval; the crack width
at the bottom side of the beam can be determined as w max = w r ( h - x c d - x c ) with x c the distance between the neutral axis and the
edge of the compressed side in the cracked section, h the height, and d the effective depth of the beam, respectively.
Based on the limit conditions, it is possible to analyze two limiting cracking configurations corresponding to the minimum
and maximum crack spacings, respectively. In this way, for each load value and for the assigned geometrical dimensions of beams
and mechanical properties of materials, the limit values of crack widths are determined. The actual values for a crack will be within
this range. It seems that the limit values for the cracking configuration can also be obtained taking into account the real structural be-
havior of reinforced concrete beams characterized by a random variance both of the mechanical properties of materials and of the
geometrical dimensions of structural elements.
2.2. Maximum crack spacing configuration: initial values. This configuration corresponds to the attainment of the con-
crete tensile strength at the tensed side of the halfway section without crack opening. Because of symmetry, cracks can open only at
a distance l max = 2l min , where l min is the distance between the cracked and halfway sections. It this case, slips are absent, u º 0, and
the stresses and strains in the constituents are obtained from the condition that the stress in concrete at the edge of the tensed side of
the beam is equal to the tensile strength of concrete.
2.3. Minimum crack spacing configuration: initial values. The minimum crack spacing corresponds to a crack opening
in the halfway section. Now, the minimum crack spacing, l min , is known, while the stress and strain at the halfway section are deter-
mined by an iterative procedure involving the equilibrium and compatibility conditions. In this case, the slip at the halfway is equal
to zero. In both cases, the initial values of the stresses and strains at the cracked section correspond to those within the section deter-
mined supposing that the tensed concrete does not take up a load.
2.4. The bond-slip law. The commercially available FRP rebars are characterized by a wide variability both from the geo-
metrical (surface treatment to improve the bond) and mechanical (resins and fibers) points of view. Therefore, significant differ-
ences in the structural behavior of concrete members reinforced with FRP rods occur. In particular, the wide variability of rebars in-
fluences the bond behavior between the reinforcement and concrete and, consequently, the static behavior of structures both in
service and at the ultimate state. In the cracking analysis, the knowledge of the bond is essential for evaluating the tension-stiffening ef-
fects. Therefore, consideration of the variability of parameters defining the bond-slip law between FRP rebars and concrete is neces-
sary for a reliable structural analysis.
The bond strength and the bond-slip law between the FRP rebars and concrete have been the subjects of numerous experi-
mental investigations. Actually, a large amount of experimental results carried out by many researchers are available; however, they
show great variations in the bond behavior both from qualitative and quantitative points of view. As a consequence, their use for de-
veloping analytical models presents a severe problem.
The experimental bond tests are of different kinds, such as pullout (eccentric or centric), splice, and beam tests [2, 3]. Each
of them imposes different stress conditions and, consequently, the results obtained depend on the test procedure adopted. In pullout
tests, the compression acting on the concrete surrounding the rebar reduces the possibility of cracking by increasing the bond
strength; in the beam tests, instead, the concrete surrounding the rebar is under tension, which causes cracking at low stresses and
thus reduces the bond strength. It seems, therefore, more realistic to carry out an appropriate bond test for each individual structural
condition (i.e., tension, bending, etc.) and to determine an analytical bond-slip law for use in structural analysis [4].
In the analysis of serviceability of FRP-reinforced beams, the results obtained from the beam tests are more realistic as they
better simulate the actual behavior of flexural members. Some of these results can furnish useful information for a reliable structural
analysis of FRP-reinforced concrete members, namely:
· the bond behavior is strongly dependent on the surface treatment of FRP rebars [2],
· the analytical representation of the bond-slip law for FRP-reinforced concrete elements can be obtained utilizing the
well-known Bertero— Popov—Eligheausen (BPE) law adopted for the traditional steel-reinforced concrete structures. The
parameters defining analytically the BPE law have to be determined on the basis of experimental results [2], and
· the bond strength derived from the pullout tests are approximately 55—85 percent higher than that obtained in the beam
tests [4].
Recently, a detailed analysis of the bond of FRP rebars to concrete has been carried out in [2], where the values of parame-
ters of the BPE bond-slip law are given based on the experimental results available. These parameters, determined from a limited
sample of experimental results, depend only on the surface conditions of FRP rebars. The procedure adopted, however, is very use-
ful, and the estimation of parameters can be improved by analyzing a wider sample of experimental data.
In the cracking analysis described here, the Bertero—Popov—Eligheausen relationship is adopted. Since this analysis is
restricted to the service conditions of beams, only the ascending branch of the law is considered. The BPE law is expressed as
a
t æ u ö
=ç ÷ ,
t1 çè u1 ÷
ø (4)

391
 1.4 b 1.4 b 0.05 r

0.04
1.2 1.2
t u t u
0.03
1.0 1.0
m=4.2467
0.02
m=2.016
0.8 0.8
0.01
m=2.948
w/wm w/wm w, mm
0.6 0.6
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0 1 2 3 4
Fig. 1 Fig. 2 Fig. 3

Fig. 1. b = t t m versus w w m (r = 0.010311, k = 0.1811, m = 2.948).

Fig. 2. b = u u m versus w w m (r = 0.010311, k = 0.1811, m = 3.6866).
Fig. 3. Crack width w versus the reinforcement ratio r with varying m (k = 0.1811).

where t1 and u1 are the peak bond stress and the slip at the peak bond stress, respectively, and a is an experimental parameter.

3. Numerical Investigation

In order to investigate the influence of geometrical and mechanical parameters on the crack width and crack spacing in
FRP-concrete beams, the above-described procedure is adopted. The analysis refers to beams with rectangular cross sections. The
parameters considered in the analysis were i) the constants of the bond-slip law (t1 , u1 ), ii) the diameter of rebars, iii) the tensile re-
inforcement ratio, iv) the cover thickness, and v) the strength of materials.
The numerical results are obtained allowing for the variations of the bond stress and slip with respect to their mean values. In
Figs. 1 and 2, dimensionless diagrams (b = t t m and u u m , versus w w m with w m as the crack width corresponding to the mean values
of the bond stress and the slip, t m and u m , respectively) are shown for a beam of rectangular cross section (r = A r bd is the reinforce-
ment ratio with A r the amount of FRP reinforcement, b and d the width and the effective depth of the section; k = c d with c the cover
thickness; m = 6M bd 2 f ct with M the applied moment and f ct the tensile strength of concrete). In the numerical analysis, the effec-
tive area of the concrete in tension is evaluated as A eff = bd eff = b ( c + 7. 5d b ).
From the results obtained it follows that the crack widths increase almost linearly with the bond strength and increase
slowly with increasing slip.
Figures 1 and 2 show that a 10% increase in t produces a 10% decrease in the crack width, whereas a 10% increase in u
leads to a 2% increase in the crack width.
The influence of the reinforcement ratio and of the cover thickness is shown in Figs. 3 and 4 for different values of the ap-
plied bending moments. As expected, the crack widths increase with the reinforcement ratio and decrease with the cover thickness.
Figures 5 and 6 show the influence of the material properties on the crack widths. It is seen that the crack widths increase
when the compressive strength of concrete increases or the elastic tensile modulus of the FRP decreases.

4. Estimation of the Bond-Slip Law Parameters

The bond stress and the corresponding slip have to be considered as mechanical properties of materials (concrete and rein-
forcement); as a consequence, like the compressive and tensile strengths, their reference values can be determined on the basis of
their statistical distributions.
For FRP-reinforced concrete structures, the experimental results available are not sufficient for a complete statistical anal-
ysis; these results, however, can furnish the reference values useful for practical purposes. Analytically, the bond-slip law has to be
defined through the evaluation of the parameters t1 , u1 , and a.
In our paper, the estimation of these parameters is carried out using the available experimental results of bond tests. Col-
lecting all the homogeneous results, i.e., those corresponding to the same FRP type (fibers and resins) with the same surface treat-

392
 0.4 k = c/d 2.0 w, mm 1.2 w, mm
m=3.5026
m=4.4285 1.0 m=3.6866
0.3 1.5
m=2.573
0.8 m=2.948
0.2 1.0 m=2.948
m=4.4285 0.6
m=2.016
0.1 0.5
m=2.016 0.4
w, mm fc, MPa E, GPa
0 0.2
0 0.5 1.0 1.5 2.0 21 28 35 42 49 56 63 70 36 38 40 42 44 46 48 50 52 54
Fig. 4 Fig. 5 Fig. 6

Fig. 4. Crack width w versus k with varying m (r = 0.010311).

Fig. 5. Crack width w versus the concrete strength with varying m (k = 0.1811).
Fig. 6. Crack width w versus the elastic modulus E of FRP with varying m (r = 0.010311, k = 0.1811).

TABLE 1. Experimental Values of the Parameters t, u, and a

t1, MPa u, mm a
Mean value 13.908 0.2638 0.2265
Coefficient of variation 0.1315 0.305 0.523

ment, it is possible to obtain several values of t1 , u1 , and a. Using the distributions of bond-slip law parameters described previ-
ously and taking into account the influence of these parameters on the cracking behavior of FRP-reinforced concrete elements, we
derive the following estimates for t1 and u1 :
t k = t m - k 0 st , uk = um + k 0 s u , (5)
where t m and u m are the mean values of the corresponding parameters, st and s u are the standard deviations of the bond stress and
the slip, respectively, and k 0 is a numerical coefficient. As a reference value for a k , the mean experimental value is assumed.

5. Comparison with the Experimental Results

The cracking behavior of FRP-reinforced concrete beams has been analyzed by many researchers and the results obtained are
available in the literature [5, 6]. These studies refer to beams reinforced with glass-FRP rebars. It is, thus, possible to make a compari-
son between the theoretical results and experimental data. A reliable estimation of the parameters characterizing the bond slip law is
necessary for correct theoretical predictions. Referring to the GFRP rebars (named C-BARTM, produced by Marshall Industries,
USA) used in experiments, the results of bond-slip tests are also available in the literature [7, 8]. In this analysis, the experimental
data were interpolated by the analytical bond-slip law (4). In Table 1, the mean values of the bond stress and slip with their coeffi-
cients of variation for each experimental curve are shown.
When using relations (5), the reference values defining the bond-slip law for the C-bar are as follows (k 0 is assumed equal
to 2): t k = 10.25 MPa, u k = 0.42476 mm, and a k = a m = 0.2265.
Using these values, the theoretical predictions were carried out and compared with those obtained experimentally. The re-
sults of comparison are shown in Figs. 7 and 8, and a good agreement between the theoretical predictions and experiments is seen
for the cases examined.

6. Conclusions

A cracking analysis of FRP-reinforced concrete beams is carried out in this paper. A nonlinear procedure based on the slip
and bond stresses is described and adopted for the prediction of the crack width and crack spacing in FRP-reinforced concrete
beams. The results obtained allow us to draw the following conclusions:

393
 7 M M
5

6
Experimental 4 Experimental
5 Theoretical
3
4 Experimental
2 Theoretical
3
w, mm w, mm
2 1
0 1 2 3 0 0.5 1.0 1.5 2.0 2.5 3.0
Fig. 7 Fig. 8
Fig. 7. Experimental and theoretical comparison [7]. Crack width versus the dimensionless applied
moment M (F2 beam) (r = 0.012165, k = 0.27586, E c = 26 GPa, d b = 12.7 mm).
Fig. 8. Experimental and theoretical comparison [8]. Crack width versus the dimensionless applied
moment M (CB2 beams) (r = 0.0071, k = 0.1879, E c = 33 GPa, d b = 14.9 mm).

·a reliable analysis of the cracking behavior of FRP-reinforced concrete beams requires a nonlinear approach able to take into
account the actual behavior of materials and the interaction between the concrete and FRP rods,
· the bond strength is the parameter influencing most significantly the cracking behavior of FRP-reinforced concrete mem-
bers. An increase in the bond strength of FRP rebars reduces the width of cracks,
· the crack width decreases with increasing reinforcement ratio and increases with increasing cover thickness, and
· for reliable estimation of the bond-slip law parameters used in the structural analysis, a better definition of the bond test pro-
cedure is needed.
Further studies, both theoretical and experimental, are necessary for a better understanding of the cracking behavior of
FRP reinforced concrete structures; in addition, design procedures suitable for use in creating such innovative structures have to be
elaborated.

REFERENCES

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