Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates
A theoretical method to predict the interfacial stresses in the adhesive layer of damaged reinforced concrete beams strengthened with externally bonded carbon fiber-reinforced polymer plate is presented. The adopted model is developed including the adherend shear deformations by assuming a linear sh...
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| Veröffentlicht in: | Проблемы прочности |
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| Datum: | 2013 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2013
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| Zitieren: | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates / T. Hassaine Daouadji // Проблемы прочности. — 2013. — № 5. — С. 104-118. — Бібліогр.: 19 назв. — англ. |
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| author | Hassaine Daouadji, T. |
| author_facet | Hassaine Daouadji, T. |
| citation_txt | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates / T. Hassaine Daouadji // Проблемы прочности. — 2013. — № 5. — С. 104-118. — Бібліогр.: 19 назв. — англ. |
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| description | A theoretical method to predict the interfacial stresses in the adhesive layer of damaged reinforced concrete beams strengthened with externally bonded carbon fiber-reinforced polymer plate is presented. The adopted model is developed including the adherend shear deformations by assuming a linear shear stress through the depth of the reinforced concrete beam, while all existing solutions neglect this effect. In addition, in the present study the anisotropic damage model is adopted to describe the damage of the reinforced concrete beams. It is shown that the damage has a significant effect on the interfacial stresses in fiber-reinforced polymer damaged reinforced concrete beam.
Предложен аналитический метод оценки межповерхностных напряжений в слоях поврежденных железобетонных балок, укрепленных наружными композитными пластинами из армированного волокнами углепластика. Предложена модель, которая, в отличие от существующих
решений, учитывает наличие межповерхностных касательных деформаций путем постулирования линейного распределения касательных напряжений по глубине железобетонной балки.
Разработана анизотропная модель разрушения, описывающая процесс повреждения железобетонной балки. Установлено, что повреждение существенно влияет на величину межповерхностных напряжений в поврежденных железобетонных балках, армированных композитными пластинами.
Запропоновано аналітичний метод оцінки міжповерхневих напружень у шарі пошкоджених залізобетонних балок, що закріплені зовнішніми композитними пластинами з армованого волокнами вуглепластика. Запропоновано модель, яка, на відміну від існуючих рішень, враховує наявність міжповерхневих дотичних деформацій шляхом постулювання лінійного розподілу дотич-
них напружень по глибині залізобетонної балки. Розроблено анізотропну
модель руйнування, що описує процес пошкодження залізобетонної балки.
Установлено, що пошкодження суттєво впливає на величину міжповерхневих напружень у пошкоджених залізобетонних балках, армованих композитними пластинами.
|
| first_indexed | 2025-12-07T13:17:02Z |
| format | Article |
| fulltext |
UDC 539.4
Analytical Analysis of the Interfacial Stress in Damaged Reinforced
Concrete Beams Strengthened by Bonded Composite Plates
T. Hassaine Daouadji
University of Tiaret (Université Ibn Khaldoun de Tiaret), Tiaret, Algerie
daouadjitah@yahoo.fr
ÓÄÊ 539.4
Àíàëèòè÷åñêèé ðàñ÷åò ìåæïîâåðõíîñòíûõ íàïðÿæåíèé â ïîâðåæ-
äåííûõ æåëåçîáåòîííûõ áàëêàõ, àðìèðîâàííûõ êîìïîçèòíûìè
ïëàñòèíàìè
Ò. Õàññàéí Äàóàäæè
Óíèâåðñèòåò èì. Èáí Õàëäóíà, Òèàðåò, Àëæèð
Ïðåäëîæåí àíàëèòè÷åñêèé ìåòîä îöåíêè ìåæïîâåðõíîñòíûõ íàïðÿæåíèé â ñëîÿõ ïîâðåæ-
äåííûõ æåëåçîáåòîííûõ áàëîê, óêðåïëåííûõ íàðóæíûìè êîìïîçèòíûìè ïëàñòèíàìè èç àðìè-
ðîâàííîãî âîëîêíàìè óãëåïëàñòèêà. Ïðåäëîæåíà ìîäåëü, êîòîðàÿ, â îòëè÷èå îò ñóùåñòâóþùèõ
ðåøåíèé, ó÷èòûâàåò íàëè÷èå ìåæïîâåðõíîñòíûõ êàñàòåëüíûõ äåôîðìàöèé ïóòåì ïîñòóëè-
ðîâàíèÿ ëèíåéíîãî ðàñïðåäåëåíèÿ êàñàòåëüíûõ íàïðÿæåíèé ïî ãëóáèíå æåëåçîáåòîííîé áàëêè.
Ðàçðàáîòàíà àíèçîòðîïíàÿ ìîäåëü ðàçðóøåíèÿ, îïèñûâàþùàÿ ïðîöåññ ïîâðåæäåíèÿ æåëåçî-
áåòîííîé áàëêè. Óñòàíîâëåíî, ÷òî ïîâðåæäåíèå ñóùåñòâåííî âëèÿåò íà âåëè÷èíó ìåæïîâåðõ-
íîñòíûõ íàïðÿæåíèé â ïîâðåæäåííûõ æåëåçîáåòîííûõ áàëêàõ, àðìèðîâàííûõ êîìïîçèòíûìè
ïëàñòèíàìè.
Êëþ÷åâûå ñëîâà: ïîâðåæäåííàÿ áåòîííàÿ áàëêà, ìåæïîâåðõíîñòíûå íàïðÿ-
æåíèÿ, óïðî÷íåíèå, êîìïîçèòíûå ïëàñòèíû.
Introduction. Advanced composite materials, e.g., fiber-reinforced polymers
(FRP), have found their new applications in the rehabilitation of reinforced
concrete structure [1, 2]. Compared with the traditional materials, composite
materials have some unique features, i.e., high strength and stiffness to weight
ratio, attractive corrosion resistance and ease of handling and application [3, 4].
Among these materials, carbon fiber-reinforced polymers (CFRP) are extensively
used because of their unparalleled characteristics [5, 6]. The transferring of stresses
from concrete to the FRP reinforcement is central to the reinforcement effect of
FRP-strengthened concrete structures. This is because the stresses are susceptible
to cause the undesirable premature and brittle failure, such as debonding of the
soffit plate from the RC beam. This debonding failure mode is brittle and prevents
the full utilization of the tensile strength of the bonded plate. It is therefore
important to understand the mechanism of this debonding failure mode and
develop sound design rules. This brittle mode of failure is a result of the high shear
and vertical normal (peeling) stress concentrations arising at the edges of the
bonded FRP strip. Hence, this limited area in the close vicinity of the bonded strip
© T. HASSAINE DAOUADJI, 2013
104 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
edge, subjected to high peeling and interfacial shear stresses, proves to be among
the most critical parts of the strengthened beam. Consequently, the determination
of interfacial stresses has been researched for the last decade for beams bonded
with either steel or advanced composite materials [7, 8]. In particular, several
closed-form analytical solutions have been developed [9, 10]. All these solutions
are for linear elastic materials and employ the same key assumption that the
adhesive is subject to normal and shear stresses that are constant across the
thickness of the adhesive layer. It is this key assumption that enables relatively
simple closed-form solutions to be obtained. In the existing solutions, two different
approaches have been employed. Roberts [11] and Roberts and Haji-Kazemi [12]
used a staged analysis approach, while Vilnay [9], Taljsten [13], Malek et al. [14],
and Smith and Teng [10] considered directly deformation compatibility conditions.
Rabinovich and Frostig [15] have presented a higher-order approximate
interfacial stress. A disadvantage, of this solution is that, unlike the other solutions,
explicit expressions are not available for the interfacial stresses, so results are not
easily obtainable, which makes it difficult to be exploited in developing a design
rule. Shen et al. [16] presented a higher-order in which explicit expressions have
been obtained, but these are much more complex than the expressions from other
analyses.
Most of the research efforts have focused on strengthening of undamaged RC
beams with externally bonded sheets, whereas the interfacial stresses in damaged
RC beams strengthened by externally bonded FRP strips have not been fully
studied yet.
The main objective of the present study is to analyze the interfacial stresses in
damaged RC beams strengthened with FRP plate. The simple approximate closed-
form solutions discussed in this paper provide a useful but simple tool for
understanding the interfacial behavior of an externally bonded FRP plate on the
damaged concrete beam with the consideration of the effect of the fiber orientations.
1. Theoretical Analysis and Solutions Procedure.
1.1. Material Properties of Damaged Plates. Voyiadjis and Kattan [17]
proposed an anisotropic damaged model, in which the elastic energy configuration
of deformed and damaged state is equivalent to the elastic energy configuration of
deformed but undamaged state. Based on this assumption, the relations of elastic
constants of damaged state and undamaged state can be expressed as
~
( ) ,E E11 11 11
21� � �
~
( ) ,E E22 22 22
21� � � (1)
where
~
,
~
E E11 22 and E E11 22, are the elastic constants of damaged and undamaged
state, respectively, and �11 and � 22 are damaged variables. Hence, the material
properties of the damaged plate can be represented by replacing the above elastic
constants with the effective ones defined in Eq. (1). A convenient way to determine
�11 and � 22 is to utilize the damage law postulated by Yu et al. [18] for concrete.
It is given as
�
�
�
22
21
2 1
�
�
�
�
�
�
�NC f
c
NC
, � �11 22� H ( )H
1 (2)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 105
Analytical Analysis of the Interfacial Stress ...
and
N
E
E E
C
f
C
C f
C
�
�2( )
, (3)
where E f
C is the tangential elastic modulus when the stress reaches its peak, EC is
the initial elastic modulus, � f
c is the failure strain, and �2 is the current state of
strain. Value of H is a constant determined by experiments, for example, when
EC � 49.49 GPa, NC � 3.65, H �3, and � � 0.2, then � � �22 2
3 65012048� . ( ) ..
f
c
1.2. Mathematical Formulation of the Present Method. A differential section
dx , can be cut out from the FRP reinforced concrete beam (Fig. 1), as shown in
Fig. 2. The composite beam is made from three materials: concrete (or reinforced
concrete), adhesive layer and FRP reinforcement. In the present analysis, linear
elastic behavior is regarded to be for all the materials; the adhesive is assumed to
play a role only in transferring the stresses from the concrete to the FRP
reinforcement, and the stresses in the adhesive layer do not vary through the
direction of the thickness.
1.2.1. Basic Equation of Elasticity. The strain �1( )x in the concrete beam
near adhesive interface can be expressed as
�1
1 1
1 1
1
1 1
( )
( ) ( ) ( )
.x
du x
dx
eM x
E I
N x
E A
� � � (4)
The laminate theory is used to estimate the strain �2 ( )x in the external FRP
reinforcement near adhesive interface. Furthermore, it is assumed that the ply
arrangement of the plate is symmetrical
�2
2
11
2
2
2 11
2
22
( ) ( )
( )
,x
du
dx
D
t
b
M x A
N x
b
� �� � � � (5)
where u x1( ) and u x2 ( ) are horizontal displacements of the concrete and the
external FRP reinforcement near interface, respectively, M x1( ) and M x2 ( ) are
Fig. 1. Simply supported beam strengthened with bonded FRP plate.
106 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
bending moments applied in the concrete and the external FRP reinforcement,
respectively, E1 is the Young modulus of the concrete, I1 is the second moment
area, e is distance from the neutral axis to the bottom of RC beam, N1 and N 2 are
axial forces applied in the concrete and the external FRP reinforcement, respectively,
b2 is the width of the plate, t2 is thickness of the external reinforcement,
[ ] [ ]� � �A A 1 is the inverse of the extensional matrix [ ],A and [ ] [ ]� � �D D 1 is the
inverse of the flexural matrix.
By adopting the equilibrium conditions of the concrete, we have:
Along x-direction:
dN x
dx
x b
1
2
( )
( ) ,� � (6)
where �( )x is shear stress in the adhesive layer.
Along y-direction:
dV x
dx
x b qn
1
2
( )
[ ( ) ],�� �� (7)
where V x1( ) is shear force applied in the concrete, � n x( ) is normal stress in the
adhesive layer, q is the uniformly distributed load, and b1 is width of RC beam.
Fig. 2. Forces in infinitesimal element of a soffit-plated beam.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 107
Analytical Analysis of the Interfacial Stress ...
Moment equilibrium:
dM x
dx
V x x b e
1
1 2
( )
( ) ( ) .� � � (8)
The equilibrium of the external FRP reinforcement along the x- and y-direction
and moment equilibrium can also be written as:
Along x-direction:
dN x
dx
x b
2
2
( )
( ) .� � (9)
Along y-direction:
dV x
dx
x bn
2
2
( )
( ) .� � (10)
Moment equilibrium:
dM x
dx
V x x b
t2
2 2
2
2
( )
( ) ( ) ,� � � (11)
where V x2 ( ) is shear force applied in the external FRP reinforcement.
1.2.2. Shear Stress Distribution along the FRP Concrete Interface. The shear
stress in the adhesive can be expressed as follows:
�( ) ( ) [ ( ) ( )],x K u x K u x u xs s� � �� 2 1 (12)
where K s is shear stiffness of the adhesive per unit length and can be derived as
K
x
u x
x
u x t t
G
ts
a a
a
a
� � �
� �( )
( )
( )
( )
,
� �
1
(13)
�u x( ) is relative horizontal displacement at the adhesive interface, Ga is the shear
modulus in the adhesive, and ta is the thickness of the adhesive.
Differentiating Eqs. (4), (5), and (12) with respect to x , respectively:
d x
dx
K A
N x
b
D
t
b
M x
N x
E A
eM
s
�( ) ( )
( )
( )
� � � � � �11
2
2
11
2
2
2
1
1 1
1
2
( )
.
x
E I1 1
�
�
�
� (14)
Assuming equal curvature in the beam and the FRP plate, the relationship
between the moments in the two adherends can be expressed as
M x RM x1 2( ) ( )� (15)
with
R
E I D
b
�
�1 1 11
2
. (16)
108 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
Moment equilibrium of the differential segment of the plated beam in Fig. 2
gives:
M x M x M x N x e t
t
T a( ) ( ) ( ) ( ) ,� � � � �
�
�
�
�
�
�
1 2
2
2
(17)
M xT ( ) is the applied moment, and N x( ) is given as follows:
N x N x N x b x dx
x
( ) ( ) ( ) ( ) .� � � �1 2 2
0
� (18)
The bending moment in RC beam, expressed as a function of the total applied
moment and the interfacial shear stress, is given as
M x
R
R
M x b x e t
t
dxT a
x
1 2
2
0
1 2
( ) ( ) ( )�
�
� � �
�
�
�
�
�
�
�
�
�
�� � (19)
and
M x
R
M x b x e t
t
dxT a
x
2 2
2
0
1
1 2
( ) ( ) ( ) .�
�
� � �
�
�
�
�
�
�
�
�
�
�� � (20)
The first derivative of the bending moment in each adherend gives:
dM x
dx
R
R
V x b x e t
t
T a
1
2
2
1 2
( )
( ) ( )�
�
� � �
�
�
�
�
�
�
�
�
��
� (21)
and
dM x
dx R
V x b x e t
t
T a
2
2
21
1 2
( )
( ) ( ) .�
�
� � �
�
�
�
�
�
�
�
�
��
� (22)
Differentiating Eq. (14):
d x
dx
K
t
b
D
dM x
dx
A
b
dN x
dx
e
E Is
2
2
2
2
11
2 11
2
2
12
�( ) ( ) ( )
�
�
� �
�
�
1
1
1 1
11dM x
dx E A
dN x
dx
( ) ( )
.�
�
�
��
�
�
��
(23)
Substitution of the shear forces [Eqs. (21) and (22)] and axial forces [Eq. (18)]
into Eq. (23) gives the following governing differential equation for the interfacial
shear stress:
d x
dx
K A
b
E A
e
t
e t
t
Es
a2
2 11
2
1 1
2 2
2 2�( )
� � � �
�
�
�
�
�
�
� � �
�
�
�
�
�
�
1 1 11 2
2 11I D b
b D x
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
��( )
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 109
Analytical Analysis of the Interfacial Stress ...
�
�
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�K
e
t
E I D b
D V xs T
2
1 1 11 2
11
2
0( ) . (24)
For simplicity, the general solutions presented below are limited to loading
which is either concentrated or uniformly distributed over part or the total span of
the beam, or both. For such loading, d V x dxT
2 2 0( ) ,� and the general solution to
Eq. (24) is given by
� � �( ) cosh( ) sinh( ) ,x B x B x m q
L
x a� � � � �
�
�
�
�
�
�
1 2 1 2
(25)
where
�2
11
2
1 1
2 2
1 1 11
2 2
� � � �
�
�
�
�
�
�
� � �
�
�
�
�
�
�
�
K A
b
E A
e
t
e t
t
E I Ds
a
�
�
�
�
�
�
�
�
�
�
�
�
�
�b
b D
2
2 11 (26)
and
m
K e
t
E I D b
D
s
1 2
2
1 1 11 2
11
2�
�
� �
�
�
�
�
�
�
�
�
�
�
�
�
��
, (27)
B1 and B2 are constant coefficients determined from the boundary conditions. In
the present study, a simply supported beam is investigated which is subjected to a
uniformly distributed load as shown in Fig. 1. The constants of integration need to
be determined by applying suitable boundary conditions.
The first boundary condition is applied bending moment at x �0. Here, the
moment at the plate end M 2 0( ) and the axial force of either the concrete beam or
FRP plate [ ( ) ( )]N N1 20 0� are zero. As a result, the moment in the section at the
plate curtailment is resisted by the beam alone and can be expressed as
M M
qa
L aT1 0 0
2
( ) ( ) ( ).� � � (28)
Applying the above boundary condition in Eq. (14):
d x
dx
m M T
�( )
( )
�
��
0
02 and m
eK
E I
s
2
1 1
� . (29)
By substituting Eq. (25) into (29), B2 can be determined as
B
qam
L a
m
q2
2 1
2
�� � �
� �
( ) . (30)
110 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
The second boundary condition requires zero interfacial shear stress at
midspan due to symmetry of the applied load. Value of B1 can therefore be
determined as
B
aqm
L a
L qm Lp p
1
2 1
2 2 2
� �
�
�
��
�
�
���
�
�
��
�
�
�
�
�
�
�
( ) tanh tanh �. (31)
For practical cases �Lp 2 10
and as a result tanh( ) .�Lp 2 1� So the
expression for B1 can be reduced to
B
aqm
L a
qm
B1
2 1
22
� � � ��
� �
( ) . (32)
Substitution of B1 and B2 into Eq. (25) gives an expression for the
interfacial shear stress at any point
�
�
�
( ) ( )x
m a
L a m
qe
m q
L
a x
x
� � �
�
�
��
� � �
�
�
�
�
�
�
�
2
1 12 2
0� �x Lp , (33)
where q is the uniformly distributed load and x , a, L, and Lp are defined in Fig. 1.
1.2.3. Normal Stress Distribution along the FRP Concrete Interface. The
normal stress in the adhesive can be expressed as follows:
� n n nx K w x K w x w x( ) ( ) [ ( ) ( )],� � �� 2 1 (34)
where K n is normal stiffness of the adhesive per unit length and can be deduced
as
K
x
w x
x
w x t t
E
tn
n n
a a
a
a
� �
�
�
��
�
�
���
� �( )
( )
( )
( )� �
1
(35)
w x1( ) and w x2 ( ) are the vertical displacements of adherend 1 and 2 respectively.
Differentiating Eq. (34) twice results in
d x
dx
K
d w x
dx
d w x
dx
n
n
2
2
2
2
2
2
1
2
� ( ) ( ) ( )
.� �
�
�
�
� (36)
Considering the moment–curvature relationships for the beam to be strengthened
and the external reinforcement, respectively:
d w x
dx
M x
E I
2
1
2
1
1 1
( ) ( )
,��
d w x
dx
D M x
b
2
2
2
11 2
2
( ) ( )
.��
�
(37)
Based on the equilibrium equations (6)–(11), the governing differential
equations for the deflection of adherends 1 and 2, expressed in terms of the
interfacial shear and normal stresses, are given as follows:
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 111
Analytical Analysis of the Interfacial Stress ...
Adherend 1:
d w x
dx E I
b x
e
E I
b
d x
dx
q
E In
4
1
4
1 1
2
1 1
2
1 1
1( )
( )
( )
.� � ��
�
(38)
Adherend 2:
d w x
dx
D x D
t d x
dxn
4
2
4 11 11
2
2
( )
( )
( )
.�� � � ��
�
(39)
Substitution of Eqs. (38) and (39) into the fourth derivation of the interfacial
normal stress obtainable from Eq. (34) yields the following governing differential
equation for the interfacial normal stress:
d x
dx
K D
b
E I
x
n
n n
4
4 11
2
1 1
�
�
( )
( )� � �
�
�
��
�
�
�� �
� � �
�
�
��
�
�
�� � �K D
t eb
E I
d x
dx
qK
E In
n
11
2 2
1 1 1 12
0
�( )
. (40)
The general solution to this fourth-order differential equation is
� � ��
n
xx e C x C x( ) [ cos( ) sin( )]� � ��
1 2
� � � �e C x C x n
d x
dx
n qx� � �
�
[ cos( ) sin( )]
( )
.3 4 1 2 (41)
For large values of x it is assumed that the normal stress approaches zero,
and as a result C C3 4 0� � . The general solution therefore becomes:
� � �
��
n
xx e C x C x n
d x
dx
n q( ) [ cos( ) sin( )]
( )
,� � � ��
1 2 1 2 (42)
where
�� � �
�
�
��
�
�
��
K b
E I
D
n
4
2
1 1
11
4 , (43)
n
eb D E I
t
D E I b1
2 11 1 1
2
11 1 1 2
2�
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
, (44)
and
n
D E I b2
11 1 1 2
1
�
� �
. (45)
112 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
The constants C1 and C 2 in Eq. (42) are written as follows:
C
K
E I
V M
n n d
dx
n
T T1 3
1 1
3
3
1
3
4
2
0 0
2
0
2
0
� � � �
�
�
�
�
�
�
[ ( ) ( )] ( )
( )
4
3
3
0
�
�
�
�
�
�
�
�
��
�d
dx
( )
, (46)
C
K
E I
M
n d
dx
n
T2 2
1 1
1
2
3
32
0
2
0
�� �
� �
�
( )
( )
, (47)
where
n b K
e
E I
D t
bn3 2
1 1
11 2
22
� �
��
�
��
�
�
��. (48)
The above expressions for the constants C1 and C 2 have been left in terms
of the bending moment M T ( )0 and shear force VT ( )0 at the end of the soffit
plate.
2. Comparison of Analytical Solutions. A comparison of the interfacial
shear and normal stresses from the different existing closed-form solutions and the
present new solution is undertaken in this section. An undamaged RC beam
bonded with a CFRP soffit plate is considered. The beam is simply supported and
subjected to a uniformly distributed load. A summary of the geometric and material
properties is given in Table 1. The span of the RC beam is 3000 mm, the distance
from the support to the end of the plate is 300 mm and the uniformly distributed
load is 50 kN/m.
Figure 3 plots the interfacial shear and normal stresses near the plate end for
the example RC beam bonded with a CFRP plate for the uniformly distributed load
case. Overall, the predictions of the different solutions agree closely with each
other. The interfacial normal stress is seen to change sign at a short distance away
from the plate end.
3. Parametric Studies. In this section, numerical results of the present
solutions are presented to study the effect of various parameters on the distributions
of the interfacial stresses in a damaged RC beam bonded with an FRP plate. A
damaged state is described through the incorporation of damage variable �11 in
the same way as is described by Shen et al. [19]. These results are intended to
demonstrate the main characteristics of interfacial stress distributions in these
strengthened beams.
T a b l e 1
Geometric and Material Properties
Material E11 , GPa E22 , GPa G12 , GPa �12 Width (mm) Depth (mm)
CFRP plate 140 10 5 0.28 b2 200� t2 4�
GFRP plate 50 10 5 0.28 b2 200� t2 4�
RC beam 30 30 – 0.18 b1 200� t1 300�
Adhesive layer 3 3 – 0.35 b2 200� ta � 4
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 113
Analytical Analysis of the Interfacial Stress ...
Fig. 3. Comparison of interfacial shear and normal stresses for an RC beam with a bonded CFRP
soffit plate [ ]016 S subjected to a uniformly distributed load.
Fig. 4. Effects of damage on the maximum interfacial shear stress of RC beam strengthened by a
CFRP or GFRP plate [ ] .016 S
Fig. 5. Effects of damage on the maximum interfacial normal stress of RC beam strengthened by a
CFRP or GFRP plate [ ] .016 S
114 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
3.1. Effect of Damage on the Maximum Interfacial Stresses. Figures 4 and 5
show the effect of damage extent on maximum shear and normal interfacial
stresses, respectively, for the two types of FRP materials. The results show that
when the damage variable �11 increases from 0 to 0.375, the maximum interfacial
stresses increase slowly. However, it can be seen that from 0.375 to 0.825, these
stresses increase rapidly.
3.2. Effect of FRP Strip Thickness. Peak shear and peeling stresses for
various thicknesses of the FRP strip appear in Fig. 6. The damage variables are
taken as � 22 � 0.12048 and � �11 223� , which are corresponding to the peak
stress, that is � �2 � f
c in Eq. (2). The results reveal that thickness of the FRP strip
significantly increases the edge peeling and shear stresses. Generally, the thickness
of FRP plate used in practical engineering is very small, as compared with one of
steel plate. Therefore, the fact of the smaller interfacial stress level and concentration
should be one of advantages of retrofitting by FRP over by steel plate.
Fig. 6. Influence of thickness of CFRP strip on edge stresses (�22 012048� . and � �11 223� ).
Fig. 7. Effect of the length of damaged region on the maximum interfacial shear stress (�22 012048� .
and � �11 223� ).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 115
Analytical Analysis of the Interfacial Stress ...
3.3. Effect of Length of Damaged Region. The influence of the length of the
damaged region on the interfacial stresses is shown in Figs. 7 and 8. The range of
the examined lengths is within 600–2400 mm for the two types of FRP strips. It is
seen that while the length of the damaged region is less than the length of the
strengthening plate (DA Lp� ), the edge peeling and shear stresses are constant.
When the two lengths are equal (DA Lp� ), these stresses attain higher values.
Consequently, it is recommended to use a strengthening plate having length,
superior to the damaged zone.
Conclusions. This paper is concerned with the prediction of interfacial shear
and normal stresses in damaged RC beams retrofitted with externally advanced
composite materials. Such interfacial stresses provide the basis for understanding
debonding failures in such beams and for development of suitable design rules.
Numerical comparison between the existing solutions and the present new solution
has been carried out. The anisotropic damage model is adopted to describe the
damage of the RC beams. The results show that the damage has a significant effect
on the interfacial stresses in FRP-damaged RC beam, especially, when the length of
damaged region is equal or superior to the plate length. Consequently, it is
recommended to use a strengthening plate having length, superior to the damaged
zone. The results reveal also that thickness of the FRP strip significantly increases
the edge peeling and shear stresses.
Ð å ç þ ì å
Çàïðîïîíîâàíî àíàë³òè÷íèé ìåòîä îö³íêè ì³æïîâåðõíåâèõ íàïðóæåíü ó øàð³
ïîøêîäæåíèõ çàë³çîáåòîííèõ áàëîê, ùî çàêð³ïëåí³ çîâí³øí³ìè êîìïîçèòíè-
ìè ïëàñòèíàìè ç àðìîâàíîãî âîëîêíàìè âóãëåïëàñòèêà. Çàïðîïîíîâàíî ìî-
äåëü, ÿêà, íà â³äì³íó â³ä ³ñíóþ÷èõ ð³øåíü, âðàõîâóº íàÿâí³ñòü ì³æïîâåðõíå-
âèõ äîòè÷íèõ äåôîðìàö³é øëÿõîì ïîñòóëþâàííÿ ë³í³éíîãî ðîçïîä³ëó äîòè÷-
íèõ íàïðóæåíü ïî ãëèáèí³ çàë³çîáåòîííî¿ áàëêè. Ðîçðîáëåíî àí³çîòðîïíó
ìîäåëü ðóéíóâàííÿ, ùî îïèñóº ïðîöåñ ïîøêîäæåííÿ çàë³çîáåòîííî¿ áàëêè.
Fig. 8. Effect of the length of damaged region on the maximum interfacial normal stress (�22 012048� .
and � �11 223� ).
116 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
Óñòàíîâëåíî, ùî ïîøêîäæåííÿ ñóòòºâî âïëèâຠíà âåëè÷èíó ì³æïîâåðõíåâèõ
íàïðóæåíü ó ïîøêîäæåíèõ çàë³çîáåòîííèõ áàëêàõ, àðìîâàíèõ êîìïîçèòíèìè
ïëàñòèíàìè.
1. T. C. Triantafillou, “Composites: a new possibility for the shear strengthening
of concrete, masonry and wood,” Compos. Sci. Technol., 58, No. 8, 1285–
1295 (1998).
2. R. J. Quantrill and L. C. Hollaway, “The flexural rehabilitation of reinforced
concrete beams by the use of prestressed advanced composite plates,” Compos.
Sci. Technol., 58, No. 8, 1259–1275 (1998).
3. U. Meier, M. Deuring, H. Meier, and G. Schwegler, “Strengthening of
structures with advanced composites,” in: J. L. Clarke (Ed.), Alternative
Materials for the Reinforcement and Prestressing of Concrete, Glasgow
(1993), pp. 153–171.
4. U. Meier, “Post strengthening by continuous fiber laminates in Europe,” in:
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5. U. Meier, “Strengthening of structures using carbon fiber/epoxy composites,”
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modes of reinforced concrete beams strengthened with prestressed carbon
composite plates,” Composites Part B, 29B, 411–424 (1998).
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No. 3, 132–137 (1998).
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bonded steel plates,” Int. J. Cem. Compos. Lightweight Concr., 10, No. 2,
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of R/C beams strengthened with FRP plate due to stress concentration at the
plate end,” ACI Struct. J., 95, No. 2, 142–152 (1998).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 117
Analytical Analysis of the Interfacial Stress ...
15. O. Rabinovich and Y. Frostig, “Closed-from high order analysis of RC beams
strengthened with FRP strips,” J. Compos. Constr., 4, No. 2, 65–74 (2000).
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Received 10. 07. 2012
118 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
T. Hassaine Daouadji
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| id | nasplib_isofts_kiev_ua-123456789-112042 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T13:17:02Z |
| publishDate | 2013 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Hassaine Daouadji, T. 2017-01-16T18:50:52Z 2017-01-16T18:50:52Z 2013 Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates / T. Hassaine Daouadji // Проблемы прочности. — 2013. — № 5. — С. 104-118. — Бібліогр.: 19 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/112042 539.4 A theoretical method to predict the interfacial stresses in the adhesive layer of damaged reinforced concrete beams strengthened with externally bonded carbon fiber-reinforced polymer plate is presented. The adopted model is developed including the adherend shear deformations by assuming a linear shear stress through the depth of the reinforced concrete beam, while all existing solutions neglect this effect. In addition, in the present study the anisotropic damage model is adopted to describe the damage of the reinforced concrete beams. It is shown that the damage has a significant effect on the interfacial stresses in fiber-reinforced polymer damaged reinforced concrete beam. Предложен аналитический метод оценки межповерхностных напряжений в слоях поврежденных железобетонных балок, укрепленных наружными композитными пластинами из армированного волокнами углепластика. Предложена модель, которая, в отличие от существующих решений, учитывает наличие межповерхностных касательных деформаций путем постулирования линейного распределения касательных напряжений по глубине железобетонной балки. Разработана анизотропная модель разрушения, описывающая процесс повреждения железобетонной балки. Установлено, что повреждение существенно влияет на величину межповерхностных напряжений в поврежденных железобетонных балках, армированных композитными пластинами. Запропоновано аналітичний метод оцінки міжповерхневих напружень у шарі пошкоджених залізобетонних балок, що закріплені зовнішніми композитними пластинами з армованого волокнами вуглепластика. Запропоновано модель, яка, на відміну від існуючих рішень, враховує наявність міжповерхневих дотичних деформацій шляхом постулювання лінійного розподілу дотич- них напружень по глибині залізобетонної балки. Розроблено анізотропну модель руйнування, що описує процес пошкодження залізобетонної балки. Установлено, що пошкодження суттєво впливає на величину міжповерхневих напружень у пошкоджених залізобетонних балках, армованих композитними пластинами. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates Аналитический расчет межповерхностных напряжений в поврежденных железобетонных балках, армированных композитным пластинами Article published earlier |
| spellingShingle | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates Hassaine Daouadji, T. Научно-технический раздел |
| title | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates |
| title_alt | Аналитический расчет межповерхностных напряжений в поврежденных железобетонных балках, армированных композитным пластинами |
| title_full | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates |
| title_fullStr | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates |
| title_full_unstemmed | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates |
| title_short | Analytical Analysis of the Interfacial Stress in Damaged Reinforced Concrete Beams Strengthened by Bonded Composite Plates |
| title_sort | analytical analysis of the interfacial stress in damaged reinforced concrete beams strengthened by bonded composite plates |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112042 |
| work_keys_str_mv | AT hassainedaouadjit analyticalanalysisoftheinterfacialstressindamagedreinforcedconcretebeamsstrengthenedbybondedcompositeplates AT hassainedaouadjit analitičeskiirasčetmežpoverhnostnyhnaprâženiivpovreždennyhželezobetonnyhbalkaharmirovannyhkompozitnymplastinami |