A New Approach for Adhesive Stress Analysis of a Beam Bonded with Composite Plate Having Variable Fiber Spacing

Composite Interfaces 18 (2011) 135–149brill.nl/ci

A New Approach for Adhesive Stress Analysis of a BeamBonded with Composite Plate Having Variable Fiber Spacing

Baghdad Krour a,b, Abdeloauhed Tounsi a,c,∗ and Sid Ahmed Meftah a,c

a Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes, Algérieb Université de Mascara, Département de Génie Civil, Algérie

c Université Sidi Bel Abbes, Département de génie civil, Faculté des Sciences de l’Ingénieur, Algérie

Received 16 June 2010; accepted 5 October 2010

AbstractExternal bonding of FRP plates or sheets has emerged as a popular method for strengthening reinforcedconcrete structures. Debonding along the FRP–concrete interface can lead to premature failure of the struc-tures. Shear and normal stresses (in the thickness direction) at interfaces (referred to as interfacial shear andtransverse normal stresses hereafter) have played a significant role in understanding this premature debond-ing failure of such hybrid structures. Extensive experimental and analytical analyses have been undertakento investigate this problem. Large discrepancies have been found from various studies. This paper reports aclosed-form formulation for interfacial stresses in plated beams. The analysis is based on the deformationcompatibility approach developed by Tounsi and Benyoucef [8] where both the shear and normal stressesare assumed to be invariant across the adhesive layer thickness. The composite plate with variable fiberspacing bonded to reinforced concrete beam is considered in the present study. In addition, an unrealisticrestriction of the same curvatures in the RC beam and FRP panel commonly used in most of the existingstudies is released in the present theoretical formulation. Compared with previously published analyticalresults, this one improves the accuracy of predicting the transverse normal stresses and the solution remainsa closed form.© Koninklijke Brill NV, Leiden, 2011

KeywordsFRP composites, interfacial stresses, fibre spacing, strengthening

1. Introduction

Engineering of modern composite materials has had a significant impact on thetechnology of design and construction. The main objective of using or selecting anymaterial in construction is to make use of its properties efficiently for best perfor-mance and durability of the structure. The merits of a material are based on factorssuch as availability, structural strength, durability and workability. Recently, exter-

* To whom correspondence should be addressed. E-mail: [email protected]

© Koninklijke Brill NV, Leiden, 2011 DOI:10.1163/092764411X567413

136 B. Krour et al. / Composite Interfaces 18 (2011) 135–149

nal bonding of composite plates or sheets to conventional materials (e.g., reinforcedconcrete, steel or wood) has been found to be a viable and effective technique inflexural strengthening and retrofitting of structures. In such a strengthening method,the adhesively bonded interface between the originally strengthened beam (e.g.,wood or concrete) and the strengthening plate (e.g., FRP plates or sheets) plays acrucial role in providing effective stress transfer from the beam to the strengthen-ing plate as well as securing the integrity and durability of the hybrid structure.Due to high stress concentrations in the adhesive layer, especially at the locationsnear the plate ends, interface debonding of the strengthening plate from the origi-nal beam member often occurs, leading to the weakening of the strengthening effectand eventually premature failure of the structure. Therefore, accurate prediction andcharacterization of the interface stress and bond strength are essential for the designof structures externally bonded with FRP composites.

Many analytically studies have been conducted to predict the interfacial stresses:see, for example, those by Vilnay [1], Roberts [2], Roberts and Haji-Kazemi [3],Taljsten [4], Malek et al. [5], Smith and Teng [6], Tounsi [7], Tounsi and Beny-oucef [8], Cai et al. [9] and Tounsi et al. [10]. However, the above existing studiesare based on the assumption that the RC beam and the FRP panel have the samecurvature. In addition, few of the studies consider composites with variable fiberspacing. Martin and Leissa [11] investigated the plane stress problem of a rectangu-lar of composite sheet with variable fiber content. Shiau and Lee [12] studied stressconcentration around holes in composite laminates with variable fiber spacing. Itproves that reducing the fiber volume ratio near the hole edges can significantlyreduce the stress concentration in that region. Recently, Meftah et al. [13] con-sidered the seismic analysis of reinforced concrete coupled shear walls structuresstrengthened by bonded composite plates having variable fiber spacing. Dynamicanalysis was performed to investigate the influence of the fiber redistribution ofthe bonded plates on the lateral deflections of RC coupled shear walls. Meftah etal. [14] also presented a finite element model for static and free vibration analysisof reinforced concrete shear-wall structures strengthened with thin composite plateshaving variable fibers spacing. Numerical results were obtained for six non-uniformdistributions of E-glass, graphite and boron fibers in epoxy matrices. The fiber redis-tributions of the bonded plates are seen to increase the frequency modes and reducesubstantially the lateral displacements. Kuo and Shiau [15] studied the bucklingand vibration of composite laminated plates with variable fiber spacing by use ofthe finite element method. Benatta et al. [16] presented a mathematical solutionfor bending of short hybrid composite beams with variable fibers spacing. Boure-mana et al. [17] proposed a new concept to minimize the thermal deformation ormatch the thermal deformation of another structural component by using a compos-ite whose fibers have a negative axial thermal expansion coefficient. By varying thefiber volume fraction within a symmetric laminated beam to create a functionallygraded material (FGM), certain thermal deformations can be controlled or tailored.Bedjilili et al. [18] presented exact solutions for shear flexible symmetric composite

B. Krour et al. / Composite Interfaces 18 (2011) 135–149 137

beams with a variable fiber volume fraction through thickness. The laminates withvariable fiber spacing along the thickness direction have been investigated recentlyby Fu et al. [19]. In this later paper, fiber volume ratio distribution functions aredefined separately in each single layer and the classic state space method as wellas differential quadrature state space method are utilized for different boundary andplied conditions.

With this in mind, the objectives of this paper are to (1) present an improve-ment to Tounsi and Benyoucef’s solution [8] to obtain a new closed-form solutionwhich releases the improper restriction of the equal curvatures of RC beam andFRP panel used by some existing models; and (2) quantitatively evaluate the effectof bonded composite plates having variable fibers spacing on interfacial stresses inhybrid structures. It is believed that the present results will be of interest to civil andstructural engineers and researchers.

2. Governing Equations

Consider a concrete beam of span L retrofitted by FRP plate of length Lp bondedtogether with an adhesive layer (Fig. 1). The beam is simply supported and sub-jected to a transverse load of density q(x), as shown in Fig. 2. It is assumed that (1)all of the materials are linear elastic; (2) since the thickness of the adhesive is small,both the normal and shear stresses in the adhesive are assumed constant across itsthickness. Note that the key assumption of invariant through-thickness stress dis-tribution in the adhesive layer has been adopted in all of the existing theoreticalanalyses, except few ‘higher-order’ solutions, in order to enable relatively simpleclosed-form solutions to be obtained.

Because the displacements at the interfaces between the respective RC beam,FRP panel and the adhesive layer are continuous, we can obtain the expressions ofadhesive layer average strains from the angular deformation:

εy = ∂v(x, y)

∂y≈ v2(x) − v1(x)

ta, (1)

γxy = ∂u(x, y)

∂y+ ∂v(x, y)

∂x≈ ut

2(x) − ub1(x)

ta+ 1

2

(∂v2(x)

∂x+ ∂v1(x)

∂x

), (2)

Figure 1. Simply supported beam strengthened with bonded FRP plate.


Figure 2. Forces in infinitesimal element of a soffit-plated beam.

where u(x, y) and v(x, y) are the horizontal and vertical displacements respec-tively at any point in the adhesive layer as defined in Fig. 2. ub

1(x) and ut2(x) are

the longitudinal displacements at the base of RC beam and the top of FRP panel,respectively, v1(x) and v2(x) are the vertical displacements of RC beam and FRPpanel, respectively, and ta is the thickness of the adhesive layer.

In existing solutions [1–10], ∂v(x, y)/∂x = 0 is used because the same curvatureof beam and plate is assumed. It is apparent that these solutions do not considerPoisson and transverse displacement effects.

From the linear stress/strain relationships, the corresponding interfacial shear (τ )and normal (σ ) stresses are:

τ(x) = Ga

[ut

2(x) − ub1(x)

ta+ 1

2

(∂v2(x)

∂x+ ∂v1(x)

∂x

)], (3)

σ(x) = Ea

ta[v2(x) − v1(x)], (4)

where Ea and Ga are the elastic moduli of the adhesive layer.


From equations (3) and (4), we obtain

τ(x) = Ga

[ut

2(x) − ub1(x)

ta+ ∂v2(x)

∂x− ta

2Ea

∂σ(x)

∂x

]. (5)

Differentiating equation (5) with respect to x gives

dτ(x)

dx= Ga

ta

[dut

2(x)

dx− dub

1(x)

dx+ ta

∂2v2(x)

∂x2− t2

a

2Ea

∂2σ(x)

∂x2

]. (6)

The strains at the base of RC beam and the top of FRP panel, considering all threecomponents of axial, bending and adherend shear deformations, are given for anRC beam as

ε1(x) = dub1(x)

dx= y1

E1I1M1(x) − N1(x)

E1A1. (7)

2.1. FRP Panel (Plate with Variable Fiber Spacing)

Based on the Kirchhoff–Love hypothesis, the displacements parallel to the x- and y-axes of an arbitrary point in the beam, denoted by u(x, y) and v(x, y), respectively,give

u(x, y) = u0(x) − y∂v0

∂x, v(x, y) = v0(x), (8)

where u0(x) and v0(x) are the displacement components of a point in the midplane.The normal resultant force N2, and bending moment M2 are related to the normal

strain ε0 = ∂u0∂x

and flexural curvature kx = ∂2v0∂x2 by

N2 = b2A11ε0 − b2B11kx,(9)

M2 = b2B11ε0 − b2D11kx

and

(A11,B11,D11) =∫ t2/2

−t2/2E(y)(1, y, y2)dz, (10)

where t2 is thickness of the bonded FRP plate and b2 is the width of the soffit plate.The FRP plate is assumed to consist of N layers of orthotropic sheets bonded to-

gether. Each layer has arbitrary fiber orientation. The fibers in each layer are alignedparallel to the longitudinal direction but distributed unevenly in the y-direction.Hence, the fiber volume fraction, Vf, is a function of non-dimensional coordinateξ = 2y

t2having its origin at the plate center. Define the fiber distribution as Kuo and

Shiau [15]

Vf(ξ) = ((Vf)in − (Vf)out)(1 − ξ2)n + (Vf)out, (11)

where (Vf)in is the fiber volume fraction at the plate center (ξ = 0) and (Vf)outis the fiber volume fraction at the edges (ξ = ±1). It is seen that more fibers arecentralized in the central portion of the plate for higher volume fraction (Vf)in and


higher-order n. The volume fraction index n controls the variation of the volumefraction.

With this variable fiber spacing, the elastic moduli E(ξ) for the composite mate-rial are also the functions of ξ . In this study, the formulas used for the calculationof these effective engineering constants are based on the rule of mixture [20]

E(ξ) = EfVf(ξ) + Em(1 − Vf (ξ)). (12)

In the above equations, the subscripts f and m are used to denote properties of thefiber and matrix, respectively.

The coupling stiffness matrix is identically zero (B11 = 0) for the symmetriclaminates

ε2(x) = dut2(x)

dx= −y2

b2D11M2(x) + N2(x)

b2A11, (13)

where N(x) and M(x) are the axial forces, bending moment in the RC beam andFRP panel, while y1 and y2 are the distances from the bottom of the beam and topof the plate to their respective centroid. A is the cross-sectional area, I is the secondmoment of area, and E is the elastic modulus.

Considering the moment–curvature relationships for the beam to be strengthenedand the external reinforcement, respectively:

d2v1(x)

dx2= −M1(x)

E1I1,

d2v2(x)

dx2= −M2(x)

b2D11. (14)

The equilibrium of the RC beam and FRP panel leads to the following relationships:For the RC beam:

dM1(x)

dx= V1(x) − b2y1τ(x) and

dV1(x)

dx= −b2σ(x) − q. (15)

For the FRP panel:

dM2(x)

dx= V2(x) − b2y2τ(x) and

dV2(x)

dx= b2σ(x). (16)

Consideration of horizontal equilibrium gives:

dN1(x)

dx= dN1(x)

dx= b2τ(x). (17)

Based on the above equilibrium equations, the governing differential equationsfor the deflection of RC beam and FRP panel, expressed in terms of the interfacialshear and normal stresses, are given as follows:

For the RC beam:

d4v1(x)

dx4= 1

E1I1b2σ(x) + y1

E1I1b2

dτ(x)

dx+ q

E1I1. (18)

For the FRP panel:

d4v2(x)

dx4= −1

D11σ(x) + y2

D11

dτ(x)

dx. (19)


Using equations (15)–(17), the third derivative of horizontal displacement can bewritten as

d3ub1(x)

dx3= y1

E1I1

(−b2σ(x) − q − b2y1

dτ(x)

dx

)− b2

E1A1

dτ(x)

dx, (20)

d3ut2(x)

dx3= −y2

D11

(σ(x) − y2

dτ(x)

dx

)+ 1

A11

dτ(x)

dx. (21)

Differentiating equation (6) with respect to x twice and using equations (18)–(21) gives

d3τ(x)

dx3= A1

dτ(x)

dx+ A2

d4σ(x)

dx4+ A3σ(x) + A4q(x), (22)

where

A1 = Ga

ta

(y2

2

D11+ b2

E1A1+ 1

A11+ tay2

D11+ y2

1b2

E1I1

), A2 = −1

2

Gata

Ea,

(23)

A3 = Ga

ta

(y1b2

E1I1− y2

D11− ta

D11

)and A4 = Gay1

taE1I1.

Substituting equations (18) and (19) into the fourth derivative of the interfacial nor-mal stress obtainable from equation (4) gives the following governing differentialequation for the interfacial normal stress

d4σ(x)

dx4+ B1σ(x) + B2

dτ(x)

dx+ B3q = 0, (24)

where

B1 = Ea

ta

(1

D11+ b2

E1I1

),

(25)

B2 = −Ea

ta

(y2

D11− y1b2

E1I1

)and B3 = Ea

taE1I1.

Equation (24) gives that

d4σ(x)

dx4= −B1σ(x) − B2

dτ(x)

dx− B3q. (26)

Substitute equation (26) into equation (18), obtained

d3τ(x)

dx3= (A1 − A2B2)

dτ(x)

dx+ (A3 − A2B1)σ (x) + (A4 − A2B3)q(x). (27)

Equation (23) gives

σ(x) = 1

(A3 − A2B1)

[d3τ(x)

dx3− (A1 − A2B2)

dτ(x)

dx− (A4 − A2B3)q(x)

]. (28)


Substitute equation (24) into equation (22), to obtain

d7τ(x)

dx7 − (A1 − A2B2)d5τ(x)

dx5+ B1

d3τ(x)

dx3

+ [B2(A3 − A2B1) − B1(A1 − A2B2)]dτ(x)

dx− (A4 − A2B3)

d4q(x)

dx4

+ [B3(A3 − A2B1) − B1(A4 − A2B3)]q(x) = 0. (29)

For simplicity, the general solutions presented below are limited to loading whichis either concentrated or uniformly distributed over part or the whole span of thebeam, or both. For such loading, the second- and higher-order datives are zero.Then, the above expression (equation (29)) can be written as

d7τ(x)

dx7 + (A2B2 − A1)d5τ(x)

dx5+ B1

d3τ(x)

dx3

+ [B2(A3 − A2B1) − B1(A1 − A2B2)]dτ(x)

dx+ [B3(A3 − A2B1) − B1(A4 − A2B3)]q(x) = 0. (30)

The general solution to equation (30) is given by

τ(x) = A5qx + C3eλx + C2e−λx + C1, (31)

where

A5 = −(

A4B1 − A3B3

A1B1 − A3B2

), λ2 = B1A1 − B2A3

B1. (32)

In deriving equation (31), it has been taken that d5τ/dx5 = 0, because d5τ/dx5

is generally of negligible significance to the final answer.Then, the general solution to the fourth-order differential equation (equation

(24)) is

σ(x) = e−βx[D1 cos(βx) + D2 sin(βx)]+ eβx[D3 cos(βx) + D4 sin(βx)] − n1

dτ(x)

dx− n2q. (33)

For large values of x it is assumed that the normal and shear stresses approach zero,and as a result C3 = D3 = D4 = 0. The general solutions therefore becomes

τ(x) = A5qx + C2e−λx + C1, (34)

σ(x) = e−βx[D1 cos(βx) + D2 sin(βx)] − n1dτ(x)

dx− n2q, (35)

where

β = 4

√B1

4, n1 = B2

B1and n2 = B3

B1, (36)

C1, C2, D1 and D2 are constant coefficients determined from the boundary con-ditions. In the present study, a simply supported beam is investigated which is


subjected to a uniformly distributed load. Using the same methodology as is de-scribed in references [6–8], the coefficients C1, C2, D1 and D2 are given by:

C1 = −1

2A5Lpq, C2 = n3 + A5q

λ,

D1 = 1

2

Ea

β3taE1I1[VT (0) + βMT (0)] − n4

2β3τ(0)

(37)

+ n1

2β3

d4τ(x)

dx4

∣∣∣∣x=0

+ βd3τ(x)

dx3

∣∣∣∣x=0

,

D2 = 1

2

Ea

β2taE1I1MT (0) − n1

2β2

d3τ(x)

dx3

∣∣∣∣x=0

,

where

n3 = Ga

ta

(y1 + ta/2)

E1I1MT (0), n4 = Eab2

ta

(y1

E1I1− y2

D11b2

). (38)

3. Results and Discussion

In this section, an RC beam bonded with a composite soffit plate is considered asillustrative examples. The beam is simply supported and subjected to a uniformlydistributed load (UDL). A summary of the geometric and material properties isgiven in Table 1. The span of RC beam is L = 3000 mm, the distance from thesupport to the end of the plate is a = 300 mm and UDL is 50 kN/m.

As a verification of the present solution, comparisons of the present model witha typical existing analytical solution [8] are made for the case of constant fiberspacing (n = 0) and is considered to be (Vf)in = 70%. With the volume fractionindex n = 0, equation (11) becomes Vf(ξ) = (Vf)in.

It can be seen from Fig. 3 that the present method agree closely with the solutionpresented by Tounsi and Benyoucef [8].

Figures 4 and 5 show the plot of edge shear and normal stresses vs. (Vf)in inFRP plated RC beam for different outer fiber volume fraction and with the volumefraction index n = 1. It is seen that decreasing the outer fiber volume fraction (Vf)outleads to significant decrease in the peak interfacial stresses. This is because; the

Table 1.Geometric and material parameters

Component Width b (mm) Depth t (mm) E (GPa) ν

RC beam b1 = 200 t1 = 300 E1 = 30 0.18Adhesive layer ba = 200 ta = 2.0 Ea = 2 0.35Composite plate b2 = 200 t2 = 4.0 Ef = 275 νf = 0.28

Em = 3.445 νm = 0.35


Figure 3. Interfacial stresses of a FRP plated RC beam.

Figure 4. Edge shear stresses in FRP plated RC beam with different outer fiber volume fraction(n = 1).

decrease of outer fiber volume fraction induces an increase in outer matrix volumefraction (Vm)out and consequently an increase in the thickness of adhesive and, asis shown by Tounsi and Benyoucef [8], the increase of the thickness of the adhesivelayer leads to significant reduction in the peak interfacial stresses. Thus the use


Figure 5. Edge normal stresses in FRP plated RC beam with different outer fiber volume fraction(n = 1).

of bonded plates containing a weak outer fiber volume fraction, especially in thevicinity of the adhesive layer, is recommended. In the same way the increase ofthe inner fiber volume fraction (Vf)in leads to an increase in the peak interfacialstresses.

Figure 5 shows also that for inner fiber volume fraction (Vf)in greater than 30%,the edge normal stresses are not significantly influenced by the outer fiber volumefraction (Vf)out.

The effect of the order n of fiber distribution function Vf on the edge stressesis displayed in Figs 6 and 7. It can be seen that for the inner fiber volume fraction(Vf)in of less than 25%, the edge stresses increase with the order of fiber distribu-tion. On the other hand, for the inner fiber volume fraction (Vf)in greater than 25%,the edge shear stress is higher for the lower order fiber distribution. However, theedge normal stress are not significantly influenced by the volume fraction index n inthe case where (Vf)in > 25%. When (Vf)in is equal to 25%, i.e., identical to (Vf)out,the edge interfacial stresses are independent of the volume fraction index n, whichcan be easily explained by equation (11).

Figures 8 and 9 show the plot of edge shear and normal stresses vs. (Vf)out inFRP plated RC beam for different volume fraction index n and with (Vf)in = 25%.In contrast with the results presented in Figs 6 and 7, it can be seen that for theouter fiber volume fraction (Vf)out less than 25%, the edge stresses decrease withthe order of fiber distribution. On the other hand, for the outer fiber volume fraction(Vf)out greater than 25%, the edge shear stress is higher for the higher-order fiber


Figure 6. Edge shear stresses in FRP plated RC beam with different volume fraction index n

((Vf)out = 25%).

Figure 7. Edge normal stresses in FRP plated RC beam with different volume fraction index n

((Vf)out = 25%).


Figure 8. Edge shear stresses in FRP plated RC beam with different volume fraction index n

((Vf)in = 25%).

Figure 9. Edge normal stresses in FRP plated RC beam with different volume fraction index n

((Vf)in = 25%).


distribution. The edge normal stress is not significantly influenced by the volumefraction index n.

4. Conclusions

A new theoretical interfacial stress analysis has been presented for simply supportedbeams bonded with a FRP strip having variable fiber spacing and subjected to auniformly distributed load. The mismatch of the curvatures in the beam and theplate is considered. The solution methodology presented here, is general in natureand may be applicable to the analysis of other types of loading.

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A New Approach for Adhesive Stress Analysis of a Beam Bonded with Composite Plate Having Variable Fiber Spacing