Proceedings of the International Symposium on Bond Behaviour of FRP in Structures (BBFS 2005)

Chen and Teng (eds)

© 2005 International Institute for FRP in Construction

175

CLOSED-FORM RIGOROUS SOLUTION FOR THE INTERFACIAL STRESSES IN PLATED BEAMS USING A TWO-STAGE METHOD

Jian Yang1 and Jianqiao Ye2 1School of Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT

2School of the Civil Engineering, The University of Leeds, UK ABSTRACT Interfacial shear and transverse normal stresses have played a significant role in understanding the premature debonding failure of reinforced concrete (RC) beams strengthened by bonding steel/composite plates/sheets at their soffits. Due to the occurrence of dissimilar materials and the abrupt change of cross section, the stress distributions become singular and hence are considerably complicated. Extensive experimental and analytical analyses have been undertaken to investigate the stress distributions. Large discrepancies have been found among various studies, particularly from experimental results due to the apparent difficulty to measure the interfacial stresses. Numerical analysis, e.g. finite element analysis (FEA), may predict more accurate results, but they demand tedious work on meshing and sensitivity analysis. Analytical solutions, in particular the closed-form ones, are unquestionably desirable in engineering practice, provided that they predict the result reliably and easily. This paper reports an improved closed-form rigorous solution for the interfacial stresses in plated beams using a two-stage method. Compared with the previous analytical results, this one is the only one which accurately predicts the transverse normal stresses in both adhesive-concrete and plate-adhesive interfaces in a closed-form. KEYWORDS FRP, RC beams, strengthening, interfacial stresses, stress concentration, debonding failure. INTRODUCTION Reinforced concrete (RC) beams can be strengthened by bonding steel or composite plate/sheet at the soffits. Numerous studies have shown that this method improves their structural behaviours efficiently (Swamy et al., 1987; Triantafillou and Plevris, 1992; Teng et al., 2001). However, a key problem emerging in the applications of this method is the premature debonding failure, i.e. the bonded plates separate from the RC beams and the strengthened beams lose their integrity. The load at which the debonding occurs is much lower than the ultimate load achieved through their composite actions and hence becomes the controlling factor when their loading capacities are evaluated. As has been well understood, the debonding failure is closely related to the high interfacial stresses at the plate ends. Hence, a reliable prediction of the interfacial stress is prerequisite. Due to the occurrence of dissimilar materials and the abrupt change of the cross sections, the stress components become singular and are considerably complicated. Extensive experimental and analytical analyses have been undertaken to investigate the stress distributions. Smith and Teng (2001) and Mukhopadhyaya and Swamy (2001) have compared the various analytical solutions available by then and a more thorough and recent review has been done by Yang (2005), in which the published experimental, numerical and analytical investigations have been reviewed. Large discrepancies have been found among various studies, particularly from experimental results due to the difficulty to measure the interfacial stresses. Numerical analysis, e.g. finite element analysis (FEA), may predict more accurate results, but they demand tedious work on meshing and sensitivity analysis. Analytical solutions, particularly the closed-form one, are unquestionably desirable in engineering practice, provided that they predict the result reliably and easily. These include the work of Roberts and Haji-Kazemi (1989), Taljsten (1997), Malek et al (1998), and Smith and Teng (2001). Shen et al. (2001) proposed a closed-form solution for the strengthened beam subjected to uniformly distributed load (UDL) in which the adhesive layer was treated as a 2-D medium. Yang et al. (2004) extended it to arbitrary loading conditions. Their solutions predicted the shear stresses agreeing well with the FEA results, while the transverse normal stresses were less accurate, in particular in the plate-adhesive (PA) interfaces.

176

This paper reports an improved closed-form rigorous solution for the interfacial stresses in plated beams using a two-stage method. Compared with the previous analytical results, this is the only one which can accurately predict the transverse normal stresses in both adhesive-to-concrete (AC) and PA interfaces while still are in the closed-form. METHOD OF SOLUTIONS Structural idealisation of the strengthened beam We consider a simply supported RC beam subjected to a uniformly distributed load (UDL) q and a pair of symmetric end moment M0. The beam has a span length 2L, and is partially strengthened by externally bonding a plate/sheet using structural adhesive. The length of the bonded area is 2l. The structural idealisation of the beam is illustrated in Figure 1. The applied loads cause the internal forces Nl, Ql and Ml at the cross-sections of the RC beam where the bonded plate ends. The sign convention for the applied loadings and the internal forces are shown in Figure 2. A global Cartesian co-ordinate system x-y is used with its origin being at the middle of the top surface of the beam (Figure 1). The bonded segment consists of three layers: the concrete beam, the adhesive layer and the plate. A local coordinate system x[i] - y[i] is adopted for each of the three layers, with the origin being at the geometrical centre of each layer (Figure 2a). For ease of reference, these three layers are denoted by superscripts [1], [2] and [3] respectively (e.g. the thicknesses of the three layers are denoted by h[1], h[2] and h[3] respectively). Similarly, superscripts (0), (1), (2) and (3) are used to denote the surfaces and interfaces (e.g. the widths the surfaces/interfaces are b(0), b(1), b(2) and b(3) respectively) (Figure 2b). Note b(0) = b(1) = b(2) from physical observation.

q

2l 2L

y xM0M0

Fig.1 Structural idealisation of the strengthened RC beam

y[1] (0)

(1)

(2)

(3)

x[1] [1] [2]

[3] y[3]

x[3]

y[2] x[2]

y

x

q

MlNl

QlMl

Nl

Ql

h[3]

h[2

]

h[1

]

b(3)

b(0)

b(2)

b(1)

(a) Strengthened segment (b) Cross section of the strengthened segment

Fig. 2 Notation used in the solution derivation The rationale and the description of the method Previous studies, e.g. Shen et al. (2001) and Yang et al. (2004) have shown that the shear stress in the adhesive layer varies negligibly along the thickness direction. FEA modellings, e.g., Teng et al (2002) and Yang (2005), showed significant transverse normal stress variation that changes the sign from tension in the AC interface to compression in the PA interface. The present analysis consists of two stages. At both stages, Fourier series expansion of the unknown stresses and the principle of the complementary energy are used. One of the unique features of this solution is that an infinite number of terms of Fourier series can be taken and the explicit summations can be obtained. In the first stage, shear stress and transverse normal stress along the middle section of the adhesive layer (MA) are expressed in Fourier series respectively, in which an approximate relationship between them is introduced based on the assumption of composite action. This will reduce the number of the unknown coefficients and lead to explicit solutions for both stress components. At the second stage, the transverse normal stress is re-expressed and joins the shear stress obtained from the first staged to derive other stress components based on the

177

equilibrium equations and the related boundary conditions. The principle of the complementary energy is used again to calculate the new transverse normal stress with improved accuracy. At both stages, the following assumptions are adopted:

1. Each individual layer is elastic, homogeneous and orthotropic. Note that the assumption of orthotropic behaviour has implications only for the shear moduli of the materials for the RC beam and the bonded plate;

2. The three layers are perfectly bonded (no slips or opening-up at the interfaces); 3. The Euler-Bernoulli beam theory is adopted for the beam and the plate, whereas the adhesive layer is

considered to be in a plane stress state; 4. The shear stress in the adhesive layer is constantly distributed in the thickness direction; 5. In the first stage, the assumption of plan section is used to define the relationship between the

transverse normal stress and shear stress along the middle section of the adhesive layer.

Shear stress in the adhesive layer The shear stress in the adhesive layer is defined by [2]

xyσ and the transverse normal stress in the MA section is defined by

0yσ . Using the equilibrium equations and boundary conditions given in Yang et al. (2004), the transverse normal stress in the adhesive layer can be calculated by

[2][2] [2]

0xy

y y

dy

dxσ

σ σ= − (1)

where y[2] is the local transverse coordinate for the adhesive layer, [2]xyσ and

0yσ are only the functions in respect of the x coordinate.

[2]xyσ and

0yσ may be expanded using Fourier series as follows,

lxmbx

mmxy

πσ sin)(]2[ ∑= ; l

xmaxm

myπσ cos)(0 ∑= (2a, b)

where ma and mb are unknown Fourier coefficients; and m = 1, 2, …∞. Note that the constant term in Equation

2b has been set to zero to satisfy the equilibrium requirement that the integration of the interfacial normal stress over the entire length of the middle section must be equal to zero. To facilitate the derivation of the explicit solution, we set up the relationship between the unknown Fourier coefficients ma and mb by

ξπl

mba mm = (2c)

where ξ is the coefficient, which was derived in Yang and Ye (2005). This leaves only one set of unknown

coefficients mb in the equations. Following the similar procedures in Yang et al. (2004), the stress components in the RC beam and bonded

plated can be obtained by

( )[ ] ( ) ⎥⎦

⎤⎢⎣⎡ −−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+−= ∑ lxmb

mlhh

hy

hm

mmx

ππ

ξσ cos1261 ]2[]1[3]1[

]1[

]1[]1[

[ ] ( )( )

[ ]l

xmbhhh

yhy

hh

mmxy

πξξσ sin234

2341 ]2[]1[

3]1[

2]1[

]1[

]1[

]1[

]2[]1[ ∑

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−++−−

+−=

( )[ ] ( )

( )( )22

3]3[)3(

]1[]3[]2[

3]3[

]1[

]3[)3(

)2(]3[ 26cos1261 qxqlM

hb

yl

xmbm

lhhh

yhb

bl

m

mmx −++⎥⎦

⎤⎢⎣⎡ −−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++−−= ∑ ππ

ξσ

[ ] ( )( )

[ ]( )

( ) ( )( )2]3[2]3[3]3[)3(

]3[]2[3]3[

2]3[

]3[

]3[

]3[

]2[

)3(

)2(]3[ 4

2

3sin234

2341 hy

hb

qxl

xmbhhh

yhy

hh

bb

mmxy −+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+++++

−−= ∑ πξξσ (3a-d)

The unknown coefficients bm may now be determined by minimising the total complementary energy of the

strengthened segment by nullifying the first-order partial derivative with respect to mb , which yields

178

( ) ( ) mmj

jj

m Pbl

mSSm

lSbjl

mlS =

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛++⎟

⎠⎞

⎜⎝⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ ∑

=

2

32

21

,2,11 2

11 ππππ L

(4)

where the constant S1, S2 , S3 and the coefficient Pm, which are related to the geometrical and material properties of the constituent layers, can be found in Yang (2005) and Pm also includes the term number m.

Rewriting Equation 4 yields

( )3

1224

4 112 S

RSm

lPmm

mb mmm ⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−

++=

παβ (5)

where

3

22

2SSl

⎟⎠⎞

⎜⎝⎛=π

β ; 3

12

2SSl

⎟⎠⎞

⎜⎝⎛=π

α ; Θ+

Θ=

14

34

22

SlSlR

ππ , ∑ ++

=Θm mm 224 2

1αβ

( )4 2 2

12

mm

m

mPm mβ α

−Θ =

+ +∑ (6a-e)

the summations of the infinite terms of series in Θ and Θ can be explicitly calculated as constants (Yang, 2005). Substitute Equations (5) and (6) into Equations (2) yields

( ) ( )( ) ⎭

⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛

++−

+⎟⎠⎞

⎜⎝⎛

++−

⎟⎠⎞

⎜⎝⎛ −= ∑∑

m

m

m

m

xy xl

mmmm

Bxl

mmm

mRlSAS

l παβ

παβππ

σ sin2

1sin2

1224224

1

32

2]2[

xCxl

Cxl

C 32

21

1 sinhsinh +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

πγπγ (7)

where 221 αββγ −+= and 22

2 αββγ −−= and the second equation in Equation (7) is derived using the summation functions of the Fourier series functions (Yang, 2005). The constant coefficients A, B, C1, C2 and C3 are given in Yang (2005). Transverse normal stress in the adhesive layer Substituting Equations (2b) and (7) into Equation (1) yields

[2] 1 1 2 21 2 3cos cosh coshy m

m

m xa y C x C x Cl l l l l

γ π γ π γ π γ ππσ ⎛ ⎞= − + +⎜ ⎟⎝ ⎠

∑ (8)

Using the same procedure as that in the preceding section, the stress components are obtained as follow

( ) ( ) ]1[2

3]1[

]1[]1[ cos112

xam

mmx l

xmam

lh

y σππ

σ +⎥⎦⎤

⎢⎣⎡ −−⎟

⎠⎞

⎜⎝⎛= ∑ ;

( ) ( ) ( )[ ] ]1[2]1[2]1[3]1[

]1[ sin42

3xya

mmxy l

xmam

lyhh

σππ

σ +−= ∑ ;

( ) ( ) ]3[]3[2

3]3[

]3[

)3(

)2(]3[ cos112

xbxam

mmx l

xmam

lh

ybb σσπ

πσ ++⎥⎦

⎤⎢⎣⎡ −−⎟

⎠⎞

⎜⎝⎛−= ∑

;( )

( ) ( )[ ] ]3[]3[2]3[2]3[3]3[)3(

)2(]3[ sin43

23

xybxyam

mxy lxmayh

hbb σσπσ ++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−= ∑

where

( )( )

( )( )

( )( )

[1] [1] [2] [1] [1] [2][2] [1][1] 1 2

1 1 1 2 22 3 2[1] [1] [1]

6 661 cosh cosh sinh 1 cosh coshxa

y h h y h hh y x lC x C x

l lh h h

γ π γ πσ γ π γ π γ π

⎧ ⎫ ⎧⎡ ⎤ ⎡ ⎤+ +−⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥= − − + + − −⎨ ⎬ ⎨⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩

( )( )

( ) ( )( )

( )( )

22 2 [1] [1][2] [1] [1] [2]

2 33 2 3[1] [1] [1]

66 3sinh

2

l x y hh y x l y h l xC

h h hγ π

⎫ ⎡ ⎤− −− −⎪ ⎢ ⎥+ + −⎬ ⎢ ⎥⎪⎭ ⎣ ⎦

( ) ( )( )

( ) ( )( )

( ) ( )( )

2 22 2[2] [1] [1][1] [1] [2] [1] [1] [2][1] [2] [1] [2][1] 1

1 1 23 [1] 3 3 [1][1] [1] [1]

3 43 34 3 4 31 1sinh sinh4 44 44

xya

h h yy h h y h hy h y hC x C

lh hh h h

γ πσ γ π

⎧ ⎫⎡ ⎤ ⎧⎡ ⎤ ⎡ ⎤−+ +⎢ ⎥+ +⎪ ⎪ ⎪⎣ ⎦⎢ ⎥ ⎢ ⎥= − − + + − −⎨ ⎬ ⎨⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩⎩ ⎭

( ) ( )( )

2 2[2] [1] [1] 2 2[1] [2] [1] [2] [1] [2] [1]2

2 33 [1] [1] [1] [1] [1][1]

3 4 3 4 3 3 1sinh sinh 3444

h h y y h y h h h l yx C xl h h h h hh

γ πγ π

⎫⎡ ⎤− ⎧ ⎫⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥ + +⎪ ⎪ ⎪⎣ ⎦ ⎜ ⎟+ + + − − −⎢ ⎥⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪⎣ ⎦ ⎝ ⎠⎩ ⎭⎭

( )( )

( )( )

( )( )

[3] [2] [3] [3] [2] [3][2] [3](2) (2)[3] 1

1 1 1 2(3) 2 3 (3) 2[3] [3] [3]

6 661 cosh cosh sinh 1xa

y h h y h hh y x lb bC x Clb bh h h

γ πσ γ π γ π

⎧ ⎫ ⎧⎡ ⎤ ⎡ ⎤+ +−⎪ ⎡ ⎤ ⎪ ⎪⎢ ⎥ ⎢ ⎥= + − + + −⎨ ⎬ ⎨⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩

( )( )

( ) ( )( )

( )( )

22 2 [3] [3][2] [3] [3] [2](2)2

2 2 33 (3) 2 3[3] [3] [3]

66 3cosh cosh sinh

2

l x y hh y x l y h l xbx Cl bh h h

γ πγ π γ π

⎫ ⎡ ⎤− −− −⎡ ⎤ ⎪ ⎢ ⎥− + + −⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎪⎭ ⎣ ⎦

179

( ) ( )( )

( ) ( )( )

( ) ( )( )

2 22 2[2] [3] [3][3] [2] [3 [3] [2] [3](2) [3] [2] (2)[3] 1

1 1 23 3 3(3) [3] (3)[3] [3] [3]

3 43 34 3 1 sinh sinh4 4 4

xya

h h yy h h y h hb y h bC x Cb h l bh h h

γ πσ γ π⎧ ⎫⎡ ⎤ ⎧⎡ ⎤ ⎡−+ +− ⎢ ⎥⎪ ⎪ ⎪⎣ ⎦⎢ ⎥ ⎢= + − + +⎨ ⎬ ⎨⎢ ⎥ ⎢⎪ ⎪ ⎪⎣ ⎦ ⎣⎩⎩ ⎭

( ) ( )( )

2 2[2] [3] [3 2 2[3] [2] (2) [3] [2] [3] [2] [3] [2] [3]2

2 3[3] 3 (3) [3] [3] [3] [3] [3][3]

3 44 3 1 3 4 3 3 1sinh sinh 34 44 44

h h yy h b y h y h h h l yx C xlh b h h h h hh

γ πγ π

⎫⎡ ⎤− ⎧ ⎫⎡ ⎤ ⎛ ⎞⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥+ − −⎪ ⎪ ⎪⎣ ⎦ ⎜ ⎟− − + + + + − −⎢ ⎥⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎥ ⎜ ⎟⎢ ⎥⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪⎣ ⎦ ⎝ ⎠⎩ ⎭⎭

( ) ( )2203]3[)3(

]3[]3[ 26 qxqlM

hby

xb −+=σ ; ( )

( )( )]3[]3[]3[]3[3]3[)3(

]3[ 222

3 hyhyhb

qxxyb +−=σ (9a-j)

By utilising the principle of complementary energy, a similar equation to Equation (4) is obtained, i.e.

( ) ( ) mmj

jj

m PaSm

lSm

lSajl

mlS ′=

⎥⎥⎦

⎤

⎢⎢⎣

⎡′+⎟

⎠⎞

⎜⎝⎛′+⎟

⎠⎞

⎜⎝⎛′+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛′ ∑

=3

2

2

41

,2,1

2

1 211

ππππ L

(10)

Likewisely, the constant '1S , '

2S , '3S and the coefficient '

mP are not listed herein for brevity but can be found in Yang (2005). Following the similar procedure as that in the first stage, we have

( )( )

( )( )

( )( )0 1 2 32 2 22 2 2

1 2 1 2

1 1 1cos cos cos

m m m

ym m m

m m mH x H x H xl l lm m m iπ π πσ

γ γ η η

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠+ + + +

∑ ∑ ∑

( )( ) ( ) ( )4 5 62 2 22 2 2

1 2 1 2 1 2

1 1 1cos cos cosm

m m m

m m mH x H x H xl l lm i m i m iπ π π

η η η η η η

− ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠+ − + + + −

∑ ∑ ∑ (11)

where i is the unit of the imaginary of a complex and

21βαη′+′

= ; 22βαη′−′

= ; 3

22

2SSl′′

⎟⎠⎞

⎜⎝⎛=′π

β ; 3

12

2SSl′′

⎟⎠⎞

⎜⎝⎛=′π

α (12a-d)

By following a complicated mathematical derivation, the transverse normal stress in the MA section can be calculated by

{2

0 02 10 1 2 1 2 4 5 6 12

3 1 2 11 12 21 22

12y R I R

F F lG F G G F F F S RS

δ δβσηη δ δ δ δ π′ ⎛ ⎞− ′ ′= + + + + −⎜ ⎟′ ⎝ ⎠

( )2 2 4 42

0 1 0 22 1 24 3 5 6 1 1 22

1 2 11 12 21 22 11 12 21 222IG lF F F F S R G Gδ γ δ γ γ γβ β

ηη δ δ δ δ π δ δ δ δ⎫⎡ ⎤ ⎪′ ′ ′ ′ ′ ′+ − + + − + + ⎬⎢ ⎥⎪⎣ ⎦ ⎭

(13)

where RG1, IG1 ,

RG2,

IG2,

1G′ and 2G′ are the functions in respect of x, and all the other variables are constants.

Due to the limit of length of this paper, they are not listed herein but can be found in Yang (2005). Using Equation (8), the transverse interfacial normal stress can be obtained by substituting ]2[y with 2/]2[h and

2/]2[h− , respectively, i.e.

⎟⎠⎞

⎜⎝⎛ ++−= 3

222

111

]2[

0)1( coshcosh

2Cx

llCx

llCh

yyπγπγπγπγσσ (14a)

⎟⎠⎞

⎜⎝⎛ +++= 3

222

111

]2[

0)2( coshcosh

2Cx

llCx

llCh

yyπγπγπγπγσσ (14b)

SOLUTION VERIFICATION BY NUMERICAL EXAMPLES The first example is an RC beam bonded with a steel plate subjected to a UDL q = 15N/mm initially analysed by Roberts and Haji-Kazemi (1989). For ease of reference, the geometrical and material properties are relisted herein as follow: L = 1200mm, l = 900mm, h[1]= h[2]= 4mm, h[3]= 150mm, b(1) = b(2) = b(3) = 100mm, ]1[]1[

yx EE = =

200GPa, ]2[]2[yx EE = = 2GPa, ]3[]3[

yx EE = = 20GPa, ]1[xyν = 0.3, ]2[

xyν = 0.25 and ]3[xyν = 0.17. This example has been used

in various studies, e.g. Shen et al. (2000), Smith and Teng (2001), Teng et al. (2002) and Yang et al. (2004), among which Teng et al.’s (2002) elastic FEA study revealed the most severe stress singularities. Hence, they are compared with the results obtained by the present solution as shown in Figure 3. The comparison of the transverse normal stress exhibits that both methods predict tensile stresses in the AC interface and compressive ones in PA interface. The peak values in AC and PA interfaces predicted by the present solution are considerably lower than those from FEA. It is deemed to be attributed to the adoption of the Euler-Bernoulli

180

theory for the beam and the plate as mentioned in the assumptions. However, the results show very good coincidence along the most bonded span expect the region within 5 mm from the plate ends, where transverse normal stress in the adhesive becomes extremely complex. The adhesive adjacent to the AC interface is in tension and that adjacent to the PA interface is in compression. From Figure (3a), it is also seen that in the AC interface there is a tension region near the plate end followed by compression one. The stress resultants in each region generate a couple which causes the bend moment in the bonded plate. A single shear stress distribution is predicted by the present solution, and as Figure (3b) shows, its peak value is slightly lower than those from FEA method for both interfaces and the MA section. The discrepancy only occurs within the region 20 mm from the plate end.

880 885 890 895 900-8

-2

4

10

16 Teng et al.(2002) FE PA Interface FE MA Section FE AC Interface

x(mm)

Present σy

(1) (PA Interface) σ y0 (MA Section)

σy(2) (AC Interface)

Nor

mal

Stre

ss (M

Pa)

880 885 890 895 900-8

-6

-4

-2

0

Shea

r Stre

ss (M

Pa)

x(mm)

Present σ[2]

xy

Teng et al.(2002) FE PA Interface FE MA Section FE AC Interface

(a) Transverse normal stress (b) Shear stress

Fig. 3 Steel plate beam under UDL

0.9900 0.9925 0.9950 0.9975 1.0000

0.0

0.1

0.2

0.3

0.4

3

1

x / l

Dim

ensi

onle

ss N

orm

al S

tress

es

Present solution Sheng et al. (2001)

1: AC interface2: MA section3: PA interface

2

0.9900 0.9925 0.9950 0.9975 1.0000

-0.100

-0.075

-0.050

-0.025

0.000

Dim

ensi

onle

ss S

hear

Stre

sses

x / l

Present solution Sheng et al. (2001) PA Interface Sheng et al. (2001) AC Interface

(a) Transverse normal stress (b) Shear stress Fig. 4 CFRP plated beam under UDL

The second verification example is taken from Shen et al. (2001). An RC beam bonded with a CFRP plate and subjected to UDL. The geometrical and material properties are: L = 1500mm, l = 1200mm, h[1]= h[2]= 2mm, h[3]= 300mm, b(1) = b(2) = b(3) = 200mm, [1]

xE = 140GPa, [1]xyG = 5GPa, ]2[]2[

yx EE = = 3GPa, ]3[]3[yx EE = = 30GPa, ]2[

xyν =

0.35 and ]3[xyν = 0.18. The results are presented in the dimensionless format in Figure 4, i.e. the x coordinate is

normalized by l, and stress values are normalized by the extreme tensile fibre stress in concrete at the midspan, which is 2 2 cqL Z where Zc is the elastic modulus of the concrete beam. As Figure 4(a) shows, Shen et al (2001) predicted tensile normal stress in PA interface, which should be compressive. The present solution has obvious improved this. In Figure (4), both solutions predict almost identical shear stresses and Shen et al’s results suggested that there was little difference in the interfacial shear stresses between the AC and PA interfaces. This conclusion can be used to validate the assumption adopted in deriving the present solution that the shear stress is uniform along the thickness direction. CONCLUSIONS In this paper, an elastic closed-form rigorous interfacial stress solution is proposed for a strengthened RC beam. Unlike most numerical methods which require either tedious pre- and post-processing work or computer coding, the present solution may offer the practical engineers an easy method to predict the interfacial stresses in the

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plated beam, which can be achieved by using spread sheet packages. Compared with other analytical solutions, this one is more reliable and encouraging. The numerical examples show that the present solution can deliver a comparable accuracy as the FEA solutions. To the authors’ best knowledge, this is the only closed-form solution being able to predict the compressive normal stress in the PA interface so far. As a closed-form solution, it can be further simplified to serve as a compact design equation (Ye and Yang, 2005). Furthermore, it also has the potential to be exploited to develop a sound strength model which may incorporate most real natures exhibited by this application through calibrating with sufficient number of test results. The method used to derive this solution is versatile and can be used to analyse other composite structures such as strengthened cast iron beams. REFERENCE Malek, A.M. Saadatmanesh, H. Ehsani, M.R. Prediction of failure load of R/C beams strengthened with FRP

plate due to stress concentration at the plate end. ACI Structural Journal 1998; 95 (1), 142-152. Mukhopadhyaya, P. and Swamy, R. N. Interface shear stress: A new criterion for plate debonding. Journal of

composites for construction 2001; 5(1). 35-43. Roberts, T.M. and Haji-Kazemi, H. Theoretical study of the behavior of reinforced concrete beams strengthened

by externally bonded steel plates. Proceedings of the Institution of Civil Engineers, Part 2 1989; 87, 39-55. Shen, H.S, Teng, J.G, Yang, J. Interfacial stresses in beams and slabs bonded with thin plate. Journal of

Engineering Mechanics ASCE 2001; 127(4), 399-406. Smith, S.T. and Teng, J.G. Interfacial stresses in plated beams. Engineering Structures 2001; 23, 857-871. Swamy, R. M. Jones, R. and Bloxham, J. W. Structural behavior of reinforced concrete beams strengthened by

epoxy-bonded steel plates. The Structural Engineer 1987; 65A(2), 59-68. T ä ljsten, B. Strengthening of beams by plate bonding. J Mater Civil Eng ASCE 1997; 9(4), 206-212. Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. FRP strengthened RC Structures 2002; Wiley, Chichester, U.K. Teng, J.G., Zhang, J.W. and Smith, S.T. Interfacial stresses in RC beams bonded with a soffit plate: a finite

element study. Constuction and Building Materials, 2002; 16(1), 1-14. Triantafillou, T. C. and Plevris, N. Strengthening of RC beams with epoxy-bonded fiber-composite materials.

Mater. Struct. 1992; 25, 201-211. Yang, J. The Stress Analysis and Strength Prediction of Plated RC Beams. Ph.D Thesis 2005; School of Civil

Engineering, the University of Leeds, UK. Yang, J., Teng, J.G. and Chen, J.F. Interfacial stresses in soffit-plated reinforced concrete beams. Proceedings of

the Institution of Civil Engineering, Structures and Buildings 2004; 157, 77-89. Yang, J, and Ye, J.Q. Interfacial shear stress in FRP-plated RC beams under symmetric loads. 2004 (submitted). Ye, J.Q. and Yang, J. (2005) An investigation on the stress transfer in concrete beams bonded with FRP plates (published in this proceedings).

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