Inter Facial Stresses in Plated Beams

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Engineering Structures 23 (2001) 857–871

www.elsevier.com/locate/engstruct

Interfacial stresses in plated beams

S.T. Smith, J.G. Teng *

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 2 May 2000; accepted 14 July 2000

Abstract

Fibre-reinforced plastic (FRP) or steel plates can be bonded to the soffit of a beam as a means of retrofitting the beam. In such

plated beams, tensile forces develop in the bonded plate and these have to be transferred to the original beam via interfacial shearand normal stresses. Consequently, debonding failure may occur at the plate ends due to a combination of high shear and normalinterfacial stresses. This paper starts with a review of approximate closed-form solutions for interfacial stresses, identifying theirassumptions and limitations, thereby clarifying the differences between these solutions. This review also establishes the need for asimilar but more accurate solution, and such a solution is presented next in the paper. This new solution is intended for applicationto beams made of all kinds of materials bonded with a thin plate, while all existing solutions have been developed focusing on thestrengthening of reinforced concrete (RC) beams, which allowed the omission of certain terms. Finally, numerical comparisonsbetween the existing solutions and the present new solution enable a clear appreciation of the effects of various parameters. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: FRP composites; Interfacial shear stresses; Interfacial normal/peeling stresses; Plate bonding; Beams; Retrofitting; Strengthening

1. Introduction

An existing beam can be retrofitted by bonding afibre-reinforced plastic (FRP) or steel plate to its soffit(Fig. 1). This plate bonding technique has been usedwidely to retrofit reinforced concrete (RC) beams, andhas also been used to retrofit beams of other materials.The technique has numerous advantages such as increas-ing the strength and stiffness of an existing beam with

Fig. 1. Soffit-plated beam.

* Corresponding author. Tel.: +852-2766-6047; fax: +852-2334-

6389.

E-mail address: [email protected] (J.G. Teng).

0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.

PII: S0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 9 0 - 0

minimal interference to the surrounding environment.Consequently, many studies have been carried out on thebehaviour and strength of such plated beams (e.g., [1–21]). In such retrofitted beams, debonding of the soffitplate from the beam is an important failure mode as itprevents the full ultimate flexural capacity of the retro-fitted beam from being achieved. It is thus important tobe able to predict the debonding failure load. Debondingfailures depend largely on the interfacial shear and nor-mal stresses between the beam and the bonded plate.

The determination of interfacial stresses has thus beenresearched for the last decade for beams bonded witheither steel or FRP plates. In particular, several relativelysimple approximate closed-form solutions for interfacialstresses have been developed [2–4,9,14,16] based on asimple assumption for the adhesive layer as discussedlater. Despite all of these studies, one striking fact is thatthe relationship between these existing solutions has notbeen established clearly in the existing literature. Thispaper therefore starts with a review of these existing sol-utions, identifying their assumptions and limitations,thereby clarifying the differences between them. Thisreview also establishes the need for a more accurate sol-ution of the same type, which is subsequently presentedin the paper. This new solution is intended for appli-



858 S.T. Smith, J.G. Teng / Engineering Structures 23 (2001) 857–871

cation to beams made of all kinds of materials bonded

with a thin plate, while all existing solutions have gener-

ally been developed focusing on the strengthening of RC

beams, which allowed the omission of certain terms.

Finally, numerical comparisons between the existing sol-utions and the present new solution enable a clear

appreciation of the effects of various parameters.It should be noted that the type of analysis discussed

in this paper does not satisfy the zero shear stress con-

dition at the end of the adhesive layer. This drawback is known to have only a limited effect in a very small

zone near the end of the plate [3]. For this condition to

be satisfied, a higher-order analysis has to be carried out.

The first such analysis has just appeared [20]. This

higher-order solution however does not provide explicit

expressions for the interfacial stresses, so numericalresults are not easily obtainable, which makes it dif ficult

for exploitation in developing a design rule. The cor-

rectness of this analysis has also been questioned [21].

In another higher-order analysis from the authors’ group

yet to be published [21], explicit expressions have been

obtained, but these are much more complex than the

expressions from the type of analysis discussed in this

paper. The simple approximate closed-form solutions

discussed in this paper provide a useful but simple toolfor understanding the interfacial behaviour and for

exploitation in developing a design rule. The present

paper is concerned only with this simple type of analysis

and, hereafter, no further reference to higher-order

interfacial stress analysis is made.

2. Review of existing solutions

2.1. Assumptions and approaches

All existing solutions are for linear elastic materials

only. The key assumption in all of these solutions is that

the adhesive layer is subject to shear and normal 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, althoughthis assumption is somewhat hidden in some of the sol-

utions.

In the existing solutions, two different approaches

have been employed. Roberts [3] and Roberts and Haji-

Kazemi [4] used a staged analysis approach, while Vil-

nay [2], Liu and Zhu [9], Taljsten [14] and Malek et al.[16] considered directly deformation compatibility con-

ditions.

2.2. Solutions based on direct deformation

compatibility consideration

Vilnay [2], Liu and Zhu [9], Taljsten [14] and Malek

et al. [16] directly considered compatibility of defor-

mations to determine the interfacial stresses. A summary

of the main differences between these analytical sol-

utions is given in Table 1. The solution of Malek et al.

[16] is general in terms of the applied load provided the

applied moment can be expressed by M (x)=a1x

2+a2x+a3, where x= x+ L0 and L0 is the distance

between the plate cut-off point and the origin of x, whilethe rest are restricted to certain loading conditions.

Interfacial shear stresses in the adhesive layer are

related to the difference between the longitudinal dis-placement at the base of the beam and that at the top of

the sof fit plate. The differences between these solutions

for interfacial shear stresses arise from the different

choices of terms for inclusion in determining these longi-

tudinal displacements. Bending deformations in the

beam and axial deformations in the sof fit plate are takeninto account in all of these solutions. Liu and Zhu’s sol-

ution [9] is the only one that considers the effects of

shear deformations of the beam, but the contributions to

shear deformations in both the beam and the plate by

the interfacial normal stresses are ignored.

Interfacial normal stresses are related to vertical defor-

mation compatibility between the beam and the bonded

plate. Vilnay [2] and Taljsten [14] derived the governing

equation in terms of the vertical displacement of thebonded plate. Liu and Zhu [9] and Malek et al. [16]

derived the governing equation in terms of the interfacial

normal stress. Nevertheless, the governing equations

derived by Vilnay and Taljsten can be reduced to those

of Liu and Zhu and Malek et al. with some differences.

The main differences between these solutions for interfa-

cial normal stresses are also summarised in Table 1.It should also be noted that the solution of Liu and

Zhu is incomplete in that the constants of integration

were not given, instead only the boundary conditions

were listed.

2.3. Roberts and Haji-Kazemi’s solution

Roberts and Haji-Kazemi’s solution [4] is for a uni-

formly distributed load (UDL) only. In the first stage of their solution, direct considerations of deformation com-

patibility lead to the determination of the interfacial

shear stress. Axial and bending deformations in both the

beam and the plate are considered. Free strains, which

may arise due to shrinkage, creep or temperature, in both

the beam and the plate are also included. If such freestrains are not present, then the boundary conditions

implemented in the first stage of analysis model the zero

axial force condition at the ends of the bonded plate. In

this first stage, the beam and the plate are assumed to

have identical vertical deflections. As a result, theinterfacial normal stress from the first stage has to be

worked out from an equilibrium consideration of the

beam. This first-stage analysis leads to a non-zero bend-



859S.T. Smith, J.G. Teng / Engineering Structures 23 (2001) 857 –871

T a b l e 1

C o m p a r i s o n o f a s s u m p t i o n s i n a n a l y t i c a l

s o l u t i o n s

T h e o r y

V i l n a y [ 2 ]

R o b e r t s [ 3 ]

R o b e r t s a n d H a j i -

L i u a n d Z h u [ 9 ]

T a l j s t e n [ 1 4 ]

M a l e k e t a l . [ 1 6 ]

P r e s e n t s o l u t i o n

K a z e m i [ 4 ]

S h e a r a n d n o r m a l s t r e s s e s L o a d c a s e s

P o i n t l o a d a t m i d -

G e n e r a l

U D L

U D L , p o i n t l o a d

a t S i n g l e p o i n t l o a d

G e n e r a l w i t h s o m e

G e n e r a l

s p a n

m i d - s p a n , t w o

l i m i t a t i o n s

s y m m e t r i c p o i n t

l o a d s

S h e a r s t r e s s

A x i a l d e f o r m a t i o n s N o

P a r t i a l l y

Y e s

N o

Y e s

P a r t i a l l y

Y e s

o f b e a m

c o n s i d e r e d *


B e n d i n g

N o

P a r t i a l l y

Y e s

N o

N o

P a r t i a l l y

Y e s

d e f o r m a t i o n s

o f



p l a t e

S h e a r d e f o r m a t i o n s N o

N o

N o

Y e s b u t i n c o r r e c t l y

N o

N o

Y e s ( i n g o v e r n i n g

e q u a t i o n s o n l y )

B o u n d a r y / c o n

t i n u i t y Z e r o s h e a r s t r e s s

–

–

C o n t i n u i t y i n s h e

a r Z e r o s h e a r s t r e s s

Z e r o s h e a r s t r e s s

C o n t i n u i t y i n s h e a r

c o n d i t i o n s a t

p o i n t

s t r e s s a n d i t s fi r s

t

s t r e s s a n d i t s fi r s t

l o a d

d e r i v a t i v e

d e r i v a t i v e

O t h e r r e m a r k s

–

* C o n s i d e r e d i n

–

O n l y g e n e r a l

–

* S t r e s s a t b a s e o f

S p e c i fi c i n t e r f a c i a l

s t a g e 1 a n a l y s i s

s o l u t i o n s a r e g i v e n

t h e o r i g i n a l b e a m

s t r e s s e x p r e s s i o n s

w h i c h i s f o r a f u l l y

w i t h b o u n d a r y

b a s e d o n f u l l y

d e v e l o p e d o n l y f o r

c o m p o s i t e s e c t i o n

c o n d i t i o n s

c o m p o s i t e a c t i o n

t h r e e l o a d c a s e s

N o r m a l s t r e s s

B e n d i n g

N o

N o

N o

Y e s

Y e s

Y e s

Y e s


o f

b e a m

S h e a r d e f o r m a t i o n s N o

N o

N o

Y e s b u t i n c o r r e c t l y

N o

N o

Y e s ( i n g o v e r n i n g

e q u a t i o n s o n l y )

A d d i t i o n a l p l a t e

Y e s

N o

Y e s

Y e s

Y e s

N o

Y e s

b e n d i n g


d u e t o

i n t e r f a c i a l s h e a r

s t r e s s e s



860 S.T. Smith, J.G. Teng / Engineering Structures 23 (2001) 857 –871

ing moment and a non-zero transverse shear force at

each plate end.

In the second stage of analysis, a bending moment and

a shear force equal to those from the first stage but

opposite in direction are applied at each plate end. Theplate is treated as a flexible beam on an elastic foun-

dation which represents the adhesive. The effect of anydeflection of the beam on interfacial normal stresses is

ignored. The second stage leads to the determination of

additional interfacial normal stresses, and also additionalinterfacial shear stresses due to the deflection of the plate

relative to the beam. The various assumptions of Roberts

and Haji-Kazemi’s solution [4] are given in Table 1.

The final interfacial shear and normal stresses are

found by combining the results from both stages; how-

ever, interfacial normal stresses from stage 1 and interfa-cial shear stresses from stage 2 are, in general, rela-

tively small.

2.4. Roberts’ solution

Roberts’ solution [3] is general in terms of loading

conditions and consists of three stages. In the first stage,

the interfacial shear stress in the adhesive layer is

determined for a fully composite section of infinitelength. For a sof fit plate of finite length, stage 1 of the

analysis produces an axial force at each end of the

bonded plate. As this force does not exist in practice,

the second stage of the analysis is to apply an equal but

opposite axial force to either end of the plate. In the

second stage the sof fit plate is treated as being an axial

member with no bending deformations on an elasticshear foundation representing the adhesive, with thebeam assumed to be rigid. At the end of the stage 2

analysis, there exist a non-zero moment and a non-zero

shear force at each end of the plate. In analysis stage 3,

equal but opposite moments and shear forces are applied

at the ends of the bonded plate with the plate assumed

to be a beam on an elastic foundation representing the

adhesive, while the beam is assumed to be rigid. Final

interfacial shear stresses are obtained by combining the

results from stages 1 and 2, while the interfacial normalstresses are given by the stage 3 analysis only.

2.5. Need for a new solution

The above review suggests that the analysis of Roberts

and Haji-Kazemi [4] is likely to be the most accuratefor interfacial shear stresses as both axial deformations

in the beam and bending deformations in the plate are

taken into account. This solution is however restricted

to a UDL only and is rather complex. Most solutions

ignore the effects of axial deformations in the beamand/or bending deformations in the plate. Such solutions

are expected to be unsuitable for cases when the bonded

plate has a significant flexural rigidity compared with the

beam, as in the case of an aluminium beam bonded with

a carbon-fibre-reinforced plastic (CFRP) plate.

For interfacial normal stresses, the solution of Taljsten

[14] is expected to be the most accurate as both bending

deformations in the beam and additional bending defor-mations of the plate due to interfacial shear stresses are

properly accounted for. This solution is for a single pointload. Liu and Zhu’s solution [9] also considers both fac-

tors, but the constants of integration in their solution are

yet to be determined. Most solutions ignore the effect of bending deformations in the beam. This omission again

may be inappropriate when the flexural rigidity of the

bonded plate becomes significant compared with that of

the beam.

While the assumptions adopted in these solutions are

reasonable for RC beams bonded with a thin plate, whichwas the intended application, the above review does sug-

gest the need for a more accurate solution, both for appli-

cations to other situations where the flexural rigidity of

the beam and that of the bonded plate are more compara-

ble, and to assess the significance of the various terms

that have been omitted by existing solutions. In addition,

it should be commented that except for the works of

Roberts [3] and Roberts and Haji-Kazemi [4], all of the

other solutions are based on the deformation compati-bility approach. From a physical point of view, interfa-

cial stresses in the adhesive layer are induced as the

adhesive layer attempts to enforce deformation compati-

bility between the RC beam and the sof fit plate which

otherwise would deform without interaction. The defor-

mation compatibility method which embodies a simple

physical aspect is thus believed to be more advantageousover the staged analysis approach of Roberts and Roberts

and Haji-Kazemi. In the following sections, a new sol-

ution that is based on the deformation compatibility

approach and fulfils the above-mentioned requirementsis presented. Governing equations and their general sol-

utions are derived for general loading conditions, while

specific interfacial stress expressions are given for three

important load cases: an arbitrarily positioned point load,

two symmetrically positioned point loads and a UDL. It

may be noted that, among the existing solutions, the onelikely to be the most accurate for the interfacial shear

stress is Roberts and Haji-Kazemi’s solution [4] which

is limited to a UDL, while Taljsten’s solution [14],

which is likely to be the most accurate for the interfacial

normal stress, is limited to a single point load.

3. Assumptions of the new solution

The derivation of the new solution below is described

in terms of adherends 1 and 2, where adherend 1 is the

beam and adherend 2 is the sof fit plate. Adherend 2 canbe either steel or FRP but not limited to these two. The

assumptions adopted in the present solution are summar-

ised below.




Linear elastic behaviour of adherends 1 and 2, as well

as of the adhesive layer, is assumed. Deformations of

adherends 1 and 2 are due to bending moments, axial

and shear forces. The adhesive layer is assumed to be

subject to stresses invariant across its thickness. This isthe key assumption which enables relatively simple

closed-form solutions to be obtained.Under normal stresses in the thickness-wise direction,

the adhesive layer will deform, so the vertical displace-

ments at the bottom of adherend 1 and the top of adherend 2 differ. As a result, the curvature of the beam

will differ from that of the sof fit plate. These thickness-

wise deformations of the adhesive are assumed to have

a negligible effect on the interfacial shear stresses. That

is, in finding the interfacial shear stresses, the curvatures

of both adherends are assumed to be the same. The sameassumption has been used by Roberts and Haji-Kazemi

[4]. This assumption is not used in the determination of

interfacial normal stresses.

3.1. Interfacial shear stress: governing differential

equation

A differential segment of a plated beam is shown in

Fig. 2, where the interfacial shear and normal stressesare denoted by t ( x) and s ( x), respectively. Fig. 2 also

shows the positive sign convention for the bending

moment, shear force, axial force and applied loading.

The shear strain g in the adhesive layer can be written as

g du( x, y)

d y

dv( x, y)

d x

, (1)

Fig. 2. Differential segment of a sof fit-plated beam.

where u( x, y) and v( x, y) are the horizontal and vertical

displacements respectively at any point in the adhesive

layer as defined in Fig. 1. The corresponding shear stress

is given as

t ( x)Ga

du( x, y)

d y

dv( x, y)

d x , (2)

where Ga is the shear modulus of the adhesive layer.

Differentiating the above expression with respect to x

gives

dt ( x)

d xGad2u( x, y)

d xd y

d2v( x, y)

d x2. (3)

The curvature of a differential element can be related to

the applied moment, M T( x), by the following

d2v( x)

d x2

1

( EI )t M T( x), (4)

where ( EI )t is the total flexural rigidity of the compositesection considering the partial interaction between the

two adherends. The adhesive layer is assumed to be sub-

jected to uniform shear stresses and therefore u( x, y)

must vary linearly across the adhesive thickness t a, then

du

d y

1

t a[u2( x)u1( x)] (5)

and

d2u( x, y)

d xd y

1

t adu2( x)

d x

du1( x)

d x, (6)

where u1( x) and u2( x) are the longitudinal displacementsat the base of adherend 1 and the top of adherend 2,

respectively, and t a is the thickness of the adhesive layer.

Eq. (3) can be rewritten as

dt ( x)

d x

Ga

t adu2( x)

d x

du1( x)

d x

t a

( EI )t M T( x). (7)

In calculating ( EI )t, interfacial shear stresses should

be considered but to do so would complicate the sol-

ution. The third term in parentheses in Eq. (7) is very

small and thus is ignored in the following derivation.The strains at the base of adherend 1 and the top of

adherend 2, considering all three components of axial,bending and shear deformations, are given as

e1( x)du1

d x

y1

E 1 I 1 M 1( x)

1

E 1 A1

N 1( x) y1

G1a A1

[q (8)

b2s ( x)]

and

e2( x)du2

d x

y2

E 2 I 2 M 2( x)

1

E 2 A2

N 2( x) (9)

y2

G2a A2

b2s ( x),




where E is the elastic modulus, G the shear modulus, b2

the width of the sof fit plate, A the cross-sectional area, I the second moment of area and a the effective shear

area multiplier, which is equal to 5/6 for a rectangular

section. The subscripts 1 and 2 denote adherends 1 and2, respectively. M ( x), N ( x) and V ( x) are the bending

moment, axial and shear forces in each adherend while y1 and y2 are the distances from the bottom of adherend

1 and the top of adherend 2 to their respective centroid.

Consideration of horizontal equilibrium gives

d N 1( x)

d x

d N 2( x)

d xb2t ( x) (10)

where

N 1( x) N 2( x) N ( x)b2 x

0

t ( x) d x. (11)

Assuming equal curvature in the beam and the sof fitplate, the relationship between the moments in the two

adherends can be expressed as

M 1( x) RM 2( x), (12)

with

R E 1 I 1

E 2 I 2. (13)

Moment equilibrium of the differential segment of theplated beam in Fig. 2 gives

M T( x) M 1( x) M 2( x) N ( x)( y1 y2t a). (14)

The bending moment in each adherend, expressed as

a function of the total applied moment and the interfacial

shear stress, is given as

M 1( x) R

( R+1) M T( x)b2

x

0

t ( x)( y1 y2t a) d x (15)

and

M 2( x)1

( R+1) M T( x)b2

x

0

t ( x)( y1 y2t a) d x. (16)

The first derivative of the bending moment in each

adherend gives

d M 1( x)

d xV 1( x)

R

( R+1)[V T( x)b2t ( x)( y1 y2t a)] (17)

and

d M 2( x)

d xV 2( x)

1

( R+1)[V T( x)b2t ( x)( y1 y2t a)]. (18)

Substituting Eqs. (8) and (9) into Eq. (7) and differen-

tiating the resulting equation once yields

d2t ( x)

d x2

Ga

t a y2

E 2 I 2

d M 2( x)

d x

1

E 2 A2

d N 2( x)

d x

y2

G2a A2

b2

ds ( x)

d x

y1

E 1 I 1

d M 1( x)

d x

1

E 1 A1

d N 1( x)

d x(19)

y1G1a A1

dqd x y1

G1a A1

b2ds ( x)d x .

Substitution of the shear forces [Eqs. (17) and (18)] and

axial forces (Eq. (11)) in both adherends into Eq. (19)

gives the following governing differential equation for

the interfacial shear stress

d2t ( x)

d x2

Gab2

t a( y1+ y2)( y1+ y2+t a)

E 1 I 1+ E 2 I 2

1

E 1 A1

1

E 2 A2

t ( x)

Ga

t a y1+ y2

E 1 I 1+ E 2 I 2V T( x)

Ga

t a

y1

G1a A1

dq

d x(20)

Gab2

a t a y1

G1 A1

y2

G2 A2ds ( x)d x

.

4. Interfacial normal stress: governing differential

equation

The governing differential equation for the interfacial

normal stress is derived in this section. When the beam

is loaded, vertical separation occurs between adherends1 and 2. This separation creates an interfacial normal

stress in the adhesive layer. The normal stress, s ( x), isgiven as

s ( x) E a

t a[v2( x)v1( x)], (21)

where v1( x) and v2( x) are the vertical displacements of

adherends 1 and 2, respectively. The equilibrium of

adherends 1 and 2, neglecting second-order terms, leads

to the following relationships.

Adherend 1:

d2v1( x)

d x2

1

E 1 I 1 M 1( x)

1

G1a A1

[qb2s ( x)], (22)

d M 1( x)

d xV 1( x)b2 y1t ( x) (23)

and

dV 1( x)

d xb2s ( x)q. (24)

Adherend 2:

d2v2( x)

d x2

1

E 2 I 2 M 2( x)

1

G2a A2

b2s ( x), (25)




d M 2( x)

d xV 2( x)b2 y2t ( x) (26)

and

dV 2( x)

d xb2s ( x). (27)

Based on the above equilibrium equations, the governing

differential equations for the deflections of adherends 1

and 2, expressed in terms of the interfacial shear and

normal stresses, are given as follows.

Adherend 1:

d4v1( x)

d x4

1

E 1 I 1b2s ( x)

1

G1a A1

b2

d2s ( x)

d x2

y1

E 1 I 1b2

dt ( x)

d x(28)

1

E 1 I 1q

1

G1a A1

d2q

d x2.

Adherend 2:

d4v2( x)

d x4

1

E 2 I 2b2s ( x)

1

G2a A2

b2

d2s ( x)

d x2(29)

y2

E 2 I 2b2

dt ( x)

d x.

Substitution of Eqs. (28) and (29) into the fourth deriva-

tive of the interfacial normal stress obtainable from Eq.

(21) gives the following governing differential equation

for the interfacial normal stress

d4s ( x)

d x4

E ab2

a t a 1

G1 A1

1

G2 A2

d2s ( x)

d x2

E ab2

t a 1

E 1 I 1

1

E 2 I 2s ( x)

E ab2

t a y1

E 1 I 1

y2

E 2 I 2dt ( x)

d x

E a

t a

1

E 1 I 1q (30)

E a

t a

1

G1a A1

d2q

d x2.

5. General solutions for the interfacial shear and

normal stresses

The governing differential equations for the interfacialshear and normal stresses [Eqs. (20) and (30)] are

coupled and hence a solution is not easily found. To

uncouple the equations, the effects of shear deformations

in both adherends are now neglected. The governing dif-

ferential equation for the interfacial shear stress then

reduces to

d2t ( x)

d x2

Gab2

t a( y1+ y2)( y1+ y2+t a)

E 1 I 1+ E 2 I 2

1

E 1 A1

1

E 2 A2

t ( x) (31)

Ga

t a y1+ y2

E 1 I 1+ E 2 I 2V T( x)0.

For simplicity, the general solutions presented below

are limited to loading which is either concentrated or

uniformly distributed over part or the whole span of the

beam, or both. For such loading, d 2V T( x)/d x2=0, and the

general solution to Eq. (31) is given by

t ( x)

B1 cosh( l x)

B2 sinh( l x)

m1V T( x), (32)

where

l2

Gab2

t a( y1+ y2)( y1+ y2+t a)

E 1 I 1+ E 2 I 2

1

E 1 A1

1

E 2 A2

(33)

and

m1Ga

t a

1

l2 y1+ y2

E 1 I 1+ E 2 I 2. (34)

The governing differential equation for the normalstress, with the effects of shear deformations neg-

lected, becomes

d4s ( x)

d x4

E ab2

t a 1

E 1 I 1

1

E 2 I 2s ( x)

E ab2

t a y1

E 1 I 1(35)

y2

E 2 I 2dt ( x)

d x

E a

t a

1

E 1 I 1q0.

The general solution to this fourth-order differential

equation is

s ( x)e− b x[C 1 cos( b x)C 2 sin( b x)] (36)

e b x[C 3 cos( b x)C 4 sin( b x)]n1dt ( x)

d xn2q.

For large values of x it is assumed that the normal stress

approaches zero, and as a result C 3=C 4=0. The generalsolution therefore becomes

s ( x)e− b x[C 1 cos( b x)C 2 sin( b x)]n1

dt ( x)

d xn2q, (37)

where

b4

E ab2

4t a 1

E 1 I 1+

1

E 2 I 2, (38)

n1 y1 E 2 I 2− y2 E 1 I 1

E 1 I 1+ E 2 I 2 (39)

and

n2 E 2 I 2

b2( E 1 I 1+ E 2 I 2). (40)

In deriving Eq. (37), it has been assumed that

d 5t /d x5=0, because d 5t /d x5 generally has negligible sig-

nificance to the final answer.




6. Application of boundary conditions

Having derived the general solutions for the interfacial

shear and normal stresses, three load cases are now con-

sidered. A simply supported beam is investigated whichis subjected to a uniformly distributed load, an arbitrarily

positioned single point load, and two symmetric pointloads as shown in Fig. 3. This section derives the

expressions of the interfacial shear and normal stresses

for each load case by applying suitable boundary con-ditions.

6.1. Interfacial shear stress for a uniformly distributed

load

By substituting the expression for the shear force ina simply supported beam subjected to a uniformly dis-

tributed load into Eq. (32), the general solution for the

interfacial shear stress for this load case can be found as

t ( x) B1 cosh( l x) B2 sinh( l x)m1q L2 xa, (41)

0 x LP

where q is the uniformly distributed load and x, a, L and

LP are defined in Fig. 1. The constants of integration

need to be determined by applying suitable boundary

conditions.

The first boundary condition is the applied bending

moment at x=0. Here, the moment at the plate end M 2(0)

Fig. 3. Load cases.

and the axial force of either the beam or the sof fit plate

[ N 1(0)= N 2(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 1(0) M T(0)qa

2

( La). (42)

Substituting Eqs. (8) and (9) with the effects of shearignored into Eq. (7) with the third term ignored, and

applying the above boundary condition, gives

dt ( x)

d x | x=0

m2 M T(0), (43)

where

m2Ga

t a

y1

E 1 I 1. (44)

By substituting Eq. (32) into Eq. (43), B2 can be determ-ined as

B2m2

l

qa

2( La)

m1

lq. (45)

The second boundary condition requires zero interfa-

cial shear stress at mid-span due to symmetry of the

applied load. B1 can therefore be determined as

B1m2

l

qa

2( La) tanh l LP

2m1

lq tanh l LP

2. (46)

For practical cases l LP /210 and as a resulttanh( l LP /2)1, so the expression for B1 can be simpli-

fied to

B1m2

l

qa

2( La)

m1

lq B2. (47)

Substitution of B1 and B2 into Eq. (41) gives anexpression for the interfacial shear stress at any point

t ( x)m2a

2( La)m1q e− l x

lm1q L

2a x. (48)

6.2. Interfacial shear stress for a single point load

Two scenarios are considered and they are: (1) the left

end of the plate is terminated to the left of the point load(ab) and (2) the left end of the plate is terminated to

the right of the point load (ab). By substituting the

specific expressions for the shear force in a simply sup-ported beam subjected to a point load into Eq. (32), the

general solution for the interfacial shear stress for this

load case can be found as




ab: t ( x) (49)

B3 cosh( l x)+ B4 sinh( l x)+m1P1−b

L, 0 x(b−a)

B5 cosh( l x)+ B6 sinh( l x)−m1Pb

L, (b−a) x LP

ab: t ( x) B7 cosh( l x) B8 sinh( l x) (50)

m1Pb

L, 0 x LP

The constants can be evaluated using boundary con-

ditions.

For ab the boundary conditions are

at x=0, M 1(0)= M T(0)=Pa1−b

L

at x= LP, M 1( LP)= M T( LP)=Pab

L

at x=(b−a), t ( x) is continuous, i.e., t 1( x) x=b−a=t 2( x) x=b−a

at x=(b−a),dt ( x)

d xis continuous, i.e.,

dt 1( x)

d x | x=b−a

=dt 2( x)

d x | x=b−a

(51)

The first two boundary conditions are applied in the

same manner as the first boundary condition of the uni-

formly distributed load case. The third and fourth bound-

ary conditions require continuity of the interfacial shear

stress and its first derivative under the point load.

For ab the boundary conditions are given as follows

and can be applied in the same manner as the first bound-ary condition for the uniformly distributed loading case

at x=0, M 1(0)= M T(0)=Pb1−a

L

at x= LP, M 1( LP)= M T( LP)=Pab

L

(52)

In all cases, both the axial force and the moment at

the ends of the sof fit plate are zero, so the beam resists

the total applied bending moment here [Eq. (42)]. Oncethe boundary conditions are applied, the constants of

integration can be found as follows

B3m2

lPa1

b

Lm1P e−k , B4

m2

lPa1

b

L,

B5m2

lPa1

b

Lm1P sinh(k ), (53)

B6m2

lPa1

b

Lm1P sinh(k ),

B7m2

lPb1

a

L, B8

m2

lPb1

a

L,

where k = l(ba). Substitution of the constants of inte-

gration into Eqs. (49) and (50) gives

ab: t ( x) (54)

m2

lPa1−

b

L e− l x+m1P1−

b

L−m1P cosh( l x) e−k , 0 x(b−a)

m2

lPa1−

b

L e− l x−m1

Pb

L+m1P sinh(k ) e− l x, (b−a) x LP

ab: t ( x)m2

lPb1

a

L e− l xm1P

b

L, 0 x LP (55)

The two existing solutions [2,14] derived also using

the deformation compatibility approach for interfacial

shear stresses for a point load case can be found by elim-

inating certain terms from the first part of Eq. (54), as

these two solutions cover only the region 0 x(ba).

Part 1 of Eq. (54) differs from that of Taljsten [14] in

two aspects: the inclusion of the bending deformationsin adherend 2 and a different implementation of the

boundary/continuity condition under the point load

which leads to the additional term in the present solutionm1P cosh( l x) ek . At the point load, Taljsten as well as

Vilnay [2] specified zero interfacial shear stress while

the present solution specifies continuity of interfacialshear stress and its first derivative. At the plate end the

effect of this extra term is generally small in comparison

with the other terms but it does lead to a more realistic

interfacial shear stress distribution under the point load.

If these two differences together with the effect of theaxial deformation of adherend 1 are eliminated from the

present solution and if b is set equal to LP /2 to represent

a point load at mid-span, the present solution reduces to

that of Vilnay.

6.3. Interfacial shear stress for two point loads

The two point loads are symmetrically positioned.

Two cases are considered: (1) the plate extends beyondthe constant moment region (ab) and (2) the plate is

terminated within the constant moment region (ab).

Using Eq. (32), the general solution for the interfacial

shear stress for this load case is given by

ab: t ( x) (56)

B9 cosh( l x)+ B10 sinh( l x)+m1P, 0 x(b−a)

B11 cosh( l x)+ B12 sinh( l x), (b−a) x LP

2

ab: t ( x) B13 cosh( l x) B14 sinh( l x), 0 x (57)

LP




For ab the boundary conditions are

at x=0, M 1(0)= M T(0)=Pa

at x= LP

2, t LP

2=0

at x=(b−a), t ( x) is continuous, i.e., t 1( x) x=b−a=t 2( x) x=b−a

at x=(b−a),dt ( x)

d xis continuous, i.e.,

dt 1( x)

d x | x=b−a

=dt 2( x)

d x | x=b−a

(58)

The first two boundary conditions are applied in the

same manner as those for the uniformly distributed load

case. The third and fourth boundary conditions requirecontinuity of the interfacial shear stress and its first

derivative under the point loads and are applied in the

same manner as for the single point load case.

For ab the boundary conditions are as follows:

at x=0, M 1(0)= M T(0)=Pb

at x= LP, M 1( LP)= M T( LP)=Pb(59)

The constants are consequently determined to be

B9m2

lPam1P e−k , B10

m2

lPa, B11

m2

lPa

m1P sinh(k ), B12m2

lPam1P sinh(k ), (60)

B13m2

lPb, B14

m2

lPb.

Substitution of these expressions into Eqs. (56) and(57) gives

ab: t ( x) (61)

m2

lPa e− l x+m1P−m1P cosh( l x) e−k , 0 x(b−a)

m2

lPa e− l x+m1P sinh(k ) e− l x, (b−a) x

LP

2

ab: t ( x)m2

lPb e− l x, 0 x LP (62)

6.4. Interfacial normal stress: general expression for

all three load cases

The constants C 1 and C 2 in Eq. (37) are determined byconsidering appropriate boundary conditions. The first

boundary condition is the zero bending moment at the

ends of the sof fit plate. Differentiating Eq. (21) twiceand substituting Eqs. (22) and (25) into the resulting

expression yields the following relationship at the plate

end (neglecting the effects of shear deformations):

d2s ( x)

d x2 | x=0

E a

t a 1

E 1 I 1 M 1(0)

1

E 2 I 2 M 2(0). (63)

Since it has been established that M 2(0)=0,

N 1(0)= N 2(0)=0 and M 1(0)= M T(0) at the end of the sof fitplate, the above relationship can be expressed as

d2s ( x)

d x2 | x=0

E a

t a

1

E 1 I 1 M T (0). (64)

Boundary condition 2 concerns the shear force at the

end of the sof fit plate in the beam and the sof fit plate.Differentiating Eq. (21) three times and substituting Eqs.

(23) and (26) into the resulting expression lead to the

following relationship at the plate end:

d3s ( x)

d x3 | x=0

E a

t a 1

E 1 I 1V 1(0)

1

E 2 I 2V 2(0) E ab2

t a y1

E 1 I 1(65)

y2

E 2 I 2t (0).

As the applied shear force at the end of the plate is

zero [i.e., V 2(0)=0], V 1(0)=V T(0). The second boundary

condition can therefore be expressed as

d3s ( x)

d x3 | x=0

E a

t a

1

E 1 I 1V T(0)n3t (0), (66)

where

n3 E ab2

t a y1

E 1 I 1

y2

E 2 I 2. (67)

Further differentiation of Eq. (37) leads to the follow-

ing expressions for the second and third derivatives of

the interfacial normal stress at the plate end

d2s ( x)

d x2 | x=0

2 b2C 2n1

d3t ( x)

d x3 | x=0

n2

d2q

d x2(68)

and

d3s ( x)

d x3 | x=0

2 b3C 12 b3C 2n1

d4t ( x)

d x4 | x=0

n2

d3q

d x3. (69)

Since the loading is limited to either uniformly distrib-uted or point loads, the second- and higher-order deriva-tives of q become zero. Substituting the boundary con-

ditions into the above two equations then leads to thedetermination of C 1 and C 2 as follows:

C 1 E a

2 b3t a

1

E 1 I 1[V T(0) b M T(0)]

n3

2 b3t (0) (70)

n1

2 b3d4t ( x)

d x4 | x=0

bd3t ( x)

d x3 | x=0

and




C 2 E a

2 b2t a

1

E 1 I 1 M T(0)

n1

2 b2

d3t ( x)

d x3 | x=0

. (71)

The above expressions for the constants C 1 and C 2 have

been left in terms of the bending moment M T(0) and

shear force V T(0) at the end of the sof fit plate. With the

constants C 1 and C 2 determined, the interfacial normalstress can then be found using Eq. (37) for all three

load cases.

7. Comparison of analytical solutions

A comparison of the interfacial shear and normal

stresses from the different closed-form solutions

reviewed earlier is undertaken in this section. Two

example problems are considered. The first is an RC

beam bonded with a glass-fibre-reinforced plastic

(GFRP), CFRP or steel sof fit plate. The second is a hol-low aluminium (AL) beam with a bonded CFRP sof fitplate. In both examples, the beams are simply supported

and subjected to a central point load or a uniformly dis-

tributed load. A summary of the geometric and material

properties is given in Table 2. The span of the RC beam

is 3000 mm, the distance from the support to the end of

the plate is 300 mm, the mid-point load is 150 kN andthe UDL is 50 kN/m. The span of the aluminium beam

is 500 mm, the distance from the support to the plateend is 50 mm, the mid-point load is 1 kN and the UDL

is 2 kN/m. The geometric properties of the aluminium

beam are taken from a study undertaken by Broughton

et al. [22] although the distance from the support to thesof fit plate has been changed from 20 to 50 mm. The

same elastic modulus for the AL beam has been used

and similar properties have been used for the adhesive

layer and CFRP sof fit plate.

7.1. RC beam with GFRP, CFRP and steel soffit

plate: mid-point load and UDL

Fig. 4 plots the interfacial shear and normal stresses

near the plate end for the example RC beam bonded with

a CFRP plate for the mid-point load case, while Fig. 5

Table 2

Geometric and material properties

Component Width (mm) Depth (mm) Young’s modulus (MPa) Poisson’s ratio

RC beam b1=200 d 1=300 E 1=30,000 –

AL beam (wall thickness=2 mm) b1=20 d 1=30 E 1=65,300 –

Adhesive layer (RC beam) ba=200 t a=2.0 E a=2000 na=0.35

Adhesive layer (AL beam) ba=20 t a=2.0 E a=2000 na=0.35

GFRP plate (bonded to RC beam) b2=200 t 2=4.0 E 2=50,000 –

CFRP plate (bonded to RC beam) b2=200 t 2=4.0 E 2=100,000 –

Steel plate (bonded to RC beam) b2=200 t 2=4.0 E 2=200,000 –

CFRP plate (bonded to AL beam) b2=20 t 2=2.0 E 2=100,000 –

Fig. 4. Comparison of interfacial shear and normal stresses for an RC

beam with a bonded CFRP sof fit plate subjected to a mid-point load.

Fig. 5. Comparison of interfacial shear and normal stresses for an

RC beam with a bonded CFRP sof fit plate subjected to a UDL.

is a similar plot for the UDL case. Overall, the predic-

tions of the different solutions agree closely with each

other. This indicates that the bending deformations in

the beam and the axial deformations in the bonded plateare the dominant actions determining the interfacial

stresses, including the peak values occurring at the plate

end, and the omission or inclusion of other terms such

as the bending deformations in the bonded plate has only

a very small effect. The results in Fig. 5 show that in

Roberts and Haji-Kazemi’s solution [4], the second stage




Table 3

Comparison of peak interfacial shear and normal stresses: mid-point load

Theory RC beam with GFRP plate RC beam with CFRP plate RC beam with steel plate AL beam with CFRP plate

t (MPa) s (MPa) t (MPa) s (MPa) t (MPa) s (MPa) t (MPa) s (MPa)

Vilnay 2.240 1.381 3.152 1.669 4.350 1.967 2.440 1.109Roberts 2.179 1.553 2.925 1.761 3.725 1.899 1.788 1.041

Taljsten 2.215 1.397 3.087 1.674 4.188 1.944 2.172 1.122

Malek et al. 2.179 1.553 2.925 1.761 3.725 1.899 1.787 1.040

Present 2.214 1.396 3.083 1.671 4.176 1.938 2.091 1.081

interfacial shear stress and the first stage interfacial nor-

mal stress are both small.

The interfacial normal stress is seen to change sign ata short distance away from the plate end. In the region

of negative interfacial normal stress, the predictions of

those solutions which take into account the additional

bending deformations in the plate due to interfacial shearstresses are similar and differ from the predictions of

those ignoring these deformations. The former includesVilnay’s [2] and Taljsten’s [14] solutions for the mid-

point load case, Roberts and Haji-Kazemi’s solution [4]

for the UDL case and the present solution for both load

cases, while the latter includes Roberts’ [3] and Malek

et al.’s [16] solutions for both load cases, as detailed inTable 1. Roberts and Haji-Kazemi’s solution ignoring

the contribution from the first stage predicts interfacial

normal stresses similar to those from Roberts’ and Malek

et al.’s solutions, indicating that the first stage contri-

bution is equivalent to the inclusion of the additional

bending deformations in the bonded plate due to interfa-cial shear stresses.

The results of the peak interfacial shear and normal

stresses (at the end of the sof fit plate) are given in Tables

3 and 4 for the example RC beam with a GFRP, CFRP

or steel sof fit plate. Table 3 shows that, for the mid-point

load case, the predictions of Roberts’ [3] and Malek et

al.’s [16] solutions are identical, while the present and

Taljsten’s [14] solutions give very similar results. Vil-

nay’s solution [2] leads to results closer to those of thelatter two solutions than those of the former two. For

Table 4Comparison of peak interfacial shear and normal stresses: UDL

Theory RC beam with GFRP RC beam with CFRP RC beam with steel AL beam with CFRP

plate plate plate plate

t (MPa) s (MPa) t (MPa) s (MPa) t (MPa) s (MPa) t (MPa) s (MPa)

Roberts and Haji-Kazemi (t — Stage 1; 2.001 1.425 2.776 1.668 3.745 1.902 2.079 1.153

s — Stage 2)

Roberts and Haji-Kazemi (Stage 1+2) 1.813 1.256 2.591 1.500 3.567 1.733 1.962 1.060

Roberts 1.945 1.386 2.604 1.567 3.302 1.683 1.552 0.910

Malek et al. 1.943 1.384 2.597 1.563 3.287 1.675 1.499 0.871

Present 1.975 1.244 2.740 1.484 3.696 1.713 1.796 0.930

the UDL case, the present solution gives results which

generally agree better with those from Roberts and Haji-

Kazemi’s solution [4] than with those from Roberts’ andMalek et al.’s solutions. The latter two again give simi-

lar results.

Relating the above observations to the different

assumptions adopted in these solutions (Table 1), theeffects of various terms can be clarified. In short, it may

be concluded that all solutions are satisfactory for RCbeams bonded with a thin plate as the rigidity of the

sof fit plate is small in comparison with the that of the

RC beam. Those solutions which consider the additional

bending deformations in the sof fit plate due to the

interfacial shear stresses give more accurate results. Thepresent solution is the only solution which covers both

point loads and uniformly distributed loads and con-

siders this effect and the effects of other parameters.

7.2. Hollow aluminium beam with CFRP sof fit plate:

mid-point load

Figs. 6 and 7 plot the interfacial shear and normal

stress distributions in the example aluminium beam

bonded with a CFRP plate subject to a mid-point load.

For this problem, the rigidity of the sof fit plate is of

much greater significance than in the RC beam problem.

The effects of bending deformations in the plate and

axial deformations in the beam are therefore expected to

become significant.For the interfacial shear stress, the difference between




Fig. 6. Comparison of interfacial shear stresses for a hollow alu-

minium beam bonded with a CFRP sof fit plate subjected to a mid-

point load.

Fig. 7. Comparison of interfacial normal stresses for a hollow alu-minium beam with a bonded CFRP sof fit plate subjected to a mid-

point load.

the results of Vilnay’s [2] and Taljsten’s [14] solutions

seen in Fig. 6 is due to Taljsten considering axial defor-

mations in the beam. The current solution considers one

extra term to Taljsten — the bending deformations of

the sof fit plate — and this accounts for the small differ-

ence in the results (Fig. 6). The solutions of Roberts [3]

and Malek et al. [16] give lower peak values but higherresults away from the plate end compared with the

present accurate solution (Fig. 6). This is due to the fact

that in these two solutions, the axial deformations in the

beam and the bending deformations in the plate are only

partially considered through the use of the transformed

section properties. These two solutions give very simi-lar results.

The interfacial shear stress at the mid-thickness point

of the adhesive layer, based on fully composite action

between the beam and the plate, is also plotted in Fig.

6. At a suitable distance from the end of the sof fit plate(approximately 60 mm), the predictions of all solutions

converge to the results obtained from this simple beam

theory. The interfacial shear stress calculated from the

current solution tends to zero under the point load. This

is due to the continuity conditions enforced under a point

load. The solutions of Vilnay [2], Taljsten [14] and

Malek et al. [16] employ various simplifications such as

rounding cosh and tanh terms in their derivation. As aresult of such simplifications, the interfacial shear stress

does not tend to zero at the point load even though zerointerfacial shear stress beneath a point load is a boundary

condition in these solutions.

The normal stress results from the solutions of Roberts[3] and Malek et al. [16] are very similar to each other

but differ from those from the solutions of Vilnay [2]

and Taljsten [14] and the present solution (Fig. 7). The

difference is mainly in the region of negative interfacial

normal stress and is due to the consideration of the plate

bending deformations induced by interfacial shearstresses in the latter three solutions.

The peak values of the interfacial stresses are given

in Table 3. Significant differences exist between the peak

values of interfacial shear and normal stresses. For the

interfacial shear stress, the largest value, which is from

Vilnay’s solution [2], is 1.37 times the lowest value

which is from Malek et al.’s solution [16]. The corre-

sponding difference for the interfacial normal stress is

smaller. The present solution, being the most accuratesolution, gives results similar to those of Taljsten’s sol-

ution [14].

7.3. Hollow aluminium beam with CFRP sof fit plate:

UDL

Figs. 8 and 9 plot the interfacial shear and normalstresses for the example aluminium beam bonded witha CFRP plate subjected to a UDL. The results from the

solutions of Roberts [3] and Malek et al. [16], for both

the interfacial shear and normal stresses, are very simi-

lar.

The reason for the difference between Roberts and

Haji-Kazemi’s solution [4] and the present solution for

Fig. 8. Comparison of interfacial shear stresses for a hollow alu-

minium beam with a bonded CFRP sof fit plate subjected to a UDL.




Fig. 9. Comparison of interfacial normal stresses for a hollow alu-

minium beam with a bonded CFRP sof fit plate subjected to a UDL.

the interfacial shear stress is believed to be due to small

differences in secondary assumptions. One of these dif-

ferences is the assumption in Roberts and Haji-Kazemi

that the applied bending moment is resisted by both the

beam and the plate without shear interaction [i.e.,

( EI )total= E 1 I 1+ E 2 I 2] in determining the curvature of the

beam at the plate end. In the present solution, the applied

moment here is assumed to be taken up by the beamonly. The difference in the interfacial normal stress is

expected to be due mainly to the difference in bending

deformations induced by the interfacial shear stress.

Even though Roberts and Haji-Kazemi and the present

solution both consider the effect of the interfacial shear

stress on the interfacial normal stress, the shear stresses

are different near the plate end and this accounts for thedifference in the interfacial normal stress near the plateend. The interfacial shear stress along the mid-thickness

line of the adhesive layer, based on a simple fully com-

posite beam analysis, is plotted and a good correlation

is evident with all solutions at a suitable distance from

the end of the plate.

The peak values of the interfacial shear and normal

stresses are given in Table 4. Significant differences

exist between the various solutions, particularly for the

interfacial shear stress. The highest value for the peak interfacial stress, which is given by Roberts and Haji-

Kazemi’s solution [4], is 1.31 times the lowest value

which is from Malek et al.’s solution [16]. For the peak

interfacial normal stress, the corresponding number is

1.22. The results of the present solution differ from those

of Roberts and Haji-Kazemi by about 10% for bothstresses.

8. Conclusions

This paper has been concerned with the prediction of

interfacial shear and normal stresses in beams strength-

ened by an externally bonded plate. Such interfacial

stresses provide the basis for understanding debonding

failures in such beams and for the development of suit-

able design rules. Six existing approximate closed-form

solutions for interfacial stresses in such beams have been

reviewed, identifying their assumptions and limitations,

thereby clarifying the differences between these sol-

utions for the first time. All of these solutions are basedon the assumption that the interfacial stresses do not vary

across the adhesive layer thickness, a key assumption

which enabled relatively simple explicit expressions for

interracial stresses to be found. A new similar but more

accurate solution was then presented. Numerical com-

parisons between the existing solutions and the present

new solution have also been carried out. The following

conclusions can be drawn from the present study.

1. All of the existing solutions include the bending

deformations in the beam and the axial deformationsin the bonded plate. These two actions dominate the

interfacial stresses in RC beams with a bonded plate.

The differences between the existing solutions are due

to the different choices in the inclusion of other terms,

of which the additional bending deformations in the

bonded plate due to interfacial shear stresses is the

most significant for RC beams.

2. The new solution includes the effects of all defor-

mation terms except shear deformations whose

inclusion would complicate the solution. Neverthe-

less, governing equations considering the effects of

shear deformations have been derived. The new sol-

ution can be applied to beams made of all kinds of

materials bonded with a plate where the rigidity of

the beam and the plate are more comparable.

3. For plated beams with a relatively stiff plate, signifi-

cantly different results have been obtained from dif-

ferent solutions. Two existing solutions give predic-

tions relatively close to those from the present

solution: Taljsten’s solution [14], which is limited to

a single point load, and Roberts and Haji-Kazemi’s

solution [4], which is for a uniformly distributed load

and significantly more complicated than the present

solution. The solutions of Roberts [3] and Malek et

al. [16], which are general in terms of loading, arethe more approximate among the available solutions,

and they give results almost identical to each other.

The present solution covers all three common load

cases (single point load, double point load and UDL),

and is based on the more direct and simpler approach

of deformation compatibility rather than the staged

approach of Roberts and Haji-Kazemi. The present

solution is thus recommended as a widely applicable

and accurate solution with due simplicity for appli-

cation to beams bonded with a sof fit plate, particularly

when the plate is relatively stiff.




Acknowledgements

Both authors wish to thank The Hong Kong Polytech-

nic University for the provision of a postdoctoral fellow-

ship to the first author.

References

[1] Jones R, Swamy RN, Charif A. Plate separation and anchorage

of reinforced concrete beams strengthened by epoxy-bonded steel

plates. The Struct Engr 1988;66(5):85–94.

[2] Vilnay O. The analysis of reinforced concrete beams strengthened

by epoxy bonded steel plates. Int J Cement Compos Lightweight

Concrete 1988;10(2):73–8.

[3] Roberts TM. Approximate analysis of shear and normal stress

concentrations in the adhesive layer of plated RC beams. The

Struct Engr 1989;67(12):229–33.

[4] Roberts TM, Haji-Kazemi H. Theoretical study of the behaviour

of reinforced concrete beams strengthened by externally bonded

steel plates. Proc Instn Civil Engrs 1989;87(Part 2):39–55.

[5] Oehlers DJ, Moran JP. Premature failure of externally platedreinforced concrete beams. J Struct Eng, ASCE

1990;116(4):979–95.

[6] Oehlers DJ. Reinforced concrete beams with plates glued to their

sof fits. J Struct Eng, ASCE 1992;118(8):2023–38.

[7] Sharif A, Al-Sulaimani GJ, Basunbul IA, Baluch MH, Ghaleb

BN. Strengthening of initially loaded reinforced concrete beams

using FRP plates. ACI Struct J 1994;91(2):160–8.

[8] Chajes MJ, Thomson TA Jr., Januszka TF, Finch WW Jr.. Flex-

ural strengthening of concrete beams using externally bonded

composite materials. Construct Bldg Mater 1994;8(3):191–201.

[9] Liu Z, Zhu B. Analytical solutions for R/C beams strengthened

by externally bonded steel plates. J Tongji Univ 1994;22(1):21–

6 (in Chinese).

[10] Quantrill RJ, Hollaway LC, Thorne AM. Experimental and ana-

lytical investigations of FRP-strengthened beam response: Part I.

Mag Concrete Res 1996;48(177):331–42.

[11] Quantrill RJ, Hollaway LC, Thorne AM. Predictions of the

maximum plate end stresses of FRP-strengthened beams: Part II.

Mag Concrete Res 1996;48(177):343–51.

[12] Arduini M, Nanni A. Parametric study of beams with externally

bonded FRP reinforcement. ACI Struct J 1997;94(5):493–501.[13] Raoof M, Zhang S. An insight into the structural behaviour of

reinforced concrete beams with externally bonded plates. Proc

Instn Civil Engrs — Struct Bldgs 1997;122:477–92.

[14] Taljsten B. Strengthening of beams by plate bonding. J Mater

Civil Eng, ASCE 1997;9(4):206–12.

[15] Buyukozturk O, Hearing B. Failure behaviour of pre-cracked con-

crete beams retrofitted with FRP. J Compos Construct, ASCE

1998;2(3):138–44.

[16] Malek AM, Saadatmanesh H, Ehsani MR. Prediction of failure

load of R/C beams strengthened with FRP plate due to stress

concentration at the plate end. ACI Struct J 1998;95(1):142–52.

[17] Saadatmanesh H, Malek AM. Design guidelines for flexural

strengthening of RC beams with FRP plates. J Compos Construct,

ASCE 1998;2(4):158–64.

[18] Swamy RN, Mukhopadhyaya P. Debonding of carbon-fibre-reinforced polymer plate from concrete beams. Proc Instn Civil

Eng — Struct Bldgs 1999;134:301–17.

[19] Raoof M, El-Rimawi JA, Hassanen MAH. Theoretical and

experimental study on externally plated RC beams. Eng Struct

2000;22(1):85–101.

[20] Rabinovich O, Frostig Y. Closed-form high-order analysis of RC

beams strengthened with FRP strips. J Compos Construct, ASCE

2000;4(2):65–74.

[21] Shen HS, Teng JG, Yang J. Interfacial stresses in beams and slabs

bonded with a thin plate. J Eng Mech, ASCE, (in press).

[22] Broughton JG, Beevers A, Hutchinson AR. Carbon-fibre-

reinforced plastic (CFRP) strengthening of aluminium extrusions.

Int J Adhes Adhes 1997;17:269–78.

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Inter Facial Stresses in Plated Beams