Analytical Mechanics Solution for Measuring the Deflection of Strengthened RC Beams Using FRP Plates

Accepted Manuscript Title: Analytical Mechanics Solution for Measuring the Deflection of Strengthened RC Beams Using FRP Plates Authors: Mohamed El-Zeadani, Raizal Saifulnaz M.R., Y.H. Mugahed Amran, F. Hejazi, M.S. Jaafar, Rayed Alyousef, Hisham Alabduljabbar PII: DOI: Article Number:

S2214-5095(19)30252-9 https://doi.org/10.1016/j.cscm.2019.e00272 e00272

Reference:

CSCM 272

To appear in: Received date: Revised date: Accepted date:

26 May 2019 9 July 2019 19 July 2019

Please cite this article as: El-Zeadani M, M.R. RS, Mugahed Amran YH, Hejazi F, Jaafar MS, Alyousef R, Alabduljabbar H, Analytical Mechanics Solution for Measuring the Deflection of Strengthened RC Beams Using FRP Plates, Case Studies in Construction Materials (2019), https://doi.org/10.1016/j.cscm.2019.e00272 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Analytical Mechanics Solution for Measuring the Deflection of Strengthened RC Beams Using FRP Plates Mohamed El-Zeadania,* [email protected], Raizal Saifulnaz M. R.a, Y. H. Mugahed Amranb, c, F. Hejazia, M. S. Jaafara, Rayed Alyousefb, Hisham Alabduljabbarb

Department of Civil Engineering, Universiti Putra Malaysia, 43400 Serdang, Malaysia

b

Department of Civil Engineering, Prince Sattam Bin Abdulaziz University, 11942 Alkharj, Saudi Arabia.

c

Department of Civil Engineering, Faculty of Engineering, Amran University, 9677 Quhal, Amran, Yemen.

Corresponding author, Tel: +60 18 355 9042, E-mail: [email protected] (M. El-Zeadani)

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Abstract Partial-interaction due to sliding between the steel bars, adhesively attached FRP plates and their bordering concrete surface, accompanied with the detachment of the FRP plates due to intermediate crack (IC) debonding make the deflection of strengthened RC beams difficult to anticipate. Previous research and design rules on determining the deflection of strengthened RC beams using FRP plates have opted for a full-interaction moment-curvature design

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technique where the deflection was measured by either deriving average effective moment of inertia and using elastic deflection equations or integrating the curvature along the beam’s length. Therefore, IC deboning of the plate

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and the slip resulting from the formation and broadening of new cracks were not directly considered. In this study, a partial-interaction moment-rotation analysis of an adhesively plated beam segment was used to derive analytical

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equations for the rotation of individual crack faces. The analytical expressions were used to compute the rotation at a crack for a given moment; subsequently, the influence of each crack to the midspan deflection of the RC beams was

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calculated. As for the uncracked region of the beam, the deflection contribution was measured by integrating the curvature over the uncracked span. The deflection results from the mechanics solution seem to compare well with

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experimental results. The analytical mechanics solution accounts for the partial-interaction between the steel bars, externally bonded FRP plate and their bordering concrete surface, and also the detachment of the external plate through IC debonding. Further, due to its generic nature and non-reliance on empirical data, the mechanics solution

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can be adopted to forecast the deflection of strengthened RC beams with novel types of reinforcement materials. Keywords: FRP strengthened beams; intermediate crack debonding; partial-interaction; moment-rotation; analytical solutions, serviceability deflection

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1.0 Introduction

Strengthening of reinforced concrete (RC) structures using adhesively attached fibre reinforced polymer (FRP) plates has gained a lot of popularity over the past two decades. This comes down to the inherent material properties of the FRP (e.g. resistance to corrosion, high strength) and ease of construction. Debonding of the externally bonded (EB) plate can be attributable to: (a) plate end (PE) debonding which starts at the end of the plate and spreads towards the center; (b) critical diagonal crack (CDC) debonding that happens when a main diagonal shear crack intersects the FRP plate close to the end and spreads from that point to the plate end; and (c) intermediate crack (IC)

2

debonding which usually originates at the maximum moment region and spreads outward towards the FRP plate end. PE debonding relies on the interface shear and normal stress between the FRP plate and bordering concrete [1 – 5]. CDC debonding depends on the shear load that is needed to form a diagonal crack and the shear capacity that the diagonal crack can resist [6, 7]. Meanwhile, IC debonding relies largely on the interface shear between the FRP plate and bordering concrete [8, 9]. It is well recognized that for RC beams with thin plates, IC debonding usually controls the analysis [10] and can be considered as the major debonding mechanism.

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Consider the half span of the symmetrically loaded FRP strengthened beam shown in Figure 1(a) where IC debonding has commenced at the point of maximum moment. Initially, the behavior of the FRP plate can be

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represented by the prism shown in Figure 1(b); that is, as the moment M on the beam intensifies, the force in the FRP plate rises causing the bond force B to build up as shown. Once the force in the FRP plate reaches the IC

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debonding force, PIC, the commensurate bond force becomes B IC and the slip at the crack becomes 𝛿max. The force PIC induces a strain in the FRP plate, 𝜀IC, known as the IC debonding strain. In the event that the plate is pulled

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further due to an increase in moment M, the bond force stays at BIC and the plate’s force remains at PIC as illustrated in Figure 1(c). Henceforth, the stretching of the plate is accommodated by debonding along distance L db-p due to 𝜀IC

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and the total extension of the plate becomes 𝛿max + 𝜀ICLdb-p. This trend holds until the total plate length has detached from the concrete surface as depicted in Figure 1(d), where the total extension of the plate is 𝛿max + 𝜀ICLdb-p-max. Any further elongation of the FRP plate past that depicted in Figure 1(d) will induce unstable detachment of the plate that

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is rapid and destructive in nature [11].

3

M (a) crack Lplate

𝛿max

(b)

PIC BIC 𝛿max + 𝜀IC Ldb-p

(c)

PIC Ldb-p

BIC

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𝛿max + 𝜀IC Ldb-p-max

(d)

ro

PIC Ldb-p-max

-p

BIC

Figure 1: Intermediate Crack Debonding of an Adhesively Plated RC Beam

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Further, when a flexural crack intercepts the reinforcement steel, slip occurs between the steel rebars and the adjoining concrete [12 – 15] such that the strain in the reinforcement steel is not the same as the strain in the

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adjacent concrete and this is known as partial-interaction [10]. The same could be said about the EB plate, where a crack intercepting the plate could induce sliding between the plate and the bordering concrete surface. These cracks

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continue to widen through slip as straining of the reinforcement alone does not cause the cracks to widen [16 – 19]. The formation of cracks in RC beams and the increase in their rotation by widening causes an increase in the midspan deflection [20]. This phenomenon of crack formation and widening due to slip, together with the plate’s IC debonding mechanism, make the deflection of FRP plated beams difficult to predict. Past research [21 – 24] and design guidelines [25, 26] on estimating the deflection of FRP strengthened RC beams have adopted a full-

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interaction design technique where neither the slip between the steel rebars and adjoining concrete, nor the slip concerning the EB plate and the bordering concrete were accounted for. These full-interaction moment-curvature approaches for measuring the service load deflection of RC beams may be divided into two categories: (a) the estimation of an effective moment of inertia (Ieff) for the entire beam which is employed with elastic deflection equations to evaluate the mid-span deflection of the beam [27 – 31]; and (b) the integration of curvature along the length of the beam [32 – 34]. In all of these approaches, design equations were adjusted with experimental findings,

4

and they can only be applied to similar RC structures from which they were derived. Therefore, measuring the deflection of RC members with novel types of reinforcement materials using the above approaches would be inappropriate. To measure the mid-span deflection induced by the formation of flexural cracks and slip between the reinforcement and adjoining concrete, researchers have idealized the tensile region of the beam as a concentrically loaded prism [35, 36]. Closed-form solutions for the load-slip relationship were derived together with expressions for the crack

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spacing and the load in the reinforcement bars to prompt additional cracks in the concrete prism [37 – 40]. After that, partial-interaction mechanics-centered methods for evaluating the deflection of unplated RC beams were

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introduced [41 – 44]. In these mechanics-centered methods, the partial-interaction behavior between the steel rebars and adjoining concrete was accounted for and the deflection contribution due to the formation and widening of

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cracks was considered as well. However, these partial-interaction design approaches only considered the deflection of unstrengthened beams; henceforth, not integrating the intermediate crack detachment of the adhesively bonded

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FRP plate in the analysis, and not directly considering the plate slip resulting from the formation of flexural cracks. Moreover, several researchers have presented partial-interaction mechanics methods for calculating the ultimate

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capacity of FRP strengthened beams using a segmental moment-rotation analysis [11, 45]. However, these mechanics-based design methods for FRP plated beams were not used directly to compute the deflection of the FRP plated beams or quantify the service behavior. Further, due to their iterative nature where the neutral axis depth had

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to be guessed first and then the section rotated until force equilibrium was achieved, the approaches presented by previous researchers were both laborious and time consuming. As an alternative, analytical expressions for the moment-rotation relationship can be derived and used directly in a discrete rotation approach to determine the deflection due to crack formation and widening of an FRP plated beam.

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Therefore, the aim of this research is to introduce an analytical mechanics solution for FRP strengthened RC beams to measure the serviceability deflection. This is achieved by first deriving an analytical solution for the rotation of individual cracks faces, 𝜃, obtained from a partial-interaction segmental moment-rotation analysis. After that, a discrete rotation approach is employed to determine the deflection due to each crack and also the deflection due to the uncracked part of the FRP strengthened beam. The load-deflection results attained from the analytical mechanics solution are then compared with code methods and experimental test findings for verification purposes. It is

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important to note that only the instantaneous deflection is accounted for in the analytical mechanics solution presented here; however, the approach can be utilized to predict the deflection under long-term loads as well by modifying the tension stiffening mechanism as presented by Visintin et al. [46]; Knight et al. [47] and Visintin et al. [48]. 2.0 Segmental Moment-rotation Analysis Consider the FRP strengthened beam segment of length Spr = 2Ldef displayed in Figure 2(a). The segment is

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subjected to a constant moment M which induces an Euler-Bernoulli rotation of sides A-A to B-B by an angle 𝜃 such that plane sections remain plane. Prior to cracking, full-interaction exists such that the strains in the tensile

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reinforcement and adjacent concrete are the same. However, after the formation of cracks, the EB plate and steel rebars slip against the neighboring concrete forcing the strains to be different from that in the neighboring concrete.

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Further, when the slip in the plate, 𝛥p, reaches the maximum debonding slip, 𝛿max, IC debonding commences. The plate in the debonded region acts as a passively prestressed tendon inducing a force at the level of the plate equal to

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PIC. Therefore, the segmental moment-rotation analysis may be practically separated into: (a) uncracked segmental

Ldef

Ldef A M

𝜃

A 𝛥p

M

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𝜃

B

b

A

B

B

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analysis; (b) cracked segmental analysis; and (c) passively prestressed cracked segmental analysis, as follows.

A

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(a)

b

b cd cd

B

𝛥rt

Spr

h

bp (b)

2cd bp

(c)

(d)

Figure 2: RC beam segment subjected to a constant moment

2.1 Uncracked Segmental Analysis Half of the RC beam segment in Figure 2(a) is depicted in Figure 3(a). The tensile steel bars strain, 𝜀rt, can be determined as follows;

6

(

𝜀 where

)

(1)

is the curvature of the section as depicted in Figure 3(b), drt is the distance to the tensile steel rebars from

the top compression fibre and dNA is the depth of the neutral axis from the top compression fibre. The force in the tensile steel rebars can be computed as follows; 𝜀

(2)

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where Ert and Art are the elastic modulus and cross-sectional area of the tensile steel rebars respectively. Substituting

(

)

ro

equation (1) into (2) gives;

(3)

(

)

(4)

re

𝜀

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In a similar way, the strain in the EB plate, 𝜀p, and the corresponding force, P p, can be determined as follows;

𝜀

(5)

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where dp is the distance to the EB plate from the top compression fibre as shown in Figure 3, Ep is the elastic modulus of the EB plate and Ap is the cross-sectional area of the plate. Substituting equation (4) into (5) gives;

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(

Ldef

Strain, 𝜀

𝜃

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drt

Force, F

Prc Pcc

𝜒 dp

𝜀rt 𝜀p (a)

Stress, 𝜎

𝜀rc

dNA

h

(6)

𝜀cc

c

M

)

Pct Prt 𝜀ct (b)

Pp (c)

Figure 3: Uncracked Segmental Analysis

7

(d)

The strain, 𝜀rc, and force, Prc, in the compressive reinforcement can be determined as follows; (

𝜀

)

(7)

𝜀

(8)

where c is the distance to the compressive steel rebars from the top compression fibre, while E rc and Arc are the elastic modulus and cross-sectional area of the compressive reinforcement respectively. Again, substituting equation

)

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(

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(7) into (8) gives; (9)

The strain in the concrete at the top compression fibre, 𝜀cc, can be computed as;

-p

𝜀

(10)

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The force contribution of the concrete in compression, P cc, where a triangular stress distribution takes place as

(11)

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shown in Figure 3(c), is given as follows;

where Ec is the elastic modulus of the concrete and b is the width of the beam as illustrated in Figure 2(b).

𝜀

(

)

(

(12)

)

(13)

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Meanwhile, the strain in the concrete at the bottom tensile fibre, 𝜀ct, and force contribution, Pct, are given by;

where h is the height of the section. Taking equilibrium of forces by adding equations (3), (6), (9), (11) and (13) yields the following expression;

(

)

(

)

(

which can be rearranged to give;

8

)

(

)

(14)

(15)

Further, taking the moment of the forces in Figure 3(d) about the top compression fibre gives;

(

)

(

(

)

)

(

)

(

) (16)

cr-FI-p,

at which cracking occurs in an RC plated segment can be determined as follows;

)

(17)

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(

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The curvature,

where 𝜀ct is the maximum tensile strain of concrete. Substituting equation (17) into equation (16) gives the full-

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interaction cracking moment, Mcr-FI-p, of an RC plated segment, and once exceeded in the RC plated beam, a crack

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forms. 2.2 Cracked Segmental Analysis

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In a cracked segmental analysis, the tensile region of the beam can be idealized as a prism of depth 2cd and width b as shown in Figure 2(c) and (d). The area of the tensile rebar in the prism shown in Figure 2(c) is equivalent to the

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area of all steel bars in the tensile region in Figure 2(b). A length of the prism in Figure 2(c) is depicted in Figure 4(a) where the prism has multiple cracks with zero slip midway between the cracks as illustrated in Figure 4(b). The load Prt induces a slip 𝛥rt at the crack face as displayed in Figure 4(a). The reinforcement load-slip (Prt - 𝛥rt) relationship can be determined from a partial-interaction analysis of a symmetrically loaded concrete prism. Similarly, a length of the FRP plated prim depicted in Figure 2(d) is illustrated in Figure 4(c). Unlike the

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reinforcement steel where multiple crack behavior controls the analysis, for the EB plate, the behavior always reverts to that of a single crack analysis despite the occurrence of multiple cracks [10]. Therefore, a single crack analysis as shown in Figure 4(c) is assumed to take place. This gives a conservative estimate of the FRP plate’s IC debonding behavior.

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Ldef

Ldef Spr

Prt

Prt 𝛥rt

2cd

𝛥rt 𝛿=0

slip

(b)

Lcrit-p

(c)

2cd

𝛥p

(d)

𝛿=0

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slip

Full-interaction boundary

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𝛥p

Pp

(a)

-p

Figure 4: Tension Stiffening Analysis

Consider the cracked segmental analysis displayed in Figure 5. The slip of the tensile rebars, 𝛥rt, can be determined

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as follows;

)

lP

𝜃(

(18)

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and the force in the tensile steel rebars, Prt, can be written as;

(19)

where Krt is a function of the bond and slip between the steel rebars and adjoining concrete, the prism length Spr, and

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the axial rigidities for the reinforcement, ErtArt, and concrete, EcAc. Substituting equation (18) into (19) gives; 𝜃(

)

(20)

Similarly, the slip of the EB plate, 𝛥p, and the force in the plate, Pp, can be given as follows; 𝜃(

)

(21)

(22) 𝜃(

10

)

(23)

Where Kp is a function of the bond and slip between the EB plate and bordering concrete, the critical length of the plate, Lcrit-p, shown in Figure 4(c), and axial rigidities of the plate, EpAp, and concrete prism, EcAc.

Ldef Strain, 𝜀

Stress, 𝜎

Force, F

c M

𝜒 =𝜃/Ldef

dNA drt

Prc Pcc

dp

𝛥rt

Prt

(b)

(c)

(d)

-p

(a)

ro

Pp

𝛥p

of

𝜃

Figure 5: Cracked Segmental Analysis

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As full-interaction exists in the compression region of the segment, the force in the compression reinforcement steel and concrete in compression are as determined before in equation (9) and (11) respectively. The curvature of the

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section, , can be written in terms of the segment rotation, 𝜃, as follows;

(24)

Substituting equation (24) into (9) and (11) gives; (

)

(25)

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(26)

Figure 5(d) shows the force profile of the cracked segment. Taking equilibrium of forces by summing equations (20), (23), (25) and (26) results in the following expression;

𝜃(

)

𝜃(

(

)

)

Solving equation (27) gives the neutral axis depth, dNA, of a cracked segment;

11

(27)

√[

]

[

][

] [

]

(28) [

]

Further, taking moment of the forces in Figure 5(d) about the top compression fibre gives the following expression;

𝜃(

)

𝜃(

(

)

)

(29)

𝜃

)

(

)

(

)

(30)

]

ro

[(

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which can be rearranged to give the rotation, 𝜃, as follows;

Equation (30) can be utilized to compute the rotation of a crack face, 𝜃, at any given moment M, which can be used

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to quantify the deflection of the RC structure.

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The rising portion of the CEB-FIP bond-slip model [49] as depicted in Figure 6(a) can be used to characterize the interaction between the steel rebars and adjoining concrete. Further, a typical load-slip (Prt – 𝛥rt) relationship for the

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concrete prism displayed in Figure 4(a) is depicted in Figure 6(b). Due to good bond conditions between the ribbed steel rebars and adjoining concrete, the value of Krt can be taken as the slope of the best fit line as depicted in Figure 6(b). Zhang et al. [50] proposed a value for Krt based on the CEB-FIP bond-slip model displayed in Figure 6(a) and

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is given as follows;

(

⁄

(31)

)

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The primary crack spacing distance, Spr, can be determined as follows [39];

[

(

) (√

)

]

(32)

where,

√

(

12

)

(33)

(34)

(

)

(35)

and 𝜏max is the maximum bondstress as shown in Figure 6(a), 𝛿1 is the slip at maximum bondstress, fct is the tensile strength of concrete, Lp is the perimeter of the tensile reinforcement in the concrete prism in Figure 2(c), and 𝜑 is a

𝜏

Prt

2 √fc

τ

τ

4

δ δ

ro

τ

of

bond coefficient which can be taken as 0.4.

𝛿

re

𝛿1 = 1 mm

-p

Krt

(a)

𝛥rt (b)

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Figure 6: Reinforcement Steel: (a) CEB-FIP Bond-slip Model; (b) Load-slip (Prt – 𝛥rt) Relationship As for the EB plate, the linear-descending bond-slip model illustrated in Figure 7(a) can be used to represent the

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interaction between the plate and bordering concrete. The maximum bondstress, 𝜏max, and maximum slip, 𝛿max, can be determined from the generic equations given by Seracino et al. [51]. Furthermore, due to low bond properties of the EB plate and the non-linearity of the load-slip (Pp – 𝛥p) relationship as shown in Figure 7(b), the value of Kp cannot be assumed to be constant. A partial-interaction analysis of the load-slip (Pp – 𝛥p) relationship of the prism in

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Figure 4(c) results in the following closed-form solution [39];

[

*

+]

(36)

where;

√

13

(37)

(38)

Substituting equation (36) into (22) and rearranging gives;

[

cc

[

]]

(39)

Initially, the slip of the EB plate, 𝛥p, has to be guessed, after which, the value of K p can be determined using

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equation (39). Next, the rotation of the half-segment, 𝜃, can be computed using equation (30). Finally, to check whether the guessed slip 𝛥p was correct, equation (21) can be used. The value of the guessed slip and that

𝜏max

PIC

(a)

𝛿max

𝛿

lP

𝛿max

-p

Pp

re

𝜏

ro

determined from equation (21) should be the same.

(b)

𝛥p

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Figure 7: EB Plate: (a) Linear-descending Bond-slip Model; (b) Load-slip (Pp – 𝛥p) Relationship 2.3 Passively Prestressed Cracked Segmental Analysis IC debonding of the EB plate commences when the force in the plate reaches the IC debonding force, P IC, as illustrated in Figure 1(b). The behavior of the FRP plate follows that illustrated in Figure 1(c) where any elongation

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is accommodated by extension of the plate and the force in the EB plate within the debonded region remains at P IC. Figure 8(a) shows an adhesively plated segment within the debonded region of the beam. The analysis follows as before except for the fact that the force in the plate now remains at PIC with increase in rotation 𝜃 as shown in Figure 8(b). The value of Kp can be determined as follows;

(40)

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Equation (40) can be substituted into equation (30) to determine the rotation of the segment, 𝜃, at which IC debonding commences.

Ldef Force, F c 𝜃

Prc Pcc

dNA drt

dp

𝛥rt

of

M

PIC

𝛥p = 𝛿max + 𝜀IC Ldb-p

(b)

-p

(a)

ro

Prt

re

Figure 8: Passively Prestressed Cracked Segmental Analysis

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3.0 Discrete Rotation Approach

At first, the beam is uncracked as depicted in Figure 9(a); that is, full-interaction between the steel rebars, EB plate

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and their bordering concrete surface exists, and the deflection at the middle of the strengthened beams can be determined using elastic deflection formula as shown in Table 1 for various loading arrangements. However, when the cracking moment Mcr-FI-p, obtained from equation (16), is first achieved at the middle of the beam, a new crack forms as illustrated in Figure 9(b). At this point, the deflection of the beam can be determined by considering the deflection of the uncracked region of the beam of length Luncr where full-interaction applies, and the rotation of

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crack A.

The deflection of the uncracked region of the beam can be computed using elastic deflection formula at distance Luncr as given in Table 1. As for the deflection contribution due to the crack, it can be computed as shown in Figure 10 for crack C. The rotation of each crack face, 𝜃C, in Figure 10(a) can be determined using equation (30) for a given moment M at the crack. The summation of the rotation of each crack face gives the total rotation of crack C shown as 2𝜃C in Figure 10(b). Moreover, from the geometry of the shape in Figure 10(b), 2𝜃C is equal to;

15

2𝜃 where angles

(41)

and β are in radians. These angles can be written as;

(42)

(43)

of

Substituting equations (42) and (43) into equation (41) gives;

which can be rewritten as; 𝛿 *

+

re

Therefore, the deflection under crack C is given by; 𝛿

𝛿 *

+

(44)

(45)

-p

2𝜃

ro

2𝜃

(46)

𝛿mid-C, can be written as;

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Further, by considering the geometry in Figure 10(b), the deflection at the middle of the beam due to Crack C; that is

(47)

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⁄

Substituting equation (46) into (47) and solving for 𝛿mid-C one gets; 𝜃

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𝛿

16

(48)

Mcr-FI-p

M

RC Beam

A crack

Luncr

L (a)

(b)

Mcr-FI-p

MA

Mcr-FI-p M M cr-FI-p A

A Luncr

C Luncr

Spr

Luncr

MA M M B cr-FI-p

MB M cr-FI-p

Luncr

Spr

lP

(e)

A

C

E

B Spr

MA

Mcr-FI-p M C

-p

A Spr

Spr

(d)

re

C

Spr

B

ro

(c) MC

A

of

FRP Plate

Spr

B Spr

Spr

Spr

(f)

ur na

Figure 9: Discrete Rotation Approach

Crack B

𝐶

𝜃 𝐶 𝐿𝐿 𝐶 𝐿𝑅 𝐶

𝐿

𝛿𝑚𝑖𝑑 2𝜃𝐶

𝐿𝐿

𝐶

𝛼

𝐶

𝛽

2𝜃C LL-C

Jo

𝜃C 𝜃C

𝛿

LR-C

L/2

(a)

(b)

Figure 10: Discrete Rotation of an Individual Crack

17

𝜃𝐶

𝛼

D

Table 1: Elastic Deflection Formula for Different Loading Arrangements Loading Arrangement

Deflection at the center of the beam

Three point bending

Defection at distance Luncr (

4

) for 2

(

(

)

) for

( (

)

4

Uniformly distributed load

) for

of

2

(

2

)

re

-p

4

ro

Four point bending

lP

After the formation of the initial crack in Figure 9(b), the next crack can occur at any point in the full interaction region; however, as most beams in practice happen to be subjected to a moment gradient, it is reasonable to assume that the next crack will occur at the point of full interaction at a distance S pr from the first crack once the moment

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reaches Mcr-FI-p. Therefore, the loading on the beam is increased such that the moment at distance S pr from the first crack is now Mcr-FI-p as presented in Figure 9(c). The deflection of the FRP plated beam is computed just prior to the formation of additional cracks by bearing in mind the rotation of crack A and the deflection of the uncracked region of the strengthened beam of length Luncr. After that, two new cracks form as shown in Figure 9(d) and the deflection

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of the beam can be computed by accounting for the rotation of every discrete crack following the method shown in Figure 10 and also accounting for the deflection of the uncracked region of the beam of length L uncr given in Table 1. The mid-span deflection contribution of cracks A, B and C in Figure 9(d) can be computed using equation (48). In a similar way, the load on the beam is increased such that the moment at distance S pr from crack B is Mcr-FI-p as shown in Figure 9(e) at which point the deflection can be determined as before. After that, two new cracks form as shown in Figure 9(f) and the analysis is continued by increasing the load and observing the number of cracks that form. In summary, to evaluate the deflection of the adhesively plated beam for a given moment, the number of cracks that

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form has to be determined first; after that, the rotation of each crack is computed using equation (30) and the midspan deflection contribution of each crack can be determined using equation (48). As for the uncracked region of the beam of length Luncr, the deflection contribution can be calculated as shown in Table 1 for the loading arrangements presented. 4.0 Validation with Experimental Results To verify the deflection results obtained from the mechanics solution, three CFRP plated RC beams subjected to

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three-point bending were considered together with four CFRP plated RC beams tested under four-point bending by Alagusundaramoorthy et al. [24]. The details of the experimental setup together with the test results and discussion

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are given as follows.

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4.1 Beam Specimen Details and Experimental Setup

Three CFRP strengthened RC beams examined in the laboratory, referred to as B1, B2 and B3 had a cross-section as

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shown in Figure 11(a). The width of the beams, b, was 200 mm while the height, h, was 300 mm. The total length of the beams was 3600 mm as shown in Figure 11(b) while the free span between the rounded supports was maintained

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at 3500 mm. The FRP strengthened beams had two 12 mm diameter compression steel rebars and two 16 mm diameter tensile steel rebars. Shear links of 8 mm diameter were set apart at 150 mm from center to center as

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illustrated in Figure 11(b) and the concrete cover to the mild-steel links was 30 mm. The CFRP plate was placed symmetrically on the tensile concrete surface of the beam as shown in Figure 11(d). The plate had a 50 mm width and a 1.2 mm thickness; meanwhile, the entire length of the plate was 3250 mm. The beams were tested under three point bending as depicted in Figure 11(b) and (c) where a hydraulic jack was used to apply a point load at the center. The deflection at the middle of the beam was recorded by means of a linear variable differential transducer (LVDT)

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located under the applied load (Figure 11(c)).

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(d)

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(c)

Figure 11: Beam Specimen Details: (a) Cross-section; (b) Longitudinal Section; (c) Experimental Setup; (d)

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Bonding of CFRP Plates

Further, four CFRP plated RC beams having different plate layout as detailed by Alagusundaramoorthy et al. [24] and shown in Figure 12 were considered as well. The beams had a width, b, equal to 230 mm and a height, h, equal to 380 mm. The beams total length was 4880 mm while the distance between the supports was chosen to be 4576

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mm as depicted in Figure 12(c). The strengthened beams had two tensile steel reinforcement of 25 mm diameter and two compressive steel reinforcement of 9 mm diameter. To resist the shear force and tie the longitudinal rebars together, 9 mm diameter steel links spread out at 150 mm along the length of the beams were provided. The concrete cover to the tensile reinforcement bars was maintained at 38 mm whereas that to the compression reinforcement bars was set at 25 mm. Two of the FRP strengthened beams, referred to as CB3-2S and CB4-2S, were reinforced with two CFRP plates as shown in Figure 12(a). The CFRP plates were 1.4 mm thick and 76 mm wide. The total length of the plates was 4270 mm as shown in Figure 12(c). Meanwhile, the remaining two beams, named CB5-3S and

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CB6-3S, had three CFRP plates attached to their tensile concrete surface as displayed in Figure 12(b). The plates had similar dimensions to those in Figure 12(a). The beams were tested under four-point flexural testing as shown in Figure 12(c) and the point loads were applied at 1830 mm from the supports. The mid-span deflection was recorded

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using an LVDT positioned at the center of the beam.

Figure 12: Alagusundaramoorthy et al. [24] RC beam sample details: (a) CB3-2S and CB4-2S Cross-section;

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(b) CB5-3S and CB6-3S Cross-section; (c) Four Point Bending Test Setup 4.2 Material Properties

For the beams tested in this study, the concrete was transported from the concrete plant to the laboratory using a concrete mixer truck. One batch of concrete was used for all specimens and the concrete compressive strength, fc,

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the splitting tensile strength, ftsp, and elastic modulus, Ec, are reported in Table 2. Similarly, the material properties for the concrete used by Alagusundaramoorthy et al. [24] are also reported in Table 2. Meanwhile, the mechanical properties for the reinforcement steel and CFRP plates used for the beams tested in this study and for those reported by Alagusundaramoorthy et al. [24] are given in Table 3 and Table 4 respectively.

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Table 2: Concrete Material Properties Beam Specimens

Compressive Strength, fc (MPa) 22.7 31

B1, B2, B3 CB3-2S, CB4-2S, CB5-3S, CB6-3S

Elastic Modulus, Ec (MPa)

Splitting Tensile Strength, ftsp (MPa) 2.8 3.4

22,717 25,385

Table 3: Mechanical Properties of Reinforcement Steel Yield Strength, fy (MPa) 535 414

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B1, B2, B3 CB3-2S, CB4-2S, CB5-3S, CB6-3S

Elastic Modulus, Er (GPa) 154 200

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Beam Specimens

Beam Specimens

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Table 4: CFRP Plate Characteristics Width, bp

Thickness, tp (mm)

Elastic modulus, Ep

(MPA)

(GPa)

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(mm)

Ultimate tensile stress

50

1.2

2,800

165

CB3-2S, CB4-2S, CB5-3S, CB6-3S

76

1.4

2,068

138

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4.3 Test Results and Discussion

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B1, B2, B3

In this section, the experimental crack spacing and mid-span deflection results are presented and compared with the theoretical results derived from the mechanics-based discrete rotation approach.

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4.3.1 Crack Spacing

The discrete rotation approach presented here depends on the development of cracks and their distance from the supports to measure the mid-span deflection. Therefore, comparing the experimental and theoretical crack spacing values could give a good indication on the adequacy of the method proposed. A summary of the number of cracks that formed just prior to failure by IC debonding of the FRP plates and also a summary of the experimental crack spacing values are given in Table 5. This includes the minimum crack spacing, Sexp-min, the maximum crack spacing, Sexp-max, and the mean crack spacing results, Sexp-mean. Moreover, the theoretical crack spacing values, Spr, as

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computed from equation (32) are also reported in Table 5. The mean crack spacing values, S exp-mean, correlate well with the theoretical crack spacing results, Spr. This is especially true for B1 where the values were almost identical. CFRP plated beam B2 mean experimental crack spacing, Sexp-mean, showed the greatest variance to the theoretical crack spacing results with about 18 mm difference. However, given the unpredictable nature of crack opening and crack spacing, it can be concluded that equation (32) gives a very a good estimate of the crack spacing. Furthermore, Figure 13 shows the development of flexural cracks as the load on beam B1 increased.

Number of cracks

B1 B2 B3

21 21 22

Experimental crack spacing, Sexp (mm) Sexp-min Sexp-max Sexp-mean 215 205 206

143.1 127.3 133.5

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88 74 76

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Theoretical crack spacing, Spr (mm)

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Specimen

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Table 5: Experimental and theoretical crack spacing results

145.3 145.3 145.3

(b) P = 20 kN

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(a) P = 15 kN

(d) P = 30 kN

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(c) P = 25 kN

(f) P = 75 kN

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(e) P = 45 kN

Figure 13: Development of Cracks in Beam B1

4.3.2 Mid-span Deflection The experimental load-deflection results for the beams considered in this study are shown in Figure 14. The deflection results were considered under service loads only; that is, up to 50% of the ultimate load to cause failure. Further, the deflection results from the analytical mechanics solution together with the deflection predictions

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determined from the ACI code approach [26] are also presented in Figure 14. Take the mechanics-based discrete rotation solution for B1, B2 and B3 depicted in Figure 14(a) and illustrated separately in Figure 14(b) up to a load of 11 kN. Initially the deflection follows the uncracked response A-B shown in Figure 14(b). At point B, a crack forms causing slip to occur between the reinforcement bars and adjacent concrete as well as the CFRP plate and the concrete adjacent to it. This slip causes the crack to broaden and eventually increases the deflection by shifting from B-C on the load-deflection response as shown in Figure 14(b). After that, with an increase in the applied load, the deflection increases by rising from C-D. At this point, two new cracks form and the opening of these cracks due to

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slip causes the deflection to move from D-E as shown. Again, as the load increases, the deflection moves from E-F until two new cracks form. This trend continues for the early stages of loading. However, at higher loads, the load-

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deflection response from the mechanics solution moves slightly to the back with the formation of new cracks as can be seen in Figure 14. This happens due to the reduction of the contribution of the uncracked region to the mid-span

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deflection (reduction in Luncr) and also due to the reduced influence of the newly formed cracks on the mid-span deflection as they are farther away from the center. Moreover, if the beams have a single crack before loading due to

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damage or shrinkage of concrete, the load-deflection response follows line A-C-D as opposed to A-B-C in Figure

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14(b). This illuminates the cause of the early inconsistency between the mechanics solution and the experimental findings in Figure 14(a). This variance is almost eliminated in the remaining beams in Figure 14(c) and (d). Further, once the slip of the plate at the crack face, 𝛥p, reaches the maximum debonding slip, 𝛿max, IC debonding

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starts and the behavior at this point is similar to that in Figure 1(b). The load on the FRP strengthened beams, P, to cause the initiation of IC debonding as derived from the analytical mechanics solution together with the corresponding plate slip, 𝛥p, are given in Table 6. These loads are also marked in Figure 14 on the load-deflection response, and any increase in the applied load will cause stable debonding as shown in Figure 1(c) until failure

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occurs. From Figure 14, it can be observed that for most of the service loads, the plate is attached to the concrete surface where the linear-descending bond-slip model in Figure 7(a) represents the plate-concrete interaction. Therefore, stable IC debonding can be considered to greatly affect the ultimate behavior of the FRP plated beams as opposed to their service behavior.

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Table 6: Applied Load to Cause Commencement of Stable IC Debonding Beam Specimen

B1, B2, B3 CB3-2S, CB4-2S CB5-3S, CB6-3S

Load, P, to Induce Commencement of Stable IC Debonding (kN) 40 124 134

Plate Slip at the Crack Face (mm) 0.152 0.123 0.123

Lastly, the mechanics approach presented here gives better predictions to the mid-span deflection than the ACI code

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method, and this is very clear especially for the beams tested by Alagusundaramoorthy et al. [24] in Figure 14(c) and Figure 14(d). This can be accredited to the fact that the analytical mechanics solution considers the effect of the IC

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debonding mechanism of the FRP strengthened beams, accounts for the partial-interaction due to slip between the adjacent concrete and steel rebars, and incorporates the deflection induced by the formation of new cracks.

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Meanwhile, the deflection results following the ACI code design approach were linear and smooth without step

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changes, and did not account for either the partial-interaction due to slip or the IC debonding of the FRP plate.

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Commencement of stable IC debonding F D B

7 Cracks

E C

5 Cracks 3 Cracks

1 Crack

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A

(b)

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(a)

Commencement of stable IC debonding

(d)

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(c)

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Commencement of stable IC debonding

Figure 14: Mid-span Deflection Results

5.0 Conclusion

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The calculation of the deflection of FRP plated RC beams has been carried out previously by determining average effective moment of inertia and using elastic deflection equations. This full-interaction approach fails to consider the deflection induced by the widening of flexural cracks due to slip between the adjacent concrete and steel bars and does not account for the IC debonding mechanism of the plated member. In this study, a partial-interaction segmental analysis of an adhesively plated RC beam was used to derive analytical expressions for the rotation of discrete cracks. Subsequently, the deflection at the middle of the beam was quantified by dividing the member into a cracked and an uncracked region. The deflection of the cracked region was evaluated by accounting for the rotation

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of every single crack obtained from the analytical expressions derived from the partial-interaction segmental analysis. Meanwhile, full-interaction was presumed to take place in the uncracked region of the beam and the deflection contribution was computed using elastic deflection formula. The mechanics-based analytical solution was found to give good estimates of the deflection under service loads for the beams considered in this study, and the approach marks the formation of new cracks on the load-deflection response by step changes in the slope as opposed to the linear load-deflection responses given by previous code methods. Furthermore, it was shown that while it is important to account for stable IC debonding in the analysis, the detachment of the plate through stable IC

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debonding affects the ultimate behavior of the plated beams more as opposed to their service behavior. For instance, for much of the service loads, the plate was attached to the neighboring concrete surface and stable IC debonding

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only commenced just prior to reaching 50% of the ultimate load. Finally, as the design approach presented does not need any empirical calibration on the member level, the solution has the potential to be used for predicting the

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deflection of strengthened RC members with novel construction materials and future research should focus on testing the approach with new types of concrete (e.g. high strength concrete, fibre reinforced concrete), different

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types of rebar material (e.g. FRP bars) and different types of FRP adhesively bonded plates (e.g. GFRP, AFRP).

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Acknowledgements

The authors gratefully acknowledge the technical and financial support of Universiti Putra Malaysia (UPM) while

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conducting this research. Further, the authors acknowledge the support of the Department of Civil Engineering, College of Engineering, Prince Sattam Bin Abdulaziz University, Saudi Arabia; and the Department of Civil Engineering, Faculty of Engineering, Amran University, Yemen. Funding: This work was supported by Putra Grant, Universiti Putra Malaysia (UPM) [grant number 9555600].

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