Viscoelastic analysis of FRP strengthened reinforced concrete beams

Composite Structures 93 (2011) 3200–3208

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Viscoelastic analysis of FRP strengthened reinforced concrete beams Chao Zhang, Jialai Wang ⇑ Department of Civil, Construction, and Environmental Engineering, The University of Alabama, Tuscaloosa, AL 35487, United States

a r t i c l e

i n f o

Article history: Available online 21 June 2011 Keywords: FRP-strengthening Interface stress Viscoelastic analysis Long-term behavior Stress redistribution Creep

a b s t r a c t External bonding of FRP plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP-concrete interface are critical to the effectiveness of this technique because high stress concentration along the FRP-concrete interface can lead to the FRP debonding from the concrete beam. Although the short-term stress distribution along the FRP-concrete interface has been studied extensively, very few studies have been conducted on the long-term stress distribution, which closely simulates the behavior of the structure during the service-life. In this study, we develop a viscoelastic solution for the long-term interface stress distribution in a FRP plate strengthened reinforced concrete beam. In this solution, the RC beam and the FRP plate are modeled as elastic materials; while the adhesive layer is modeled as a viscoelastic material using the Standard Linear Solid model. Closed-form expressions of the interface stresses and deflection of the beam are obtained using Laplace transform and calculated using the Zakian’s numerical method. The validation of this viscoelastic solution is verified by finite element analysis using a subroutine UMAT based on the Standard Linear Solid model. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforced polymers (FRPs) have emerged as important structural materials in the last three decades [1]. Among their many applications in civil infrastructure, retrofitting/rehabilitating reinforced concrete (RC) structures is most popular due to many advantages of FRP, such as high corrosion resistance, high strength-to-weight ratio, and easy of handling. In this application, FRP plates/fabrics/strips are either externally bonded (EB) [2] or near-surface mounted (NSM) on RC structures [3]. To improve the strengthening efficiency, prestress can be applied to FRPs [4]. This technique has evolved into one of primary techniques to address the deterioration of civil infrastructure system caused by severe environmental exposures, natural extreme events, excessive use, and intentional attacks. Stresses along the FRP-concrete interface are critical to the success of this technique because the high interface stress concentration can lead to debonding along the FPR-concrete interface, which has been shown to be one of the most common failure modes of the FRP-strengthened RC structures. Extensive studies have been conducted, and various models have been proposed to estimate the stresses along the FRP-concrete interface [5]. A comprehensive review on these studies was given by Smith and Teng [5] and a recent study by Wang and Zhang [6]. However, all these existing studies are limited to the elastic analysis, which only gives the instant response of the structures.

⇑ Corresponding author. Tel.: +1 205 348 6786; fax: +1 205 348 0783. E-mail address: [email protected] (J. Wang). 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.06.006

Epoxy, the most widely used adhesive in bonding FRP, exhibits viscoelastic properties [7–11]. Its material properties vary with time under different situations, especially in the regions of high stress concentration. Such variation of material properties can induce redistributions of stresses and additional deformation, which could be significant during the service life of the structure and cause potential failure of the strengthening, as demonstrated recently by Meshgin et al. [12]. Meshgin et al. [12] found that the creep of epoxy could result in failure at the interfaces due to the combined effect of relatively high shear stress to ultimate shear strength ratio and a thick layer of epoxy. For this reason, the time-dependent behavior of the FRP strengthened concrete structures has become the focus of a number of recent studies [12– 18] both experimentally and numerically. All of these studies show that the concrete-FRP interface exhibits significant time-dependent behavior, and that the shear stress to the shear strength ratio within the adhesive layer is a primary factor affecting the longterm behavior of the FRP-concrete interfaces [18]. Several rheological models were proposed to simulate the creep behaviors observed in the tests [12,13,17,18]. Based on these rheological models, numerical methods [13,18] have been proposed to simulate the time-dependent behaviors of the FRP-strengthened RC structures. Numerical methods are usually very time-consuming in simulating the time-dependent behavior of structures because sufficient small step must be used to avoid error accumulation. Analytical solutions are much more efficient. The existing analytical solutions of the interface stresses of FRPstrengthened RC beams [5,19–24] are based on the classical adhesively bonded joint model of Goland and Reissner [25] (G–R model).

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In this model, the strengthened beam and the FRP plate were modeled as two beams connected by a linear elastic adhesive layer, which is modeled as a layer of continuously distributed spring with shear and normal stiffnesses. However, all these analytical solutions are limited to elastic case. Delale and Erdogan [26] developed a viscoelastic solution for a symmetric adhesive joint by treating two adherends as elastic simple beams and the adhesive as a linearly viscoelastic spring. Mirman and Knecht [8] proposed another simple viscoelastic model to study the creep behavior of the adhesive layer, in which the peeling stress and the creep deformation of the adhesive layer were ignored. So far no rigorous viscoelastic solutions for the interface stresses and deflection of the FRP-strengthened RC beams are available. The major objective of this research is to fill this gap through developing viscoelastic solutions of the interface stresses and the deflection of a FRP-strengthened RC beam. These new solutions can be used as an efficient tool to evaluate the long-term behavior of the FRP-strengthened RC beams. 2. Viscoelastic model of a FRP plate strengthened RC beam Consider a simply supported RC beam (beam 1) with thickness of h1 strengthened by a thin FRP plate (beam 2) with thickness of h2 through external bonding with a thin adhesive layer with thickness of h0, as shown in Fig. 1. A uniform load with magnitude of q is applied to the RC beam. Since the creep of the concrete and the FRP plate occurs within a much longer time period compared with the adhesive [15], only the adhesive layer is modeled as viscoelastic material in this study. Both the FRP plate and the RC beam are modeled as elastic material. It should be pointed out that this simplified model is only a first step to understand the adhesive creeping effect of the FRP-strengthened RC beams since the creep behaviors of both the FRP plate and the RC beams are not considered. A notable recent development in the FRP strengthening technique is that the FRP plate is prestressed before bonding to the RC beam. This technique can make better use of the tensile strength of the FRP plate [27] and reduce the crack development in the RC beam [28]. Although a number of experiments have been conducted to study the short-term behaviors of the RC beams strengthened by this technique [29–31], their long-term behaviors have not been addressed sufficiently, especially the prestress loss due to the creep of adhesive layer, which is critical to the success of the prestressing technique. To make this study also applicable to the prestressed FRP strengthened RC beams, a prestress of N0 is applied to the FRP plate in the formulation thereafter. This prestress is applied as following: First the FRP plate is pretensioned; then the prestressed FRP plate is bonded to the tension side of the RC beam; finally, cut the FRP plate at two far ends once the adhesive achieved its fully bonding strength.

U 1 ðx1 ; z1 ; tÞ ¼ u1 ðx; tÞ þ z1 /1 ðx; tÞ; U 2 ðx2 ; z2 ; tÞ ¼ u2 ðx; tÞ þ z2 /2 ðx; tÞ  u0 ðx; tÞ

where subscript i = 1, 2, representing the beams 1 and 2 in Fig. 1. The xi and zi axels are local coordinates and xi is located at the neutral axis of the beam i and x1 = x2 = x  ui(x, t), /i(x, t), and wi(x, t) are the axial displacement, rotation and deflection at the neutral axis of beam i. u0(x, t) is the initial displacement of the FRP plate before the pretension is released. Ui(xi, zi, t) and Wi(xi, zi, t) are the axial and transverse displacements of the beam i. The strains along the neutral axes of these two beams can be written as

@u1 ðx; t Þ ; @x @/ ðx; tÞ ; ji ðx; tÞ ¼ i @x

e01 ðx; tÞ ¼

@u2 ðx; tÞ @u0 ðx; tÞ  ; @x @x @w ðx; tÞ : cixy ðx; tÞ ¼ /i ðx; tÞ þ i @x

e02 ðx; tÞ ¼

ð2Þ

The constitutive equations for the beam i can be written as:

@u1 ðx; tÞ ; @x   @u2 ðx; t Þ @u0 ðx; t Þ ;  N2 ðx; tÞ  N0 HðtÞ ¼ C 2 @x @x @/ ðx; t Þ M i ðx; tÞ ¼ Di i ;  @x  @wi ðx; t Þ ; Q i ðx; tÞ ¼ Bi /i ðx; tÞ þ @x

N1 ðx; tÞ ¼ C 1

ð3Þ

where Ni(x, t), Qi(x, t) and Mi(x, t) are the resultant axial force, the transverse shear force, and the bending moment of the beam i, respectively. H(t) is the Heaviside step function. N0 is the prestress applied to the FRP plate. Ci, Bi and Di are the axial, shear and bending stiffness coefficients of the beam i, respectively. For plain stress condition, we have

C i ¼ Ei bhi ;

3

Di ¼ Ei bhi =12;

Bi ¼ jGi bhi ;

ð4Þ

where Ei and Gi are the longitudinal modulus and the transverse shear modulus, respectively; j is the shear correction coefficient chosen as 5/6 in this study; b is the width of the beam. The bending stiffness of the RC beam D1 can be modified to account for a cracked section. Considering an infinitesimal free body diagram of the FRPstrengthened RC beam shown in Fig. 2, the following equilibrium equations can be established

@N1 ðx; t Þ ¼ bsðx; tÞ; @x @M 1 ðx; t Þ h1 ¼ Q 1 ðx; tÞ  bsðx; tÞ; @x 2 @Q 1 ðx; tÞ ¼ brðx; tÞ þ bqHðtÞ; @x

2.1. Shear deformable beam theory Both the RC beam and the FRP plate are modeled as Timoshenko beams, beam 1 and 2, respectively. The displacement fields can then be written as

Fig. 1. An FRP strengthened RC beam.

ð1Þ

W i ðxi ; zi ; tÞ ¼ wi ðx; tÞ;

Fig. 2. Free body diagram of a FRP-strengthened RC beam.

ð5aÞ

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2 rx   6 sij ¼ 4 s

@N2 ðx; t Þ @M 2 ðx; t Þ h2 ¼ bsðx; tÞ; ¼ Q 2 ðx; tÞ  bsðx; tÞ; @x @x 2 @Q 2 ðx; tÞ ¼ brðx; tÞ; @x

0

ð5bÞ

where s(x, t) and r(x, t) are the interface stress within the adhesive layer, which are assumed to be constant through the thickness of the adhesive layer. By combining the first and third equation of (5a) and (5b), the following equations can be given by

3

2

3

2

3

s 0 0 s 0 rx  s s 0 6 7 6 r 07 rs 0 7 5  40 s 05 ¼ 4s 5: 0

0 0 s

0

0

0

s ð11bÞ

NT ðx; tÞ ¼ N1 ðx; tÞ þ N2 ðx; tÞ;

ð7aÞ

Q T ðx; tÞ ¼ Q 1 ðx; tÞ þ Q 2 ðx; tÞ;

ð7bÞ

A number of viscoelastic models have been proposed to simulate the adhesive layer [7,12,32]. Meshgin et al. [12] found that a Standard Linear Solid (SLS) Model can properly simulate the long-term creep behavior of the FRP-concrete interface. Therefore, this model is employed in this study. As shown in Fig. 3, a SLS model consists of a Maxwell Model and a Hookean spring in parallel. The notations g and k1 represent the coefficient of viscosity and the elastic stiffness for the Maxwell model, and k2 is the elastic stiffness of the lone spring. Generally, these coefficients will change with environmental factors such as temperature, moisture, and loading conditions. However, for the sake of simplicity, these coefficients are assumed constants in this study. Then the differential operator of the adhesive can be expressed as

ð7cÞ

P1 ¼ a1 þ

@M1 ðx; tÞ h1 @N1 ðx; tÞ þ ; @x @x 2 @M2 ðx; tÞ h2 @N2 ðx; tÞ Q 2 ðx; tÞ ¼  : @x @x 2

Q 1 ðx; tÞ ¼

ð6aÞ ð6bÞ

Define

M T ðx; tÞ ¼ M 1 ðx; tÞ þ M 2 ðx; tÞ þ N1 ðx; tÞ

h1 þ h2 ; 2

where NT(x, t), QT(x, t), MT(x, t) are essentially the resulting forces of the FRP-strengthened RC beam with respect to the midplane of the FRP plate, as shown in Fig. 2.

d ; dt

Q 1 ¼ b1 þ b2

where a1 ¼ kg2 ¼ t10 ; retardation time.

d ; dt

b1 ¼ k1gk2 ¼ kt01 ;

ð12Þ b2 ¼ k1 þ k2 ; and t 0

is

the

2.3. Governing equation 2.2. Viscoelastic interface model Following the assumption most commonly used in G–R model, the shear and normal stresses are assumed constant through the thickness of the adhesive layer. Then the average strains within the adhesive layer can be established as [26]



cxy ðx;tÞ ¼ u0 ðx;tÞ þ u1 ðx; tÞ 

 h1 h2 h0 ; /1 ðx;t Þ  u2 ðx; tÞ  /2 ðx; t Þ 2 2 ð8aÞ

ey ðx; tÞ ¼ ðw1 ðx; tÞ  w2 ðx; tÞÞ=h0 ;

ð8bÞ



 @u1 ðx;tÞ h1 @/1 ðx;tÞ @u2 ðx;tÞ h2 @/2 ðx;tÞ @u0 ðx;tÞ ex ðx;tÞ ¼  þ þ  2: 2 2 @x @x @x @x @x ð8cÞ

The constitutive equations of linear isotropic viscoelastic materials can be expressed by means of differential operators in following forms

P1 ðsij Þ ¼ Q 1 ðeij Þ;

ði; jÞ ¼ 1; 2; 3;

ð9aÞ

P2 ðsÞ ¼ Q 2 ðeÞ;

Inserting the deviatoric stress and strain components given by Eqs. (10) and (11) into Eq. (9) yields

  P1 ð2rx  rÞ ¼ Q 1 2ex  ey  ez ;

ð13aÞ

P1 ð2r  rx Þ ¼ Q 1 ð2ey  ex  ez Þ;

ð13bÞ

 P1 ðrx þ rÞ ¼ Q 1 ð2ez  ex  ey Þ; 1 P1 ðsÞ ¼ Q 1 ðcxy Þ; 2 P2 ðrx þ rÞ ¼ Q 2 ðex þ ey þ ez Þ:

ð13cÞ



P2  1;

Q 2  3K;

ð14Þ

where K is the bulk modulus of the adhesive and given by Ea Ga K ¼ 3ð3G . a Ea Þ By substituting Eq. (13e) into Eqs. (13a) and (13c), we can eliminate ez

ð9bÞ



ex þ ey þ ez =3; s ¼ ðrx þ ry Þ=3;

ð10aÞ ð10bÞ

Then the deviatoric strain tensor can be obtained as

2

3

2

3

2

3

e 0 0 ex cxy =2 0 ex  e cxy =2 0 7 6 7 6 7 6 ½eij  ¼ 4 cxy =2 ey 0 5  4 0 e 0 5 ¼ 4 cxy =2 ey  e 0 5: 0 0 e 0 0 0 0 ez ez  e ð11aÞ By definingry = r, sxy = s, we can obtain the deviatoric stress tensor as

ð13eÞ

Noting that Eq. (13b) can be obtained by adding Eqs. (13a) and (13c), it is not considered in the following derivation. Since that most viscoelastic materials behave elastically under a hydrostatic stress state, we can assume

where P1, Q1, P2 and Q2 are differential operators; sij and eij (i, j = 1, 2, 3) are the deviatoric components of the stress and strain tensors, respectively; e and s are the volumetric strain and hydrostatic component of the stress tensor, respectively. Considering plane stress condition, the volumetric strain and hydrostatic stress are given by

e¼

ð13dÞ

Fig. 3. Standard Linear Solid model.

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ð2Q 1 þ 3KP 1 Þ Q 1 ð2ex  ey Þ  P1 ð6Kðex þ ey Þ  3rÞ    ð6KP1 þ Q 1 Þ Q 1 ðex þ ey Þ  3KP 1 ðex þ ey Þ ¼ 0:

ð15Þ

Substituting the expression of the differential operator Eq. (12) into Eq. (15) and after rearranging, we have

@ @2 a10 þ a11 þ a12 2 @t @t

!

! @ @2 ex þ b10 þ b11 þ b12 2 ey @t @t ! @ @2 þ c10 þ c11 þ c12 2 r ¼ 0; @t @t

Applying Laplace transform to Eq. (18) and after rearranging, we have

E11 ðsÞ

@ 4 N1L ðx; sÞ @ 4 M 1L ðx; sÞ @ 2 N1L ðx; sÞ þ E12 ðsÞ þ E13 ðsÞ 4 4 @x @x @x2 þ E14 ðsÞ

@ 2 M 1L ðx; sÞ þ E15 ðsÞN1L ðx; sÞ @x2

þ E16 ðsÞM 1L ðx; sÞ þ E17 ðsÞMTL ðx; sÞ þ E18 ðsÞq ¼ 0;

ð16Þ

ð21Þ

where 2

a10 ¼ 3b1  9a1 b1 K;

where N1L(x, s), M1L(x, s), MTL(x, s) are Laplace transforms of N1(x, t), M1(x, t), and MT(x, t), respectively; and

a11 ¼ 6b1 b2  9b1 K  9a1 b2 K;

2

a12 ¼ 3b2  9b2 K; b10 ¼ b12 ¼

2 3b1 2 3b2

  h1  1 c10 þ c11 s þ c12 s2 ; E12 ðsÞ ¼ c10 þ c11 s þ c12 s2 ; b 2b    h1 1 1  b10 þ b11 s þ b12 s2 E13 ðsÞ ¼ þ 2h0 B1 B2    1 4 4 h2 ðh1 þ h2 Þ  a10 þ a11 s þ a12 s2 ;   þ 8 C1 C2 D2

E11 ðsÞ ¼  18a1 b1 K;

b11 ¼ 6b1 b2  18b1 K  18a1 b2 K;

 18b2 K;

c10 ¼ 6a1 b1 þ 9a21 K;

c11 ¼ 6b1 þ 6a1 b2 þ 18a1 K;

c12 ¼ 6b2 þ 9K

Substituting Eq. (2) into Eq. (16) gives

! @ @2 a10 þ a11 þ a12 2 @t @t   1 @u1 ðx; tÞ h1 @/1 ðx; tÞ 1 @u2 ðx; tÞ h2 @/2 ðx; t Þ 1 @u0 ðx; tÞ  þ þ  2 @x @x 2 @x @x 2 @x 4 4 !  2 @ @ w1 ðx;t Þ  w2 ðx; tÞ þ b10 þ b11 þ b12 2 @t h0 @t ! @ @2 þ c10 þ c11 þ c12 2 rðx;tÞ ¼ 0: ð17Þ @t @t Differentiating both sides of Eq. (17) twice and inserting Eq. (3) into Eq. (17) yields 0 1

! 1 h2 ðh1 þh2 Þ @ 2 N1 ðx;t Þ 1 @x2 @ @ 2 B 2C 1  2C2  8D2 C

2 a10 þ a11 þ a12 2 @ A @ M1 ðx;t Þ h1 h2 h2 @t @t  4D þ þ bqH ð t Þ 2 4D2 4D2 @x 1 0 1

2

2 ! 1 @ M1 ðx;t Þ 1 þ h21 @ N@x1 2ðx;tÞ  h12Dþh2 2 N 1 ðx; tÞ @x2 1 @ @ 2 B B1 þ B2 C

þ b10 þ b11 þ b12 2 @ A h0 @t @t  D11 þ D12 M 1 ðx; tÞ  B12 bqHðtÞ þ D12 M T ðx; tÞ ! ! 1 @ @2 @ 4 M 1 ðx; tÞ h1 @ 4 N 1 ðx; tÞ ¼ 0: ð18Þ þ c10 þ c11 þ c12 2 þ b @t @x4 @x4 2 @t

   1 1 1  b10 þ b11 s þ b12 s2 þ h0 B1 B2   1 h1 h2  ða10 þ a11 s þ a12 s2 Þ; þ 4 D1 D2 1 ðh1 þ h2 Þðb10 þ b11 s þ b12 s2 Þ; E15 ðsÞ ¼  2D2 h0   1 1 1 E16 ðsÞ ¼  ðb10 þ b11 s þ b12 s2 Þ; þ h0 D 1 D 2

E14 ðsÞ ¼

1 ðb10 þ b11 s þ b12 s2 Þ; D2 h0 b bh2 E18 ðsÞ ¼  ðb10 þ b11 s þ b12 s2 Þ þ ða10 þ a11 s þ a12 s2 Þ: B 2 h0 s 4D2 s E17 ðsÞ ¼

Similarly, by applying Laplace transform to Eq. (20) and after rearranging, we have, 2

M1L ðx; sÞ ¼

By inserting Eq. (8a) into Eq. (13d), we have

P1 ðs; tÞ ¼

 1 h1 h2 Q 1 ðu1 ðx; t Þ  /1 ðx; tÞ  u2 ðx; t Þ  /2 ðx; tÞ 2 2 2 þu0 ðx; tÞÞ=h0 :

P1

! ¼

D11 ðsÞ ¼ ð19Þ

1 Q fC 11 N1 ðx; tÞ þ C 12 M 1 ðx; tÞ þ C 13 NT ðx; tÞ 2 1 þ C 14 M T ðx; tÞ þ C 15 N0 ðx; tÞg:

where

    1 1 1 h2 ðh1 þ h2 Þ 1 h1 h2 ; C 12 ¼  ; þ þ  h0 C 1 C 2 4D2 2h0 D1 D2 1 h2 1 ¼ ; C 14 ¼  ; C 15 ¼ : C 2 h0 C 2 h0 2D2 h0

C 11 ¼ C 13

ð22Þ

where NTL(x, s) is the Laplace transform of NT(x, t), and

Differentiating both sides of Eq. (19) twice and substituting Eq. (3) yields

1 @ 2 N1 ðx; t Þ b @x2

D12 ðsÞ d N 1L ðx; sÞ D13 ðsÞ þ N1L ðx; sÞ 2 D11 ðsÞ D11 ðsÞ dx D14 ðsÞ D15 ðsÞ D16 ðsÞ NTL ðx; sÞ þ M TL ðx; sÞ þ N0 ; þ D11 ðsÞ D11 ðsÞ D11 ðsÞs

ð20Þ

1 C 12 ðb1 þ b2 sÞ; 2

1 D13 ðsÞ ¼  C 11 ðb1 þ b2 sÞ; 2 1 D15 ðsÞ ¼  C 14 ðb1 þ b2 sÞ; 2

D12 ðsÞ ¼

1 ða1 þ sÞ; 2b

1 D14 ðsÞ ¼  C 13 ðb1 þ b2 sÞ; 2

1 D16 ðsÞ ¼  C 15 ðb1 þ b2 sÞ: 2 By substituting Eq. (22) into Eq. (21), we can eliminate M1L(x, s) and obtain the governing equation in term of N1L(x, s)

F 11 ðsÞ

@ 6 N1L ðx; sÞ @ 4 N1L ðx; sÞ @ 2 N1L ðx; sÞ þ F 12 ðsÞ þ F 13 ðsÞ @x6 @x4 @x2 þ F 14 ðsÞN1L ðx; sÞ þ F 15 ðsÞNTL ðx; sÞ þ F 16 ðsÞM TL ðx; sÞ þ F 17 ðsÞN0 =s þ F 18 ðsÞq ¼ 0;

ð23Þ

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C. Zhang, J. Wang / Composite Structures 93 (2011) 3200–3208

where

F 11 ðsÞ ¼

D12 ðsÞE12 ðsÞ ; D11 ðsÞ

F 12 ðsÞ ¼ E11 ðsÞ þ

D13 ðsÞE12 ðsÞ D12 ðsÞE14 ðsÞ þ ; D11 ðsÞ D11 ðsÞ

sL ðx; sÞ ¼

6 1X 1 @N1CL ðx; sÞ ; ci ðsÞRi ðsÞeRi ðsÞx þ b i¼1 b @x

rL ðx; sÞ ¼

1 b 

D13 ðsÞE14 ðsÞ D12 ðsÞE16 ðsÞ þ ; D11 ðsÞ D11 ðsÞ D13 ðsÞE16 ðsÞ F 14 ðsÞ ¼ E15 ðsÞ þ ; D11 ðsÞ F 13 ðsÞ ¼ E13 ðsÞ þ

i¼1

þ

D14 ðsÞE16 ðsÞ D15 ðsÞE16 ðsÞ ; F 16 ðsÞ ¼ E17 ðsÞ þ ; D11 ðsÞ D11 ðsÞ bD15 ðsÞE14 ðsÞ ; F 17 ðsÞ ¼ E18 ðsÞ þ D11 ðsÞs

M1 ðL; tÞ ¼ 

bL qHðtÞ; 2

bLa qHðtÞ; 2

N1 ðL; tÞ ¼ 0; Q 1 ðL; tÞ ¼

ð31aÞ

bL bLa qHðtÞ; M 1 ðL; tÞ ¼  qHðtÞ: 2 2

ð31bÞ

Here we assume that the external forces are applied instantly and kept constant with time.

ð24Þ

i¼1

2.4. Numerical method for inverse Laplace transform

where Ri(s)(i = 1, 2, . . . , 6) are six roots of the characteristics equation of Eq. (23); ci(s)(i = 1, 2, . . . , 6) are coefficients to be determined by boundary conditions; and

F 15 ðsÞ F 16 ðsÞ F 17 ðsÞ NTL ðx; sÞ  M TL ðx; sÞ  q F 14 ðsÞ F 14 ðsÞ F 14 ðsÞ F 18 ðsÞ  N0 : F 14 ðsÞs

N1CL ðx; sÞ ¼ 

ð25Þ

N X

K i Fðsi Þ;

ð32Þ

i¼1

where the values of Ki, si, and N are determined by a particular method. A simple implementation of Zakian’s algorithm is given by

6

X ci ðsÞ M 1L ðx; sÞ ¼ D12 ðsÞR2i ðsÞ þ D13 ðsÞ eRi ðsÞx D11 ðsÞ i¼1

þ M 1CL ðx; sÞ;

An explicit formula developed by Zakian [33,34] is used to evaluate the inverse Laplace transform numerically. According to the Zakian’s algorithm, the inverse Laplace transverse of F(s) can be written as

f ðtÞ ¼

By inserting Eq. (24) into Eq. (22), we have

f ðtÞ ¼ ð26Þ

where

5 a

2X i Re K i F ; t i¼1 t

ð33Þ

where constants ai and Ki are given by Zakian [33] and listed in Table 1. 2.5. Deflection

M 1CL ðx; sÞ ¼

1 D11 ðsÞ

 @N2 ðx; sÞ D12 ðsÞ 1CL 2 þ D13 ðsÞN1CL ðx; sÞ þ D14 ðsÞNTL ðx; sÞ @x  þ D15 ðsÞM TL ðx; sÞ þ D16 ðsÞN0

Once the resultant forces and interface stresses are obtained, the deflection and rotation of the RC beam can be calculated using constitutive equation Eq. (3). In the segment x < L, we have

ð27Þ

 D13 ðsÞ h1 R3i ðsÞ þ Ri ðsÞ þ Ri ðsÞ ci ðsÞeRi ðsÞx D11 ðsÞ D11 ðsÞ 2

/1 ðx; tÞ ¼

Z

ð34Þ

x

aL Z x

þ Q 1CL ðx; sÞ ð28Þ where

@M 1CL ðx; sÞ h1 @N1CL ðx; sÞ þ Q 1CL ðx; sÞ ¼ @x @x 2

M 1 ðxÞ ¼ bqxðx=2  ðL þ aÞÞ:

M 1 ðxÞ dx þ /1 ða  L; tÞ; D1   Q 1 ðxÞ w1 ðx; tÞ ¼  /1 ðx; tÞ dx þ w1 ðL  a; tÞ: B1 aL

6  X D12 ðsÞ i¼1

Q 1 ðxÞ ¼ bqðL þ a  xÞ;

Then the rotation and deflection in this segment can be calculated by

Substituting Eqs. (24) and (26) into Eq. (6) gives

Q 1L ðx; sÞ ¼

1 @Q 1CL ðx; sÞ q  : b @x s

N1 ðL; tÞ ¼ 0; Q 1 ðL; tÞ ¼ 

By solving Eq. (23), we can obtained the Laplace transform of the axial force in the RC beam as

ci ðsÞeRi ðsÞx þ N1CL ðx; sÞ;

 D13 ðsÞ 2 h1 Ri ðsÞ þ R2i ðsÞ ci ðsÞeRi ðsÞx D11 ðsÞ 2

For a simply supported beam shown in Fig. 1, the following boundary conditions can be used to determine ci(s)

D16 ðsÞE16 ðsÞ : F 18 ðsÞ ¼ D11 ðsÞ

N1L ðx; sÞ ¼

D11 ðsÞ

R4i ðsÞ þ

ð31Þ

F 15 ðsÞ ¼

6 X

6  X D12 ðsÞ

ð30Þ

ð29Þ

The Laplace transforms of the interface shear and normal stresses are then obtained as

ð35Þ

Table 1 Constants for ai and Ki used in the Zakian’s Method. i

ai

1 2 3 4 5

12.83767675 + j 12.22613209 + j 10.93430308 + j 8.776434715 + j 5.225453361 + j

Ki 1.666063445 5.012718792 8.409673116 11.92185389 15.72952905

36902.08210 + j 61277.02524  j 28916.56288 + j 4655.361138  j 118.7414011  j

196990.4257 95408.62551 18169.18531 1.901528642 141.3036911

C. Zhang, J. Wang / Composite Structures 93 (2011) 3200–3208

In the segment L 6 x 6 L, M1(x, t) and Q1(x, t) are determined in the above sections. Then

Z

M 1 ðx; t Þ /1 ðx; tÞ ¼ dx þ /1 ðL; tÞ; D1  ZLx  Q 1 ðx; tÞ w1 ðx; tÞ ¼  /1 ðx; tÞ dx þ w1 ðL; tÞ: B1 L

Z

ð36Þ

/1 ð0; tÞ ¼ 0;

ð37Þ

b2 ¼ 2G0 ;

ð43Þ

ð38Þ

It needs to emphasize again that the above analytical solution is a simplified model of the real FRP-strengthened RC beams. Besides the creep behavior of the RC beam, many other complicated behaviors of the RC beams, such as the nonlinear behavior cracking, are not considered in this study. A more complicated model considering these complicated factors will be established in our next study. 3. Validation of the analytical solution As verifications, in this section, a simply supported RC beam strengthened by an FRP plate shown in Fig. 1 is studied using the present analytical method and finite element analysis. As shown in Fig. 1, a uniformly distributed load q = 0.1 N/mm2 is applied to the RC beam. The geometry of the structure shown in Fig. 1 is given as: a = 300 mm, L = 1200 mm, h1 = 300 mm, h2 = 4 mm, h0 = 2 mm, and b = 200 mm. The material properties of the RC beam, the FRP plate, and the adhesive layer are listed in Table 2. By following a procedure similar to Delale and Endogen [26], we can relate the viscoelastic properties in SLS model to the conventional material properties such as shear modulus. To this end, we can consider a shear stress s = s0H(t) applied to SLS model shown in Fig. 3. Then

    @ 1 @ a1 þ s0 HðtÞ ¼ b1 þ b2 cðtÞ: @t 2 @t

ð39Þ

Applying Laplace transform on the both sides of Eq. (39) and after rearranging yields

1 1 a1

cL ðsÞ ¼ s0 1þ ; 2 b1 þ b2 s s

ð40Þ

where cL(s) is the Laplace transform of shear strain c(t). By applying inverse Laplace transform on the both sides of Eq. (40), we have

  b1 1 s b a b cðtÞ ¼ 0 a1 þ 1 1 2 eb2 t : 2 b1 b2

k1 ¼ 2G1 :

k2 ¼ k1 þ k2  k1 ¼ b2  2G1 ¼ 2ðG0  G1 Þ:

t 0 ¼ 5 days:

3.1. An FRP plate strengthened RC beam without prestressing Figs. 5 and 6 show the interface shear and normal stresses obtained by the present method and FEA for the FRP strengthened RC beam shown in Fig. 1. There is no prestress applied to the FRP plate. The stresses of finite element solution are extracted from the middle plane of the adhesive layer. It can be observed that the present model agrees very well with FEA except a small zone near the free edge of the adhesive layer. In this small zone, FEA shows that shear stress reduces to zero at the free edge; while the present analytical model shows that the shear stress reaches its maximum at the free edge. This discrepancy is caused by the inherent shortcoming of G–R model of the adhesive layer, which is essentially a two-parameter elastic foundation model. As illustrated in detail by Wang and Zhang [6], the two-parameter elastic foundation model of the adhesive layer cannot satisfy the zeroshear stress boundary condition at the free edge of the adhesive layer. To overcome this shortcoming, a three-parameter elastic

ð42Þ

b2

Table 2 Material properties of the FRP strengthened RC beam. RC beam

Adhesive layer

FRP plate

Young’s modulus (MPa) Poisson ratio

30,000 0.18

2000 0.3

100,000 0.35

ð44Þ

To obtain accurate finite element analysis solution, high order plane stress element (eight-node biquadratic element (CPS8)) is employed to mesh the structure shown in Fig. 1. This element can be used where stress concentrations exist and provide the best resolution of the stress gradients at the lowest cost. Due to symmetry, only half of the beam is modeled as shown in Fig. 4. Very fine mesh is used at the plate end to ensure interface stresses at the plate end to be accurately captured. The smallest size of elements placed within the adhesive layer is 0.25 mm, which is one fourth of the thickness of the adhesive layer. The viscoelastic behavior of adhesive layer is implemented in a commercially available finite element analysis software ABAQUS through a user subroutine UMAT. The user subroutine UMAT provides a way to program any mechanical constitutive model, which is not included in the ABAQUS’s material library. For a particular mechanical constitutive model, the stress and strain tensors and material Jacobian matrix will be updated at the end of the time increment.

Considering two extreme cases, t = 0 and t = 1, we have

Materials

ð45Þ

In the following examples, the initial shear modulus G0 is chosen as the elastic shear modulus of the adhesive shown in Table 1. The ultimate shear modulus and the retardation time are chosen as

ð41Þ

s0 1 b1 b s0 1 b1 b1

¼ 2; ¼ : ¼ ¼ 2 cð1Þ 2 ða1 þ 0Þ 2a1 cð0þ Þ 2 a1 þ b1 a1 b2

ð44Þ

Then

G1 ¼ G0 =3;

L

M 1 ð xÞ dx þ /1 ða  L; tÞ; D1 aL   Z L Q 1 ðxÞ  /1 ðx; tÞ dx: w1 ðL; tÞ ¼ B1 La /1 ðL; tÞ ¼

s0 s0 : ; G1 ¼ cð1Þ cð0þ Þ

Eq. (42) can be rewritten as

The integration constants in Eqs. (35) and (36) can be determined by the boundary and continuity conditions. Considering the symmetry of the structure shown in Fig. 1, the following boundary conditions and continuity conditions can be obtained

w1 ðL  a; tÞ ¼ 0;

Define the initial and ultimate shear moduli as

G0 ¼

x

3205

Fig. 4. Finite element model of simply supported RC beam.

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C. Zhang, J. Wang / Composite Structures 93 (2011) 3200–3208

2.5

0

50

100

150

200

Shear stress (MPa)

0

-0.4

FEA (t = 0.00001) FEA (t = 5)

-0.8

FEA (t = 10) Present (t = 0.00001) Present (t = 5)

-1.2

Max. normal stress (MPa)

Distance from the plate end (mm)

1 mm 0.5 mm

1.5 1 0.5

Time (Day) 0

Present (t = 10)

0

-1.6

5

10

15

20

25

30

Fig. 8. Maximum interface normal stress reducing with time (day).

Fig. 5. Redistribution of interface shear stress with time (day).

0.8

Normal stress (MPa)

2 mm

2

FEA (t = 0.00001)

0.6

FEA (t = 5) FEA (t = 10)

0.4

Present (t = 0.00001) Present (t = 5) Present (t = 10)

0.2

0 0

20

40

60

80

Distance from the plate end (mm)

-0.2

Fig. 9 shows that the axial force transferred from the RC beam to the FRP plate reduces with time. As a result, the bending moment in the RC beam increases with time, as shown in Fig. 10. This trend is more clearly demonstrated in Figs. 11 and 12, in which the axial force in the FRP plate and the bending moment of the RC beam at a distance of 25 mm from the FRP plate end are presented, respectively. This trend suggests that the strengthening effect of the FRP plate reduces with time due to the creep of the adhesive layer. Figs. 11 and 12 also show that more force can be transferred to the FRP plate more effectively if a thinner adhesive is used. However, this could increase the interface stress significantly, as shown in Figs. 7 and 8. Fig. 13 shows the creep deflection at the middle span of the beam. As shown in this figure, the creep deflection increases very fast in the first several days and slows down after a certain time. The thickness of the adhesive layer plays an important role in the deflection of the beam. When thicker adhesive layer is used,

Fig. 6. Redistribution of interface normal stress with time (day).

Axial forces (kN)

8 foundation model proposed by Wang and Zhang [35] should be used. As anticipated, significant interface stress redistributions with time are observed from both figures. Both the shear and normal interface stress concentrations are alleviated due to the creep deformation of the adhesive layer. Figs. 7 and 8 show that the maximum interface shear and normal stresses reduce with time. In these two figures, three thicknesses of the adhesive layer are considered. It can be seen that the maximum interface stresses are higher for thinner adhesive layer. The reducing in maximum interface stresses with time is desirable for preventing the interface debonding. But it may reduce the efficiency of the interface layer in transferring loading from the RC beam to the FRP plate. This can be seen more clearly in Figs. 9 and 10.

2

Distance from the plate end (mm) 0

50

100

150

Fig. 9. Variation of the axial force in the FRP plate with time (day).

Distance from the plate end (mm) 0

5

10

15

20

25

-7.5

30

0

Time (Day)

50

100

150

-8

-1

2 mm 1 mm 0.5 mm

-3 Fig. 7. Maximum interface shear stress reducing with time (day).

Moment (kN-m)

Max. shear stress (MPa)

4

0

0

-2

t = 0.00001 t=1 t=5 t = 10 t = 30

6

-8.5 -9 -9.5 -10

t = 0.00001 t=1 t=5 t = 10 t = 30

-10.5 Fig. 10. Variation of the moment in the RC beam with time (day).

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C. Zhang, J. Wang / Composite Structures 93 (2011) 3200–3208

7 2 mm 1 mm

Axial forces (kN)

6

0.5 mm 5

4

Time (Day) 3 0

5

10

15

20

25

30

duces much higher stress concentration. The maximum interface shear and normal stresses in the prestressed case are more than twice as high as those in the unprestressed case. Fig. 16 shows the axial force in the FRP plate. Due to pretensioned force, the axial force in the FRP plate is much higher than in previous case without prestress force (Fig. 9). Consequently, the moment in the RC beam is lower than in the previous case of without prestension force (Fig. 10), as demonstrated in Fig. 17. Fig. 18 shows the creep deflection at the middle span of the beam. Similar to the previous case (Fig. 13), higher instant deflection and creep deformation are observed for the thicker adhesive layer. Fig. 18 also clearly demonstrates the effect of the prestension force in reducing the deflection of the RC beam. Compared to the un-prestressed case

Fig. 11. Axial forces of the FRP plate at 25 mm from the FRP plate end varying with time (day).

Distance from the plate end (mm) 0

Time (Day) 5

10

15

20

Moment (kN-m)

-7.8 -7.9

25 30 2 1 0.5 mm

-8

Shear stresses (MPa)

0

50

100

150

200

0

-7.7

-1 t = 0.00001 t=1

-2

t=5 t = 10

-3

t = 30

-8.1 -4

-8.2 Fig. 12. Moments in the RC beam at 25 mm from FRP plate end varying with time (day).

Fig. 14. Redistribution of interface shear stress with time (day) in the prestressed FRP plate strengthened RC beam.

3.2. Prestressed FRP plate strengthened RC beam In this section, the same beam considered in the above section is examined with a pretension load of 10 kN applied to the FRP plate. Figs. 14 and 15 show the interface stress redistributions in the adhesive layer with time. Similar trend as in the case without prestress can be observed in these two figures. The creep deformation of the adhesive layer can actually alleviate the stress concentration within the adhesive layer. Compared with the case without prestress (Figs. 5 and 6), prestress applied to the FRP plate intro-

Normal stresses (MPa)

2 the instant deflection due to elastic deformation and the creep deflection due to the viscoelastic behavior of the adhesive are higher too, as shown in Fig. 13.

t=1 t=5

0.8

t = 10 t = 30

0.4 0

0

20

40

60

80

100

Distance from the plate end (mm)

Fig. 15. Redistribution of interface normal stress with time (day) in the prestressed FRP plate strengthened RC beam.

18 10

20 Time (day)

30

-1.41948

2 mm 1 mm 0.5 mm

Fig. 13. Deflection at the mid-span increasing with time.

15

Axial forces (kN)

0 Deflection (mm)

t = 0.00001

1.2

-0.4

-1.41664

-1.42231

1.6

t = 0.00001 t=1 t=5 t = 10 t = 30

12 9 6 3

Distance from the plate end (mm) 0 0

50

100

150

200

Fig. 16. Variation of the axial force in the prestressed FRP plate with time (day).

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C. Zhang, J. Wang / Composite Structures 93 (2011) 3200–3208

References

Distance from the plate end (mm) 0

50

100

150

Moment (kN-m)

-7 -7.5 t = 0.00001 -8

t=1 t=5

-8.5

t = 10 t = 30

-9 Fig. 17. Variation of the moment in the RC beam with time (day).

Time (day) -1.316

Deflection (mm)

0

10

20

30

-1.3165 -1.317 -1.3175 2 mm -1.318

1 mm 0.5 mm

-1.3185 Fig. 18. Deflection at the mid-span increasing with time in the prestressed FRP plate strengthened RC beam.

(Fig. 13), the instant deflection and creep deformation at the middle span for the prestressed case are lower. 4. Conclusion A viscoelastic analytical solution is developed to investigate the long term behavior of FRP plate strengthened RC beams. In this solution, only the adhesive layer is modeled as a viscoelastic material using Standard Linear Solid model. Closed-form solution of interface stresses, resultant forces, the deflections and rotations in the RC beam and the FRP plate, are obtained in Laplace transformed shape. An efficient numerical method, Zakian’s algorithm, is used to conduct inverse Laplace transform. The prestressed force in the FRP plate is also considered in this solution. To verify the present analytical solution, the analytical solution is compared with finite element analysis using a subroutine UMAT. Very good agreement has been achieved by these two methods. The redistributions of interfacial stresses, resultant beam forces, and creep deflection of the structure can be predicted by the present solution easily. As a simplified model, the analytical solution obtained in this study can be further improved by considering the creep behaviors of the FRP plate and the RC beams, and the nonlinear behavior of the RC beam. Nevertheless, this study provides an efficient analytical tool to evaluate the long-term behavior of FRP plate strengthened RC beams. Acknowledgement This study was supported by the National Science Foundation (Grant No.: CMMI-0927938 under program director Dr. Grace Hsuan).

[1] Hollaway LC. A review of the present and future utilization of FRP composites in the civil infrastructure with reference to their important in-service properties. Constr Build Mater 2010;24:2419–45. [2] ACI Committee 440. Design and construction of externally bonded FRP systems for strengthening concrete structures (ACI 440.2R-02). Farmington Hills, MI: American Concrete Institute; 2002. p. 45. [3] De Lorenzis L, Nanni A. Characterization of FRP rods as near-surface mounted reinforcement. J Compos Constr 2001;5:114–21. [4] Triantafillou TC, Deskovic N, Deuring M. Strengthening of concrete structures with prestressed fiber reinforced plastic sheets. ACI Struct J 1991;89:235–44. [5] Smith ST, Teng JG. Interfacial stresses in plated RC beams. Eng Struct 2001;23:857–71. [6] Wang J, Zhang C. Three-parameter, elastic foundation model for analysis of adhesively bonded joints. Int J Adhes Adhes 2009;29:495–502. [7] Brinson HF. Viscoelastic constitutive modelling of adhesives. Composites 1982;13:377–82. [8] Mirman BA, Knecht S. Creep strains in an elongated bond layer. IEEE Trans Compon Hybr Manuf Technol 1990;13:914–28. [9] Savoia M, Ferracuti B, Mazzotti C. Long-term creep deformation of FRP-plated r/c tensile members. J Compos Constr 2005;9:63–72. [10] Ferrier E, Hamelin P. Influence of time–temperature loading on carbon epoxy reinforcement for concrete structure. ACI Publications SP-188-44; 1999. p. 491–500. [11] Dean G. Modeling non-linear creep behavior of an epoxy adhesive. Int J Adhes Adhes 2007;27:636–46. [12] Meshgin P, Choi KK, Reda Taha MM. Experimental and analytical investigations of creep of epoxy adhesive at the concrete-FRP interfaces. Int J Adhes Adhes 2009;29:56–66. [13] Diab H, Wu Z. Nonlinear constitutive model for time-dependent behavior of FRP-concrete interface. Compos Sci Technol 2007;67:2323–33. [14] Diab H, Wu Z, Iwashita K. Short and long term bond performance of prestressed FRP sheet anchorages. Eng Struct 2009;31:1241–9. [15] Choi K-K, Meshgin P, Reda Taha MM. Shear creep of epoxy at the concrete-FRP interfaces. Composites Part B 2007;38:772–80. [16] Ascione F, Berardi VP, Feo L, Giordani A. An experimental study on the longterm behavior of CFRP pultruded laminates suitable to concrete structures rehabilitation. Compos Part B: Eng 2008;39:1147–50. [17] Ferrier E, Michel L, Jurkiewiez B, Hamelin P. Creep behavior of adhesives used for external FRP strengthening of RC structures. Constr Build Mater 2011;25:461–7. [18] Choi K-K, Reda Taha MM, Masia MJ, Shrive PL, Shrive NG. Numerical investigation of creep effects on FRP-strengthened RC beams. J Compos Constr 2010;14:812–22. [19] Roberts TM, Haji-Kazemi H. Theoretical study of the behaviour of reinforced concrete beams strengthened by externally bounded steel plates. Proc Inst Civil Eng (Part 2) 1989;87:39–55. [20] Roberts TM. Shear and normal stresses in adhesive joints. ASCE J Eng Mech 1989;115:2460–79. [21] Lau KT, Dutta PK, Zhou LM, Liu D. Mechanics of bonds in an FRP bonded concrete beam. Compos Part B: Eng 2001;32:491–502. [22] 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;95:42–52. [23] Wang J. Mechanics and fracture of hybrid material interface bound. Ph.D. Thesis. OH: Department of Civil Engineering, The University of Akron; 2003. [24] Yang J, Teng JG, Chen JF. Interfacial stresses in soffit-plated reinforced concrete beams. Proc Inst Civil Eng: Struct Build 2004;157:77–89. [25] Goland M, Reissner E. The stresses in cemented joints. J Appl Mech 1944;66:A17–27. [26] Delale F, Erdogan F. Viscoelastic analysis of adhesively bonded joint. J Appl Mech 1981;48:331–8. [27] Wight RG, Green MF, Erki MA. Prestressed FRP sheets for poststrengthening reinforced concrete beams. J Compos Constr 2001;5:214–20. [28] Al-Emrani M, Kliger R. Analysis of interfacial shear stresses in beams strengthened with bonded prestressed laminates. Composites: Part B 2006;37:265–72. [29] Xue W, Tan Y, Zeng L. Flexural response predictions of reinforced concrete beams strengthened with prestressed CFRP plates. Compos Struct 2010;92:612–22. [30] Nordin H, Taljsten B. Concrete beams strengthened with prestressed near surface mounted CFRP. J Compos Constr 2006;10:60–8. [31] Quantrill RJ, Hollaway LC. The flexural rehabilitation of reinforced concrete beams by the use of prestressed advanced composite plates. Compos Sci Technol 1998;58:1259–75. [32] Wu Z, Diab HM. Constitutive model for time-dependent behavior of FRPconcrete interface. J Compos Constr 2007;11:477–86. [33] Zakian V. Rational approximation to transform function matrix of distributed system. Electr Lett 1970;6:474–6. [34] Zakian V. Optimization of numerical inversion of Laplace transform. Electr Lett 1970;6:677–9. [35] Wang J, Zhang C. A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate. Int J Solids Struct 2010;47:998–1006.