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Modeling SRP-concrete interfacial bond behavior and strength
Engineering Structures 187 (2019) 220–230
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Modeling SRP-concrete interfacial bond behavior and strength a
b
a
b,⁎
F. Ascione , M. Lamberti , A. Napoli , A.G. Razaqpur , R. Realfonzo a b
a
T
Dept. of Civil Engineering, University of Salerno, Italy College of Environmental Science and Engineering, Nankai University, Tianjin, China
ARTICLE INFO
ABSTRACT
Keywords: Steel reinforced polymer Concrete Bond-slip models Pull-out tests Bond strength
Relatively recently a steel fabric/laminate has been proposed for externally strengthening concrete structures, using a polymeric resin. The steel fiber-polymer composite system is termed Steel Reinforced Polymer (SRP). To determine the ultimate load capacity of an SRP retrofitted concrete structure, one must accurately predict the SRP-concrete interface debonding load, which requires a robust local bond-slip model. Many design guidelines recommend mode II interfacial fracture energy limit as the failure criterion for FRP-concrete interface. For SRP strengthened members, a suitable constitutive law and failure criterion have not been established yet. Consequently, in this study the applicability of five existing bond-slip interface models for FRP-concrete interface to SRP-concrete interface is examined. The models’ parameters are calibrated for SRP-concrete interface using an experimental database by the present authors and compared with the values suggested by the original authors for FRP-concrete interface. The database involves results of tests on concrete prisms bonded to SRP strips. The experimental interfacial bond-slip relationship for the former interface is observed to have a more precipitous descent after the peak stress than predicted by the existing models; consequently, a new model is proposed here to capture this phenomenon. All the models are calibrated using a classical technique which minimizes the difference between the measured and computed interfacial shear stress values at different slip levels. The results indicate that all the models predict relatively well the slope of the ascending branch of the shear stress-slip curve, but they give substantially different values for the maximum shear stress attainable and noticeably different descending branch profiles. Among these, overall, the proposed model is in relatively better agreement with the experimental results.
1. Introduction Relatively recently, new retrofit techniques for concrete structures has been developed to externally strengthen/repair deficient structures in a manner akin to applying the familiar Fibre Reinforced Polymers (FRP) laminates [1–15]. The new techniques involve natural [16–28] or artificial [29–37] fibers. Among artificial fibers, great attention has been devoted to continuous High Tensile Strength Steel (HTSS) micro wires, twisted into small diameter cords or strands that are laid in a polymer matrix to form a unidirectional or bidirectional composite laminate, termed Steel-wire Reinforced Polymer (SRP). The SRP can be externally bonded to a substrate, similarly to FRP, via wet lay-up, using either epoxy or polyester resin. One of the salient advantages of SRP over FRP is the high elastic modulus of steel wires versus glass, aramid or low modulus carbon fibres. Furthermore, if brittle interfacial debonding can be averted, SRP retrofitted structures would exhibit relatively ductile behaviour and higher energy
dissipation at failure. Understanding of the behaviour of SRP-concrete interface and controlling the parameters that govern its behaviour are key to the prevention of unexpected separation of the SRP from the substrate. Among these, the interfacial bond-slip response plays a crucial role insofar as interfacial debonding initiation and ultimate failure are concerned. The quantification of the interfacial response requires a reliable local bond-slip constitutive law or model. In structures strengthened in flexure, the stress state at the SRPconcrete interface is similar to the one in single-lap joint shear (SLJS) test. In SLJS test, a laminate strip is bonded to a concrete prism and the bonded strip is subjected to direct tension [2]. This similarity can be exploited to develop a bond-slip model for SRP retrofitted beams and other similar components using results of SLJS test. The latter test not only provides the ultimate load of the SRP-concrete interface, but it also furnishes a local bond-slip response. In the present discussion, the term “interface” refers to the adhesive and a thin layer of the adjoining
Corresponding author. E-mail addresses: [email protected] (F. Ascione), [email protected] (M. Lamberti), [email protected] (A. Napoli), [email protected] (A.G. Razaqpur), [email protected] (R. Realfonzo). ⁎
https://doi.org/10.1016/j.engstruct.2019.02.050 Received 10 July 2018; Received in revised form 14 January 2019; Accepted 22 February 2019 0141-0296/ © 2019 Published by Elsevier Ltd.
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F. Ascione, et al.
concrete at the SRP-concrete joint, rather than an actual physical joint. It is this thin concrete layer that is assumed to undergo slip between the SRP laminate, and the concrete prism body. Local bond-slip response from SLJS test is commonly determined either from the axial strain distribution of the SRP laminate, obtained from readings of closely spaced strain gauges [4] or from load-displacement (slip at loaded end) curves [9]. Each method has its advantage and disadvantage. Specifically, in the first method the shear stress at a specified location along the interface can be determined by principles of mechanics while the corresponding slip can be computed by numerically integrating the measured axial strains of the laminate. The latter method is simple to apply but may not always yield an accurate local bond-slip response in some cases due to the abrupt changes in the measured local strain values caused by the discrete nature of concrete cracks, the heterogeneity of concrete and the unevenness of the underside of the partially debonded SRP laminate. In the second method, the local bond-slip response is determined indirectly from the load-slip curve, but it can be shown that quite different local bond-slip curves will lead to similar load-displacement curves. In the current investigation the former procedure will be used. The authors are not aware of any previous studies concerning the bond-slip model for SRP retrofitted concrete elements. Therefore, the objective of this paper is to derive such a model. Towards this end, first five existing bond-slip interface models [38–43], originally developed for FRP-concrete interface, are calibrated for application to SRP-concrete interfaces using a database of SLJS test results on SRP retrofitted concrete prisms performed previously by the authors [44,45]. A similar attempt was made in [46] where the reliability and adaptability of the FRP–concrete bond–slip laws to predict the structural behavior of RC beams externally strengthened in flexure with Steel Reinforced Grout (SRG) or SRP was evaluated. As observed in [44,45], SRP-concrete interface tends to exhibit a more precipitous descent in its bond-slip curve than the FRP-concrete interface curve. Consequently, to capture this characteristic, a new bond-slip model is proposed here and its results are compared with those of the calibrated existing models. All the models are characterized by curves possessing an ascending and a descending branch, with linear and/or non-linear variation. Each model is calibrated using the mean squared error (MSE) technique, which minimizes the difference between the experimental and corresponding computed values of interfacial shear stress at different slip values. To gauge the relative accuracy of the calibrated models, the experimentally observed maximum debonding stress values are compared with their corresponding compute or predicted values by using error norms based on the MSE technique
bonded length Load
(a)
150 SRP
(b)
200
Load
bf
400 Fig. 1. Test specimen geometry (a) Elevation, (b) Plan.
each strip type are reported in Table 1, where the symbols γ, ts, fs,u, Es, and εs,u designate the mass density, equivalent design thickness, average tensile strength, Young’s modulus, and ultimate strain, respectively. The width of the steel strip, bf, was fixed equal to 100 mm in all the tests. Furthermore, two different concrete surface preparations, viz. grinded and bush hammered, were used. The bush hammered concrete surface was heavily roughened to expose the aggregates while the grinded surface was mildly sanded to remove laitance. Finally, to evaluate the effective transfer length for each strip density, five bonded lengths were examined: 100, 150, 200, 300 and 350 mm. However, in the current analyses the results for bonded lengths of 100 mm and 150 mm are not used because these lengths were found to be less than the minimum required bonded length, resulting in rather low failure loads. The relevant mechanical properties of the two component epoxy resin, as reported by the manufacturer, are shown in Table 2, where the symbols fm,u, τm,u and Em,c designate the tensile strength, shear strength and secant modulus in compression, respectively. Fig. 2 schematically illustrates the typical test set-up. Tests were performed using a Schenck universal testing machine with 630 kN capacity. Tests were conducted by holding the SRP-concrete block assembly firmly in place between the top and bottom machine platens by means of a stiff steel jig while the steel strip free end was gripped by the machine jaws and subjected to tension. The load was applied via displacement control at the rate of 0.01 mm/s. To measure the SRP strip axial displacement and strain, the instrumentation illustrated in Fig. 2 was used. The slip at the loaded end was measured in two ways: first, two potentiometers, designated as LDT 1 and LDT 2, were placed to the left and right of the strip to measure its axial displacement at the location of the start of the bonded length; second, the same displacement was also indirectly measured by two non-contact laser sensors, designated as Laser 1 and Laser 2. The overall extension of the loaded end of the strip was monitored via the vertical displacement of the testing machine crosshead. To measure the SRP longitudinal strain variation along and across the bonded length, several electrical-resistance strain gauges were attached to the outer face of its bonded length, with the actual number of gauges being dependent on the bonded length Lf. As stated earlier, for the purpose of this study, the bonded lengths were 200, 250, 300 and 350 mm, which are greater than the corresponding effective bond length as demonstrated in [44]. In Fig. 3, each strain gauge designation and its precise location are depicted. On specimens with Lf = 200 mm, three gauges were applied along the centerline of the strip while another three gauges each were placed to the left and the right of the centerline, for a total of nine gauges. For bonded lengths greater than 200 mm, only the number of gauges along the centerline is increased (4 for 250 mm, 6 for 300 mm and 7 for 350 mm) while the side gauges
2. Experimental database A database containing the results of 99 SLJS tests (part of a wider experimental program), involving uniaxial steel wire strips, epoxy bonded to concrete prisms (SRP system), is used. The tests were performed by the authors in a previous experimental program and reported in [44,45], but for ease of reference and completeness, the testing program and its results will be recapped. Each prism had 200 × 150 mm2 cross-section and 400 mm length as depicted in Fig. 1. The concrete compressive strength (fcm,28) was obtained by testing at the age of 28 days concrete cubes with side length of 150 mm. The prisms are divided into two strength groups: the first group made of normal strength concrete (NSC) and the second group made of higher strength concrete (HSC). The NSC concrete strength varied between 13 and 26 MPa while that of HSC varied between 39 and 45 MPa. Note, in the current context the terms NSC and HSC are used to simply separate the two concrete classes tested here rather than referring to established concrete strength classifications. Four different steel strip densities were tested [47,48]: low density (L), low-medium density (LM), medium density (M) and high density (H). The relevant properties of 221
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F. Ascione, et al.
Table 1 Properties of dry steel fabric. Tape Density
Density γ (g/m2)
Thickness ts (mm)
Strength, fs,u (MPa)
Elastic ModulusEs (GPa)
Ultimate Strain εs,u (%)
Low (L) Low-Medium (LM) Medium (M) High (H)
670 1200 2000 3300
0.084 0.164 0.254 0.381
3191.0 3138.3 3085.7 3064.6
182.1 182.7 183.4 182.2
2.19 2.14 2.09 2.09
the unloaded end of the SRP strip, and the axis x running along its length, is adopted as depicted in Fig. 3. The position of i-th strain gauge is denoted by x i (with i = 1, 2, ....,t ), where t is the total number of strain gauges for the given bonded length considered, while the interval between two consecutive strain gauges is represented by x = xi + 1 x . In order to evaluate the local bond-slip relationship, the following assumptions are made:
Table 2 Properties of epoxy mineral adhesive (Geolite® Gel). Tensile Strength fm,u (MPa)
Shear Strength τm,u (MPa)
Elastic Modulus Em,c (MPa)
> 14
> 12
> 5300
were kept the same as in the specimen with Lf = 200 mm. The test data originally reported in [44,45], are summarized in Tables 3 and 4 for the two groups of specimens with different surface preparation. The symbol fcm represents the concrete compressive strength, Fd the debonding initiation load, Fmax the maximum load resisted by the specimen during the test, sd and smax, the slip values associated with Fd and Fmax, respectively, and su, the slip at failure. The first column in the tables identifies each specimen. The identifier has the generic form “xY-bf-nf#z” where: “x”, “Y” , “bf”, “nf” and “#z” denote, respectively, the SRP provided bonded length in centimeter (20, 25, 30 or 35), SRP density (L = low; LM = low-medium; M = medium; H = high), the width of the steel strip in centimeter, the number of steel fiber layers and the order of the replicate test specimen in a group of nominally identical specimens. Note: bf and nf varied during the whole experimental program although for all tests here presented they assume the value of 100 mm and 1, respectively; the data for the HSC concrete specimens were also excluded due to the relatively small number of these specimens;
- no slip between the steel strip and concrete at x = 0 (Fig. 3); - strains in concrete are negligible relative to the steel strip ( c = 0 ); - linear variation of strain in the SRP strip between two consecutive strain gauges. The average shear stress, calculated using av (x )
=
Es ts (
i+1
sav (x ) =
s (x ) = s (x i ) +
(2)
( i+1 (x i + 1
i)
xi )
xi )2
(x 2
BAR 2
BAR 2
SRP
LVDT 2 LS 2 Steel Plate
Steel Plate
i
× (x
SRP
Steel Plate LVDT 1-2 Concrete
bw
200
150 Fig. 2. Schematics of the Test Set-up.
222
BAR 3
LS 1-2 bonded length
LS 1
+
xi )
(3)
The interfacial fracture energy, Gexp , is computed using the integral in Eq. (4), which is obtained by considering the constitutive relation of steel (Ns = Es As s ) and concrete (Nc = Ec Ac c ) and the equilibrium of horizontal forces acting on the SRP
400
400
bonded length
(1)
s (x i + 1) + s (x i ) 2
SRP
Concrete
i)
x
The local slip at a point x along the bonded length is evaluated using
The strain gauges bonded on the SRP strip allowed for recording the strain distribution along the interface. Using the strain distribution profile, the local bond-slip relation of SRP-concrete interface is evaluated applying the following procedure. A reference coordinate system, composed of the origin O, located at
LVDT 1
within an interval of length x is
where i + 1 and i are the strain values recorded at the (i + 1) and i gauges, respectively, while, Es and ts are the elastic modulus and the thickness of the SRP strip, respectively. The average local slip, sav (x ) , evaluated at midpoint of two consecutive strain gauges is calculated using
3. Bond-slip experimental evaluation
BAR 1
av (x ) ,
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x (mm)
x (mm)
x (mm)
x (mm)
185 170 140 110 80 50
235 220 190 160 130 100
285 270 240 210 180 150
335 320 290 260 230 200
55
105
155
60
110
15
65 20
b)
a)
c)
d)
Fig. 3. Strain gauge locations on steel strip: (a) bond length of 200 mm, (b) bond length of 250 mm, (c) bond length of 300 mm and (d) bond length of 350 mm. Table 3 Database of pull tests: grinded surface. Test
fcm,28 (MPa)
Fd (kN)
Fmax (kN)
σmax (MPa)
sd (mm)
smax (mm)
su (mm)
20L#1 20L#2 20L#3 20L#4 30L#1 30L#2 30L#3 30L#4 20M#1 20M#2 20M#3 20M#4 20M#5 30M#1 30M#2 30M#3 30M#4 30M#5 30H#1 30H#2 30H#3 30H#4 35H#1 35HD#2 35HD#3 35HD#4
26 26 26 22 19 26 26 22 26 26 19 22 13 19 19 19 22 22 19 26 26 22 26 26 22 13
9.09 12.16 11.35 12.72 13.57 9.36 11.15 14.50 14.28 18.71 20.75 24.06 11.36 18.47 19.24 17.50 20.02 22.06 20.95 25.33 18.50 25.55 12.12 13.85 25.49 20.96
9.78 12.82 11.90 13.77 14.72 10.83 12.95 16.78 14.28 18.71 21.05 24.06 12.97 19.46 19.24 20.93 21.58 23.79 21.11 25.33 19.22 26.09 16.19 16.63 32.65 22.04
1164.4 1525.7 1417.1 1639.1 1752.4 1289.5 1542.2 1998.1 562.4 736.5 828.7 947.4 510.8 766.2 757.6 824.0 849.8 936.5 554.0 664.9 504.5 684.7 424.9 436.4 857.1 578.5
0.27 0.47 0.45 0.58 0.62 0.28 0.54 0.64 0.49 0.58 0.25 0.96 0.12 0.57 0.29 0.39 0.71 0.48 0.49 0.32 – 0.06 1.06 0.80 0.30 0.33
0.61 0.54 0.59 1.52 1.92 0.78 2.13 2.45 0.49 0.58 0.69 0.96 0.65 0.95 0.29 0.97 1.34 0.88 0.51 0.32 0.25 0.65 2.19 1.91 1.02 0.94
1.08 1.27 1.46 1.60 2.34 1.58 2.34 2.45 0.89 0.97 0.69 1.34 0.65 1.33 0.87 0.97 1.34 1.17 0.98 0.84 1.29 0.85 2.22 1.91 1.33 1.01
Gexp =
0
(s ) ds =
Lf 0
Es ts (
s
c )d s
=
Lf 0
Es ts s d s =
the peak shear stress. The ascending and descending branches may be linear or nonlinear. Cosenza et al. [38] considered a non-linear ascending branch and a linear descending branch, Monti et al. in [39] proposed both branches to be linear (bilinear law) while Dai et al. [40–41] and Ferracuti et al. [42–43] assumed both branches to be nonlinear. Here a new model is proposed by the authors, which is characterized by nonlinear ascending and descending branches, and which is believed to predict better the overall shear-slip response compared to the existing models. In the following, a brief description of the above models is given, and their associated unknown parameters are identified. These parameters must be calibrated using experimental data together with a suitable error minimization technique. The first interface model considered is the one proposed by Cosenza et al. [38], which is characterized by a non-linear ascending branch and a linear descending one. The mathematical expressions for the two branches are given by Eqs. (5a) and (5b), respectively.
2 Fmax 2Es ts bf2
(4) 4. Local bond-slip interface models and their calibration To numerically capture the interaction between concrete and its internally, or externally, bonded reinforcement, an analytical bond-slip constitutive model is required. Hence, a number of bond-slip models exists for both reinforcing situations, but due to the similarity between externally bonded FRP-concrete interface and SRP-concrete interface, in the following, the experimental data in Tables 3 and 4 will be compared with the computed values obtained using some popular bond-slip models in the literature for FRP-concrete interfaces [38–42]. It is important to mention that most models are characterized by an ascending branch, followed by a descending or softening branch after 223
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Table 4 Database of pull tests: bush hammered surface specimens. Test
fcm,28 (MPa)
Fd (kN)
Fmax (kN)
σmax (MPa)
sd (mm)
sm (mm)
su (mm)
20LD#1 20LD#2 30LD#3 30LD#4 25LMD#1 25LMD#2 25LMD#3 20MD#1 20MD#2 20MD#3 25MD#1 25MD#2 30MD#4 30MD#5 30MD#6 30HD#3 30HD#4 30HD#5 35HD#1 35HD#2 35HD#3
26 26 26 26 13 13 13 26 26 22 13 13 22 22 13 26 22 13 26 26 22
7.53 10.85 12.04 17.14 24.16 20.73 25.37 22.02 19.83 20.22 26.39 25.71 20.73 22.34 16.00 17.44 24.06 19.68 20.51 21.82 26.15
14.66 14.74 14.15 17.36 25.61 24.16 26.86 22.02 19.83 21.23 26.86 25.71 21.21 22.60 16.45 22.93 28.33 20.33 30.16 21.82 27.34
1745.3 1754.8 1683.9 2066.5 1524.5 1438.3 1598.9 867.0 780.7 835.7 1057.6 1070.0 834.9 889.6 647.5 601.9 743.6 533.7 791.5 572.8 717.5
0.34 0.50 2.02 0.60 0.53 0.43 0.99 2.55 – 0.57 0.43 – 1.41 0.54 0.56 0.49 0.28 0.76 1.09 0.48 0.65
1.96 1.49 3.81 1.95 1.02 1.17 1.16 2.55 – 0.76 0.74 – 2.20 0.94 1.02 0.95 0.93 0.90 1.51 0.48 1.13
2.07 1.52 3.95 2.53 1.16 1.17 1.59 2.59 – 0.85 1.60 – 2.20 1.18 1.02 0.95 0.93 0.90 1.52 1.04 1.23
(s ) =
max
s smax
(s ) =
max
1
if
p
s
0
s
smax smax
smax
if
In the latter equation, B is an interfacial material constant. For this model, the set of unknown parameters is { max, B} . The fifth non-linear bond-slip interface model, proposed by Ferracuti et al. [42,43], is represented by Eq. (9). Once again, a single expression is presented for capturing both branches.
(5a)
smax
s
su
(5b)
In the above equations, and all subsequent equations, unless specified differently, (s ) is the interfacial shear stress, s is the slip, max and smax are the maximum shear stress and the corresponding slip, su is the ultimate slip, and p are experimentally calibrated parameters. Thus, for the calibration of this model, the unknown parameters are { max, smax , , p} . Note, slip su can be solved for by letting (s ) equal to zero in Eq. (5b). Next, the bi-linear interface model, originally proposed by Monti et al. [39] for the case of FRP-concrete system, is presented. Its ascending branch is characterized by Eq. (6a) and its descending branch by Eq. (6b).
(s ) = (s ) =
max
max
s
if 0
smax su su
s
s if
smax
smax
smax
(s ) =
su
(6b)
max ,
(s ) =
(s ) =
max
s smax
max e
if
(s smax )
0
if
s
smax
smax
s
max (e
( Bs )
e(
2Bs ) )
(s ) =
max
s smax
(s ) =
max
smax s
(¯ th (s )
min
( )
n s smax
K
if
0
s
(9)
if
smax
smax
s
¯ exp (s )) 2
su
(10a) (10b)
(11)
k=1
(7b)
In Eq. (11), and represent the theoretical and the experimental shear stress, respectively, normalized with respect to the square root of fcm,28 [11], while N is the total number of tests (twentysix for grinded surface and twenty-one for bush-hammered surface). Furthermore, for each i-th test, the values corresponding to increments of s = 0.1 mm (considering that the tests were performed in displacement control at the rate of 0.01 mm/s) were selected from s = 0 to s = su. For bonded length of 300 or 350 mm, the number of curves per each test is equal to six because between the seven strain gauges applied there were six intervals and a curve was constructed for each interval. For the 200 and 250 mm bonded lengths, the total number of curves per
¯ th (s )
It is important to mention that the expression for the ascending branch is the same as that introduced in [38]. In Eq. (7) and the two parameters that govern the shape of the ascending and descending branch, respectively. The parameter can only assume a value between 0 and 1. The set of unknown parameters in the latter case is { max, smax , , } . The fourth model was proposed by Dai et al. [41] where a single expression is presented for both branches as indicated by Eq. (8).
(s ) = 4
1) +
N
(7a)
su
n
1 and K > 1 are shape defining parameters In Eqs. (10) 0 while the complete set of unknown parameters is { max, smax , , K } . Thus, the proposed model has the same number of unknown parameters as the Cosenza et al. and one of Dai et al. models. The set of unknown parameters for the six interface models can be calibrated by applying the Mean Squared Error (MSE) minimization technique, which is expressed as Eq. (11).
The set of unknown parameters for the calibration of this model is smax , su} . The third model considered was proposed by Dai et al. [40], and its ascending and descending branches are represented by Eqs. (7a) and (7b), respectively.
{
s smax (n
In Eq. (9) n is a parameter mainly governing the softening branch and must be greater than 2. In this case, the set of unknown parameters is { max, smax , n} . The last model considered is proposed by the authors. It treats both the ascending and descending branches to be non-linear and are characterized by Eqs. (10a) and (10b), respectively.
(6a)
s
max
(8) 224
¯ exp (s )
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2.50
2.00
1.50
1.00
0.50
0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.35 s [mm]
0.40
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
a) grinded surface specimens 2.50
2.00
1.50
1.00
0.50
0.00 0.00
0.05
0.10
0.15
0.20
0.25 0.30 s [mm]
0.35
0.40
b) bush hammered surface specimens Fig. 4. Experimental bond-slip curve for SRP-to-concrete.
each test is three and four, respectively. Figs. 4(a) and 4(b) show the experimental shear stress-slip relationships for the specimens with grinded and bush-hammered surface finish, respectively. In these graphs, the precipitous drop in the shear stress after the peak stress can be observed, especially for specimens with high interfacial shear strength. As mentioned earlier, the proposed model was motivated by the latter observation. The total number of ( , s ) points used were 15,986 and 13,239 in the case of grinded and bush-hammered surface treatment, respectively. To reduce the number of unknown parameters, the following constraint was imposed: the average experimental interface fracture enav ergy, Gexp , is equal to the theoretically computed fracture energy, Gth , i.e. av Gexp = Gth
Equality of the two fracture energy values in Eq. (12) is assumed if they differ by no more than 0.2%. The theoretical fracture energy is computed using
Gth =
(s ) ds
0
(13)
After some algebraic manipulations, it can be verified that Eq. (13) leads to the following expressions for the six interface constitutive laws described earlier where the normalization with respect to the square root of fcm,28 has been introduced
(12) 225
th G¯Cosenza et
al
th G¯Monti et
=
al
= ¯max
¯max su 2
smax s + max 1+ 2p
(14) (15)
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Table 5 Minimization process: values of unknown parameters. Authors
Parameters
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Note: ¯max =
max /
Surface
{smax , , p},{¯max } {¯max , smax },{su} {smax , , },{¯max } {B},{¯max } {¯max , n},{smax } {smax , , K },{¯max }
Grinded
Bush Hammered
{0.024, 1.000, 0.056},{0.475} {0.475, 0.024},{0.444} {0.027, 1.000, 5.684},{0.555} {10.678},{0.562} {0.563, 2.595},{0.044} {0.045, 0.907, 1.377},{0.735}
{0.035, 1.000, 0.076},{0.616} {0.616, 0.035},{0.494} {0.038, 1.000, 4.868},{0.678} {8.616},{0.655} {0.664, 2.643},{0.057} {0.052, 1.000, 1.348},{0.868}
fcm,28 with fcm,28 in MPa; smax and su in mm.
1.00 0.90 0.80 0.70
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Ascione et al. (pp)
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
a) grinded surface specimens 1.00 0.90 0.80 Cosenza et al. [38]
0.70
Monti et al. [39]
0.60
Dai et al. [40] Dai et al. [41]
0.50
Ferracuti et al. [42]
0.40
Ascione et al. (pp)
0.30 0.20
0.10 0.00 0.00
0.05
0.10
0.15
0.20
0.25 0.30 s [mm]
0.35
0.40
0.45
0.50
b) bush hammered surface specimens Fig. 5. Bond-slip curve for SRP-to-concrete based on MSE minimization technique.
226
0.55
0.60
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1.00 0.90 0.80 0.70
fcm 19 MPa fcm 22 MPa fcm 26 MPa
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
a) grinded surface specimens 1.00 0.90 0.80 0.70
fcm 13 MPa fcm 22 MPa fcm 26 MPa
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
b) bush hammered surface specimens Fig. 6. Effect of concrete strength on interfacial bond-slip relationship according to the proposed model. Table 6 Values of the proposed model parameters as function of concrete strength. fcm
13 19 22 26
Parameters
{smax , , K },{¯max }
Note: ¯max in
Surface Grinded
Bush Hammered
–
{0.057, 0.724, 1.986},{0.832} –
{0.054, 0.881, 1.832},{0.777} {0.048, 0.869, 1.496},{0.730} {0.042, 0.819, 1.155},{0.702}
th G¯Dai et
al [26]
= ¯max
th G¯Dai et
al [27]
=
smax 1 + 1+
2 ¯max B
th G¯Ferracuti et al = gf (n) ¯max smax ,
{0.058, 0.984, 1.772},{0.882} {0.042, 1.000, 0.984},{0.799}
th G¯ proposed = ¯max
N / mm , fcm,28 in MPa, smax in mm.
227
(16) (17)
gf (n) =
smax s + max 1+ K 1
2 1 n
1 n
1
sin
2 n
1
(18) (19)
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1.00 0.90 0.80 0.70
Grinded
0.60
Bush Hammered
0.50 0.40 0.30 0.20 0.10 0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.35 s [mm]
0.40
0.45
0.50
0.55
0.60
Fig. 7. Bond-slip curves for SRP concrete interfaces based on the proposed model.
5. Discussion and comparisons
Table 7 Overall mean square error of the calibrated models. Authors
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Set of parameters
{¯max , {¯max , {¯max , {¯max , {¯max , {¯max ,
smax , smax , smax , smax , smax , smax ,
, p,} su} , } B} n} , K}
The bond-slip curves obtained via the minimization technique for all the above constitutive models are depicted in Fig. 5. Besides the general shape, the magnitudes of the three key parameters; namely, the maximum bond stress, the corresponding slip and the ultimate slip, are used to assess the relative accuracy of each model. It is important to note that the six models differ substantially with respect to the predicted value of maximum shear stress and the shape of the descending branch. For the minimization technique, assuming α equal to one, the partially nonlinear model of Cosenza et al. [38] essentially reverts to the bilinear model of Monti et al. [39]. Among all the models, the latter two models furnish the lowest value of ¯max , viz. 0.47 in the case of grinded specimens; 0.62 in the case of bush hammered specimens. On the contrary, the non-linear model proposed by the authors furnishes the highest value of ¯max , viz. 0.73 in the case of grinded specimens; 0.87 in the case of bush hammered specimens. All the other models investigated here give values of ¯max ranging between the preceding maximum and minimum values. Furthermore, the Cosenza et al., Monti et al. and Dai et al. [40] models are characterized by the same ascending linear elastic relationship for both surface treatments. The model proposed by the authors is characterized by a nearly linear ascending branch but exhibiting a lower stiffness in the case of grinded surface. Considering the descending branch, the models by Cosenza et al. and Monti et al. furnish the highest value for the experimental (s ) among all the models considered while the nonlinear model proposed by the authors gives the lowest value. The precipitous drop in the shear stress after the peak stress can be observed, especially for specimens with grinded surface treatment. The present model, through the parameter K, captures the effect of concrete strength on the descending branch shape as illustrated in Fig. 6. The values of the parameters associated with the curves in Fig. 6 are reported in Table 6. For the grinded surface treatment, the 13 MPa prisms results were neglected because as shown in Table 3, only two such specimens were tested. For the same reason, the results for the 19 MPa prisms with bush-hammered surface treatment were also neglected. It may be noticed in Fig. 6 that increasing the concrete
Surface Grinded
Bush Hammered
[%]
[%]
2.68 2.68 2.57 2.80 2.61 2.74
2.72 2.72 2.66 2.87 2.71 2.82
Table 8 Comparison with bond-slip law proposed in [46]. Authors
Grinded max
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Bush Hammered max
[MPa]
[MPa]
2.769 2.769 3.236 3.276 3.282 4.285
3.591 3.591 3.819 3.819 3.871 5.061
In Eqs. (14)–(19) G¯ th and ¯max represent the interfacial theoretical fracture energy and the maximum shear stress normalized with respect to the square root of fcm,28 , respectively. The results of the minimization process are summarized in Table 5, where for each bond-slip model considered and for both types of surface treatment, the values of the unknown set of independent parameters (see first bracket in column 2) and the dependent parameter (see second bracket in column 2) are reported in the first and second bracket, respectively, in columns 3 and 4.
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strength, increases the (s ) values on the descending branch. In order to predict the effect of concrete surface finish on the shear stress-slip response of the SRP-concrete interface, in Fig. 7 the normalized shear stress-slip curves of the two types of interfaces, predicted by the proposed model (see last row in Table 5), are plotted. It can be seen that the bush hammered treatment has little effect on the shape of the curve, but it leads to higher shear strength and higher slip at failure; or overall to higher fracture energy. To determine the relative accuracy of the calibrated models, with reference to Eq. (11), the corresponding error norms using Eq. (20) are computed for each model and the results are shown in Table 7.
finish. 7. The proposed model can be used to capture the effect of the concrete strength on the shape of the descending branch of the shear stressslip curve while none of the existing models can capture this effect. 8. All factors considered, the proposed model predicts the interface shear stress-slip response for SRP retrofitted concrete substrates better than the existing models. References [1] Teng JG, Chen JF, Smith ST, Lam L. FRP-strengthened RC structures. UK: John Wiley & Sons; 2002. [2] Yao J, Teng JG, Chen JF. Experimental study on FRP-to-concrete bonded joints. Compos B Eng 2005;36:99–113. [3] De Lorenzis L, Miller B, Nanni A. Bond of fiber-reinforced polymer laminates to concrete. ACI Mater J 2001;98:256–64. [4] Nakaba K, Toshiyuki K, Tomoki F, Hiroyuki Y. Bond behavior between fiber-reinforced polymer laminates and concrete. ACI Struct J 2001;98:359–67. [5] Yuan H, Teng JG, Seracino R, Wu ZS, Yao J. Full-range behavior of FRP-to-concrete bonded joints. Eng Struct 2004;26:553–64. [6] Gonzales-Libreros JH, Sneed LH, D’Antino T, Pellegrino C. Behavior of RC beams strengthened in shear with FRP and FRCM composites. Eng Struct 2017;150:830–42. [7] Capozucca R. Double-leaf masonry walls under in-plane loading strenghtened with GFRP/SRG strips. Eng Struct 2016;128:453–73. [8] Mazzotti C, Savoia M, Ferracuti B. An experimental study on delamination of FRP plates bonded to concrete. Constr Build Mater 2008;22(7):1409–21. [9] Faella C, Martinelli E, Nigro E. Direct versus indirect method for identifying FRP-toconcrete interface relationships. J Comp Const 2009;13(3):226. 226-223. [10] Ascione L, Feo L. Modeling of composite/concrete interface of RC beams strengthened with composite laminates. Comp Part B Eng 2000;31:535–40. [11] Lu XZ, Teng JG, Ye LP, Jiang JJ. Bond-slip models for FRP sheets/plates bonded to concrete. Eng Struct 2005;2:920–37. [12] Wang X, Ghiassi B, Oliveira DV, Lam CC. Modeling the nonlinear behaviour of masonry walls strengthened with textile reinforced mortars. Eng Struct 2017;134:11–24. [13] Bakis CE, Bank LC, Brown VL, Cosenza E, Davalos JF, Lesko JJ, et al. Fiber reinforced polymer composites for construction – state-of-the-art review. J Compos Constr 2002;6(2):73–87. [14] Lu XZ, Ye LP, Teng JG, Jiang JJ. Meso-scale finite element model for FRP sheets/ plates bonded to concrete. Eng Struct 2005;27(4):564–75. [15] Camata G, Spacone E, Zarnic R. Experimental and nonlinear finite element studies of RC beams strengthened with FRP plates. Comp Part B Eng 2007;38:277–88. [16] Singha K. A short review on basalt fiber. Int J Textil Sci 2012;4(1):19–28. [17] King MFL, Srinivasan V, Purushothaman T. Basalt fiber: an ancient material for innovative and modern application. Middle East J Sci Res 2014;22(2):308–12. [18] Hafsa J, Rajesh Mishra. A green material from rock: basalt fiber a review. J Textil Inst 2016;107(7):923–37. [19] Fiore V, Scalici T, Di Bella G, Valenza A. A review on basalt fibre and its composites. Compos B Eng 2015;74:74–94. [20] Shen D, Ji Y, Yin F, Zhang J. Dynamic bond stress-slip relationship between basalt FRP sheet and concrete under initial static loading. J Compos Construct 2014;19(6):1–11. [21] Shen D, Shi H, Ji Y, Yin F. Strain rate effect on effective bond length of basalt FRP sheet bonded to concrete. Construct Build Mater 2015;82:206–18. [22] Shi J, Zhu H, Wu Z, Seracino R, Wu G. Bond behavior between basalt fiber reinforced polymer sheet and concrete substrate under the coupled effects of freezethaw cycling and sustained load. J Compos Construct 2012;17(4):530–42. [23] Diab Hesham M, Farghal Omer A. Bond strength and effective bond length of FRP sheets/plates bonded to concrete considering the type of adhesive layer. Compos B Eng 2014;58:618–24. [24] Gopinath S, Murthy AR, Iyer NR, Prabha M. Behaviour of reinforced concrete beams strengthened with basalt textile reinforced concrete. J Ind Textil 2014;44(6):924–33. [25] Nerilli F, Marino M, Vairo G. A numerical failure analysis of multi-bolted joints in FRP laminates based on basalt fibers. Procedia Eng 2015;109:492–506. [26] Nerilli F, Vairo G. Strengthening of reinforced concrete beams with basalt-based FRP sheets: an analytical assessment. AIP Publish 2016;1:1738. [27] Gholkar Akshay P, Jadhav HS. Experimental study of the flexural behaviour of damaged RC beams strengthened in bending moment region with basalt fiber reinforced polymer (BFRP) sheets. Int J Eng Res Appl 2014;4(7):142–5. [28] Sim J, Park C, Moon DY. Characteristics of basalt fiber as a strengthening material for concrete structures. Compos B Eng 2005;36(6–7):504–12. [29] Barton B, Wobbe E, Dharani LR, Silva P, Birman V, Nanni A, et al. Characterization of reinforced concrete beams strengthened by steel reinforced polymer and grout (SRP and SRG) composites. Mater Sci Eng A 2005;412:129–36. [30] Carloni C, Ascione F, Camata G, De Felice G, De Santis S, Lamberti M, Napoli A, Realfonzo R, Santandrea M, Stievanin E, Cescatti E, Valluzzi MR. An overview of the design approach to strengthen existing reinforced concrete structures with SRG. American Concrete Institute. ACI Special Publication 2018. (SP-326). [31] Huang X, Birman V, Nanni A, Tunis G. Properties and potential for application of steel reinforced polymer and steel reinforced grout composites. Compos Part B Eng 2005;36:73–82.
N
MSE =
1 min (¯ th (s ) N k=1
¯ exp (s )) 2
(20)
With reference to the values in Table 7, the overall or average mean square error of all the models is practically the same. This indicates that the minimization process used to derive the values of the unknown parameters for each model achieves practically the same variance and bias. Finally, the proposed bond-slip model is compared with the one proposed in [46]. In the latter, the authors recalibrated four bi-linear bond–slip models for FRP strengthened interfaces to evaluate the interfacial behavior of RC beams externally strengthened with SRP. For beams with concrete strength of 34 MPa, they found an optimized value of 5 MPa for the maximum shear stress. For the sake of comparison, the maximum shear stresses were evaluated by multiplying the normalized ones, reported in Table 5 for both surface treatments, for the compressive concrete strength equal to 34 MPa. The obtained values are collected in Table 8. Note, the type of surface treatment was neither considered nor reported in [46]. It can be observed in the latter table that the maximum shear stress values predicted by the proposed model are closest to the 5 MPa reported in [46]. Based on this comparison and the more accurate prediction of the descending branch of the shear-slip curve mentioned earlier, it is believed that the model herein proposed is more suitable for application to SRP retrofitted concrete interfaces than the existing models. 6. Conclusions Results of a comprehensive experimental program are used to investigate the applicability and accuracy of five interfacial shear-slip models, previously proposed for FRP-concrete interface, to SRP-concrete interface. In addition, a new model is proposed for SRP-concrete interface as part of this study. All the models are characterized by shear stress-slip curves with an ascending and a descending branch. The mean squared error technique is used to calibrate each model’s parameters. The results support the following conclusions: 1. All the models predict well the slope of the ascending branch, but they give different values for the maximum shear stress resisted by the interface. 2. The descending branch of the curves provided by the six models differ substantially from each other. 3. Among all the models, the model proposed in this study provides a more accurate representation of the descending branch. 4. Among all the models, the proposed model predicts the highest shear stress resisted by the interface, which compares better with the corresponding experimental values than the values predicted by the other models. 5. Based on the percent error provided by the minimization technique, all the models provide practically the same overall level of accuracy in terms of fracture energy. 6. The accuracy of the models is not affected by the concrete substrate surface finish, but the bush-hammered finish gives higher shear strength and higher slip at failure compared to the grinded surface 229
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F. Ascione, et al. [32] Ceroni F, Leone M, Rizzo V, Bellini A, Mazzotti C. Influence of mortar joints of FRP materials bonded to different mansonry substrates. Eng Struct 2017;153:550–68. [33] Lopez A, Galati N, Alkhrdaji T, Nanni A. Strengthening of a reinforced concrete bridge with externally bonded steel reinforced polymer (SRP). Compos Part B Eng 2007;38:429–36. [34] Alabdulhady MY, Sneed LH, Carloni C. Torsional Behaviour of RC beams strenghtened with PBO-FRCM composites – an experimental study. Eng Struct 2017;136:393–405. [35] Ascione F, Lamberti M, Napoli A, Realfonzo R. SRP/SRG strips bonded to concrete substrate: experimental characterization. American Concrete Institute. ACI Special Publication 2018. (SP-326). [36] Napoli A, de Felice G, De Santis S, Realfonzo R. Bond behaviour of steel reinforced polymer strengthening systems. Compos Struct 2016;152:499–515. [37] De Santis S, de Felice G, Napoli A, Realfonzo R. Strengthening of structures with steel reinforced polymers: a state-of-the-art review. Compos Part B: Eng 2016;104:87–110. [38] Cosenza E, Manfredi G, Realfonzo R. Behaviour and modeling of bond FRP rebars to concrete. J Compos Constr 1997;1(2):40–51. [39] Monti M, Renzelli M, Luciani P. FRP adhesion in uncraked and cracked concrete zones. Proc. of 6th international symposium on FRP reinforcement for concrete structures. Singapore: World Scientific Publications. 2003. p. 183–92.
[40] Dai JG, Ueda T. Local bond stress-Slip relations for FRP sheets-concrete interfaces. In: Shield CK, editor. FRPRCS-6. Conference Proceedings. Singapore. 2003. p. 8–10. [41] Dai JG, Ueda T, Sato Y. Development of the nonlinear bond stress-slip model of fiber reinforced plastics sheet-concrete interfaces with a simple method. J Compos Constr 2005;80:52–62. [42] Ferracuti B, Savoia M, Mazzotti C. Interface law for FRP-concrete delamination. Compos Struct 2007;80:523–31. [43] Savoia M, Mazzotti C, Ferracuti B. Mode II fracture energy and interface law for FRP-Concrete bonding with different concrete surface preparations. Proceedings of the 6th International Conference on Fracture Mechanics of Concrete and Concrete Structures 2. 2007. p. 1249–57. [44] Ascione F, Lamberti M, Napoli A, Realfonzo R. Bond behaviour of steel FRP/FRCM systems on concrete substrates: an experimental investigation. Proc. of 17th ANIDIS conference. Pistoia. 2017. p. 17–21. [45] Ascione F, Lamberti M, Napoli A, Razaqpur G, Realfonzo R. An experimental investigation on the bond behavior of steel reinforced polymers on concrete substrate. Compos Struct 2017;181:58–72. [46] Bencardino F, Condello A. SRG/SRP–concrete bond–slip laws for externally strengthened RC beams. Compos Struct 2015;132:804–15. [47] Hardwire LLC, web site: < www.hardwirellc.com > [accessed Nov 2017]. [48] Kerakoll Spa, web site: < www.kerakoll.com > [accessed Nov 2017].
230
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Modeling SRP-concrete interfacial bond behavior and strength a
b
a
b,⁎
F. Ascione , M. Lamberti , A. Napoli , A.G. Razaqpur , R. Realfonzo a b
a
T
Dept. of Civil Engineering, University of Salerno, Italy College of Environmental Science and Engineering, Nankai University, Tianjin, China
ARTICLE INFO
ABSTRACT
Keywords: Steel reinforced polymer Concrete Bond-slip models Pull-out tests Bond strength
Relatively recently a steel fabric/laminate has been proposed for externally strengthening concrete structures, using a polymeric resin. The steel fiber-polymer composite system is termed Steel Reinforced Polymer (SRP). To determine the ultimate load capacity of an SRP retrofitted concrete structure, one must accurately predict the SRP-concrete interface debonding load, which requires a robust local bond-slip model. Many design guidelines recommend mode II interfacial fracture energy limit as the failure criterion for FRP-concrete interface. For SRP strengthened members, a suitable constitutive law and failure criterion have not been established yet. Consequently, in this study the applicability of five existing bond-slip interface models for FRP-concrete interface to SRP-concrete interface is examined. The models’ parameters are calibrated for SRP-concrete interface using an experimental database by the present authors and compared with the values suggested by the original authors for FRP-concrete interface. The database involves results of tests on concrete prisms bonded to SRP strips. The experimental interfacial bond-slip relationship for the former interface is observed to have a more precipitous descent after the peak stress than predicted by the existing models; consequently, a new model is proposed here to capture this phenomenon. All the models are calibrated using a classical technique which minimizes the difference between the measured and computed interfacial shear stress values at different slip levels. The results indicate that all the models predict relatively well the slope of the ascending branch of the shear stress-slip curve, but they give substantially different values for the maximum shear stress attainable and noticeably different descending branch profiles. Among these, overall, the proposed model is in relatively better agreement with the experimental results.
1. Introduction Relatively recently, new retrofit techniques for concrete structures has been developed to externally strengthen/repair deficient structures in a manner akin to applying the familiar Fibre Reinforced Polymers (FRP) laminates [1–15]. The new techniques involve natural [16–28] or artificial [29–37] fibers. Among artificial fibers, great attention has been devoted to continuous High Tensile Strength Steel (HTSS) micro wires, twisted into small diameter cords or strands that are laid in a polymer matrix to form a unidirectional or bidirectional composite laminate, termed Steel-wire Reinforced Polymer (SRP). The SRP can be externally bonded to a substrate, similarly to FRP, via wet lay-up, using either epoxy or polyester resin. One of the salient advantages of SRP over FRP is the high elastic modulus of steel wires versus glass, aramid or low modulus carbon fibres. Furthermore, if brittle interfacial debonding can be averted, SRP retrofitted structures would exhibit relatively ductile behaviour and higher energy
dissipation at failure. Understanding of the behaviour of SRP-concrete interface and controlling the parameters that govern its behaviour are key to the prevention of unexpected separation of the SRP from the substrate. Among these, the interfacial bond-slip response plays a crucial role insofar as interfacial debonding initiation and ultimate failure are concerned. The quantification of the interfacial response requires a reliable local bond-slip constitutive law or model. In structures strengthened in flexure, the stress state at the SRPconcrete interface is similar to the one in single-lap joint shear (SLJS) test. In SLJS test, a laminate strip is bonded to a concrete prism and the bonded strip is subjected to direct tension [2]. This similarity can be exploited to develop a bond-slip model for SRP retrofitted beams and other similar components using results of SLJS test. The latter test not only provides the ultimate load of the SRP-concrete interface, but it also furnishes a local bond-slip response. In the present discussion, the term “interface” refers to the adhesive and a thin layer of the adjoining
Corresponding author. E-mail addresses: [email protected] (F. Ascione), [email protected] (M. Lamberti), [email protected] (A. Napoli), [email protected] (A.G. Razaqpur), [email protected] (R. Realfonzo). ⁎
https://doi.org/10.1016/j.engstruct.2019.02.050 Received 10 July 2018; Received in revised form 14 January 2019; Accepted 22 February 2019 0141-0296/ © 2019 Published by Elsevier Ltd.
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concrete at the SRP-concrete joint, rather than an actual physical joint. It is this thin concrete layer that is assumed to undergo slip between the SRP laminate, and the concrete prism body. Local bond-slip response from SLJS test is commonly determined either from the axial strain distribution of the SRP laminate, obtained from readings of closely spaced strain gauges [4] or from load-displacement (slip at loaded end) curves [9]. Each method has its advantage and disadvantage. Specifically, in the first method the shear stress at a specified location along the interface can be determined by principles of mechanics while the corresponding slip can be computed by numerically integrating the measured axial strains of the laminate. The latter method is simple to apply but may not always yield an accurate local bond-slip response in some cases due to the abrupt changes in the measured local strain values caused by the discrete nature of concrete cracks, the heterogeneity of concrete and the unevenness of the underside of the partially debonded SRP laminate. In the second method, the local bond-slip response is determined indirectly from the load-slip curve, but it can be shown that quite different local bond-slip curves will lead to similar load-displacement curves. In the current investigation the former procedure will be used. The authors are not aware of any previous studies concerning the bond-slip model for SRP retrofitted concrete elements. Therefore, the objective of this paper is to derive such a model. Towards this end, first five existing bond-slip interface models [38–43], originally developed for FRP-concrete interface, are calibrated for application to SRP-concrete interfaces using a database of SLJS test results on SRP retrofitted concrete prisms performed previously by the authors [44,45]. A similar attempt was made in [46] where the reliability and adaptability of the FRP–concrete bond–slip laws to predict the structural behavior of RC beams externally strengthened in flexure with Steel Reinforced Grout (SRG) or SRP was evaluated. As observed in [44,45], SRP-concrete interface tends to exhibit a more precipitous descent in its bond-slip curve than the FRP-concrete interface curve. Consequently, to capture this characteristic, a new bond-slip model is proposed here and its results are compared with those of the calibrated existing models. All the models are characterized by curves possessing an ascending and a descending branch, with linear and/or non-linear variation. Each model is calibrated using the mean squared error (MSE) technique, which minimizes the difference between the experimental and corresponding computed values of interfacial shear stress at different slip values. To gauge the relative accuracy of the calibrated models, the experimentally observed maximum debonding stress values are compared with their corresponding compute or predicted values by using error norms based on the MSE technique
bonded length Load
(a)
150 SRP
(b)
200
Load
bf
400 Fig. 1. Test specimen geometry (a) Elevation, (b) Plan.
each strip type are reported in Table 1, where the symbols γ, ts, fs,u, Es, and εs,u designate the mass density, equivalent design thickness, average tensile strength, Young’s modulus, and ultimate strain, respectively. The width of the steel strip, bf, was fixed equal to 100 mm in all the tests. Furthermore, two different concrete surface preparations, viz. grinded and bush hammered, were used. The bush hammered concrete surface was heavily roughened to expose the aggregates while the grinded surface was mildly sanded to remove laitance. Finally, to evaluate the effective transfer length for each strip density, five bonded lengths were examined: 100, 150, 200, 300 and 350 mm. However, in the current analyses the results for bonded lengths of 100 mm and 150 mm are not used because these lengths were found to be less than the minimum required bonded length, resulting in rather low failure loads. The relevant mechanical properties of the two component epoxy resin, as reported by the manufacturer, are shown in Table 2, where the symbols fm,u, τm,u and Em,c designate the tensile strength, shear strength and secant modulus in compression, respectively. Fig. 2 schematically illustrates the typical test set-up. Tests were performed using a Schenck universal testing machine with 630 kN capacity. Tests were conducted by holding the SRP-concrete block assembly firmly in place between the top and bottom machine platens by means of a stiff steel jig while the steel strip free end was gripped by the machine jaws and subjected to tension. The load was applied via displacement control at the rate of 0.01 mm/s. To measure the SRP strip axial displacement and strain, the instrumentation illustrated in Fig. 2 was used. The slip at the loaded end was measured in two ways: first, two potentiometers, designated as LDT 1 and LDT 2, were placed to the left and right of the strip to measure its axial displacement at the location of the start of the bonded length; second, the same displacement was also indirectly measured by two non-contact laser sensors, designated as Laser 1 and Laser 2. The overall extension of the loaded end of the strip was monitored via the vertical displacement of the testing machine crosshead. To measure the SRP longitudinal strain variation along and across the bonded length, several electrical-resistance strain gauges were attached to the outer face of its bonded length, with the actual number of gauges being dependent on the bonded length Lf. As stated earlier, for the purpose of this study, the bonded lengths were 200, 250, 300 and 350 mm, which are greater than the corresponding effective bond length as demonstrated in [44]. In Fig. 3, each strain gauge designation and its precise location are depicted. On specimens with Lf = 200 mm, three gauges were applied along the centerline of the strip while another three gauges each were placed to the left and the right of the centerline, for a total of nine gauges. For bonded lengths greater than 200 mm, only the number of gauges along the centerline is increased (4 for 250 mm, 6 for 300 mm and 7 for 350 mm) while the side gauges
2. Experimental database A database containing the results of 99 SLJS tests (part of a wider experimental program), involving uniaxial steel wire strips, epoxy bonded to concrete prisms (SRP system), is used. The tests were performed by the authors in a previous experimental program and reported in [44,45], but for ease of reference and completeness, the testing program and its results will be recapped. Each prism had 200 × 150 mm2 cross-section and 400 mm length as depicted in Fig. 1. The concrete compressive strength (fcm,28) was obtained by testing at the age of 28 days concrete cubes with side length of 150 mm. The prisms are divided into two strength groups: the first group made of normal strength concrete (NSC) and the second group made of higher strength concrete (HSC). The NSC concrete strength varied between 13 and 26 MPa while that of HSC varied between 39 and 45 MPa. Note, in the current context the terms NSC and HSC are used to simply separate the two concrete classes tested here rather than referring to established concrete strength classifications. Four different steel strip densities were tested [47,48]: low density (L), low-medium density (LM), medium density (M) and high density (H). The relevant properties of 221
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Table 1 Properties of dry steel fabric. Tape Density
Density γ (g/m2)
Thickness ts (mm)
Strength, fs,u (MPa)
Elastic ModulusEs (GPa)
Ultimate Strain εs,u (%)
Low (L) Low-Medium (LM) Medium (M) High (H)
670 1200 2000 3300
0.084 0.164 0.254 0.381
3191.0 3138.3 3085.7 3064.6
182.1 182.7 183.4 182.2
2.19 2.14 2.09 2.09
the unloaded end of the SRP strip, and the axis x running along its length, is adopted as depicted in Fig. 3. The position of i-th strain gauge is denoted by x i (with i = 1, 2, ....,t ), where t is the total number of strain gauges for the given bonded length considered, while the interval between two consecutive strain gauges is represented by x = xi + 1 x . In order to evaluate the local bond-slip relationship, the following assumptions are made:
Table 2 Properties of epoxy mineral adhesive (Geolite® Gel). Tensile Strength fm,u (MPa)
Shear Strength τm,u (MPa)
Elastic Modulus Em,c (MPa)
> 14
> 12
> 5300
were kept the same as in the specimen with Lf = 200 mm. The test data originally reported in [44,45], are summarized in Tables 3 and 4 for the two groups of specimens with different surface preparation. The symbol fcm represents the concrete compressive strength, Fd the debonding initiation load, Fmax the maximum load resisted by the specimen during the test, sd and smax, the slip values associated with Fd and Fmax, respectively, and su, the slip at failure. The first column in the tables identifies each specimen. The identifier has the generic form “xY-bf-nf#z” where: “x”, “Y” , “bf”, “nf” and “#z” denote, respectively, the SRP provided bonded length in centimeter (20, 25, 30 or 35), SRP density (L = low; LM = low-medium; M = medium; H = high), the width of the steel strip in centimeter, the number of steel fiber layers and the order of the replicate test specimen in a group of nominally identical specimens. Note: bf and nf varied during the whole experimental program although for all tests here presented they assume the value of 100 mm and 1, respectively; the data for the HSC concrete specimens were also excluded due to the relatively small number of these specimens;
- no slip between the steel strip and concrete at x = 0 (Fig. 3); - strains in concrete are negligible relative to the steel strip ( c = 0 ); - linear variation of strain in the SRP strip between two consecutive strain gauges. The average shear stress, calculated using av (x )
=
Es ts (
i+1
sav (x ) =
s (x ) = s (x i ) +
(2)
( i+1 (x i + 1
i)
xi )
xi )2
(x 2
BAR 2
BAR 2
SRP
LVDT 2 LS 2 Steel Plate
Steel Plate
i
× (x
SRP
Steel Plate LVDT 1-2 Concrete
bw
200
150 Fig. 2. Schematics of the Test Set-up.
222
BAR 3
LS 1-2 bonded length
LS 1
+
xi )
(3)
The interfacial fracture energy, Gexp , is computed using the integral in Eq. (4), which is obtained by considering the constitutive relation of steel (Ns = Es As s ) and concrete (Nc = Ec Ac c ) and the equilibrium of horizontal forces acting on the SRP
400
400
bonded length
(1)
s (x i + 1) + s (x i ) 2
SRP
Concrete
i)
x
The local slip at a point x along the bonded length is evaluated using
The strain gauges bonded on the SRP strip allowed for recording the strain distribution along the interface. Using the strain distribution profile, the local bond-slip relation of SRP-concrete interface is evaluated applying the following procedure. A reference coordinate system, composed of the origin O, located at
LVDT 1
within an interval of length x is
where i + 1 and i are the strain values recorded at the (i + 1) and i gauges, respectively, while, Es and ts are the elastic modulus and the thickness of the SRP strip, respectively. The average local slip, sav (x ) , evaluated at midpoint of two consecutive strain gauges is calculated using
3. Bond-slip experimental evaluation
BAR 1
av (x ) ,
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x (mm)
x (mm)
x (mm)
x (mm)
185 170 140 110 80 50
235 220 190 160 130 100
285 270 240 210 180 150
335 320 290 260 230 200
55
105
155
60
110
15
65 20
b)
a)
c)
d)
Fig. 3. Strain gauge locations on steel strip: (a) bond length of 200 mm, (b) bond length of 250 mm, (c) bond length of 300 mm and (d) bond length of 350 mm. Table 3 Database of pull tests: grinded surface. Test
fcm,28 (MPa)
Fd (kN)
Fmax (kN)
σmax (MPa)
sd (mm)
smax (mm)
su (mm)
20L#1 20L#2 20L#3 20L#4 30L#1 30L#2 30L#3 30L#4 20M#1 20M#2 20M#3 20M#4 20M#5 30M#1 30M#2 30M#3 30M#4 30M#5 30H#1 30H#2 30H#3 30H#4 35H#1 35HD#2 35HD#3 35HD#4
26 26 26 22 19 26 26 22 26 26 19 22 13 19 19 19 22 22 19 26 26 22 26 26 22 13
9.09 12.16 11.35 12.72 13.57 9.36 11.15 14.50 14.28 18.71 20.75 24.06 11.36 18.47 19.24 17.50 20.02 22.06 20.95 25.33 18.50 25.55 12.12 13.85 25.49 20.96
9.78 12.82 11.90 13.77 14.72 10.83 12.95 16.78 14.28 18.71 21.05 24.06 12.97 19.46 19.24 20.93 21.58 23.79 21.11 25.33 19.22 26.09 16.19 16.63 32.65 22.04
1164.4 1525.7 1417.1 1639.1 1752.4 1289.5 1542.2 1998.1 562.4 736.5 828.7 947.4 510.8 766.2 757.6 824.0 849.8 936.5 554.0 664.9 504.5 684.7 424.9 436.4 857.1 578.5
0.27 0.47 0.45 0.58 0.62 0.28 0.54 0.64 0.49 0.58 0.25 0.96 0.12 0.57 0.29 0.39 0.71 0.48 0.49 0.32 – 0.06 1.06 0.80 0.30 0.33
0.61 0.54 0.59 1.52 1.92 0.78 2.13 2.45 0.49 0.58 0.69 0.96 0.65 0.95 0.29 0.97 1.34 0.88 0.51 0.32 0.25 0.65 2.19 1.91 1.02 0.94
1.08 1.27 1.46 1.60 2.34 1.58 2.34 2.45 0.89 0.97 0.69 1.34 0.65 1.33 0.87 0.97 1.34 1.17 0.98 0.84 1.29 0.85 2.22 1.91 1.33 1.01
Gexp =
0
(s ) ds =
Lf 0
Es ts (
s
c )d s
=
Lf 0
Es ts s d s =
the peak shear stress. The ascending and descending branches may be linear or nonlinear. Cosenza et al. [38] considered a non-linear ascending branch and a linear descending branch, Monti et al. in [39] proposed both branches to be linear (bilinear law) while Dai et al. [40–41] and Ferracuti et al. [42–43] assumed both branches to be nonlinear. Here a new model is proposed by the authors, which is characterized by nonlinear ascending and descending branches, and which is believed to predict better the overall shear-slip response compared to the existing models. In the following, a brief description of the above models is given, and their associated unknown parameters are identified. These parameters must be calibrated using experimental data together with a suitable error minimization technique. The first interface model considered is the one proposed by Cosenza et al. [38], which is characterized by a non-linear ascending branch and a linear descending one. The mathematical expressions for the two branches are given by Eqs. (5a) and (5b), respectively.
2 Fmax 2Es ts bf2
(4) 4. Local bond-slip interface models and their calibration To numerically capture the interaction between concrete and its internally, or externally, bonded reinforcement, an analytical bond-slip constitutive model is required. Hence, a number of bond-slip models exists for both reinforcing situations, but due to the similarity between externally bonded FRP-concrete interface and SRP-concrete interface, in the following, the experimental data in Tables 3 and 4 will be compared with the computed values obtained using some popular bond-slip models in the literature for FRP-concrete interfaces [38–42]. It is important to mention that most models are characterized by an ascending branch, followed by a descending or softening branch after 223
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Table 4 Database of pull tests: bush hammered surface specimens. Test
fcm,28 (MPa)
Fd (kN)
Fmax (kN)
σmax (MPa)
sd (mm)
sm (mm)
su (mm)
20LD#1 20LD#2 30LD#3 30LD#4 25LMD#1 25LMD#2 25LMD#3 20MD#1 20MD#2 20MD#3 25MD#1 25MD#2 30MD#4 30MD#5 30MD#6 30HD#3 30HD#4 30HD#5 35HD#1 35HD#2 35HD#3
26 26 26 26 13 13 13 26 26 22 13 13 22 22 13 26 22 13 26 26 22
7.53 10.85 12.04 17.14 24.16 20.73 25.37 22.02 19.83 20.22 26.39 25.71 20.73 22.34 16.00 17.44 24.06 19.68 20.51 21.82 26.15
14.66 14.74 14.15 17.36 25.61 24.16 26.86 22.02 19.83 21.23 26.86 25.71 21.21 22.60 16.45 22.93 28.33 20.33 30.16 21.82 27.34
1745.3 1754.8 1683.9 2066.5 1524.5 1438.3 1598.9 867.0 780.7 835.7 1057.6 1070.0 834.9 889.6 647.5 601.9 743.6 533.7 791.5 572.8 717.5
0.34 0.50 2.02 0.60 0.53 0.43 0.99 2.55 – 0.57 0.43 – 1.41 0.54 0.56 0.49 0.28 0.76 1.09 0.48 0.65
1.96 1.49 3.81 1.95 1.02 1.17 1.16 2.55 – 0.76 0.74 – 2.20 0.94 1.02 0.95 0.93 0.90 1.51 0.48 1.13
2.07 1.52 3.95 2.53 1.16 1.17 1.59 2.59 – 0.85 1.60 – 2.20 1.18 1.02 0.95 0.93 0.90 1.52 1.04 1.23
(s ) =
max
s smax
(s ) =
max
1
if
p
s
0
s
smax smax
smax
if
In the latter equation, B is an interfacial material constant. For this model, the set of unknown parameters is { max, B} . The fifth non-linear bond-slip interface model, proposed by Ferracuti et al. [42,43], is represented by Eq. (9). Once again, a single expression is presented for capturing both branches.
(5a)
smax
s
su
(5b)
In the above equations, and all subsequent equations, unless specified differently, (s ) is the interfacial shear stress, s is the slip, max and smax are the maximum shear stress and the corresponding slip, su is the ultimate slip, and p are experimentally calibrated parameters. Thus, for the calibration of this model, the unknown parameters are { max, smax , , p} . Note, slip su can be solved for by letting (s ) equal to zero in Eq. (5b). Next, the bi-linear interface model, originally proposed by Monti et al. [39] for the case of FRP-concrete system, is presented. Its ascending branch is characterized by Eq. (6a) and its descending branch by Eq. (6b).
(s ) = (s ) =
max
max
s
if 0
smax su su
s
s if
smax
smax
smax
(s ) =
su
(6b)
max ,
(s ) =
(s ) =
max
s smax
max e
if
(s smax )
0
if
s
smax
smax
s
max (e
( Bs )
e(
2Bs ) )
(s ) =
max
s smax
(s ) =
max
smax s
(¯ th (s )
min
( )
n s smax
K
if
0
s
(9)
if
smax
smax
s
¯ exp (s )) 2
su
(10a) (10b)
(11)
k=1
(7b)
In Eq. (11), and represent the theoretical and the experimental shear stress, respectively, normalized with respect to the square root of fcm,28 [11], while N is the total number of tests (twentysix for grinded surface and twenty-one for bush-hammered surface). Furthermore, for each i-th test, the values corresponding to increments of s = 0.1 mm (considering that the tests were performed in displacement control at the rate of 0.01 mm/s) were selected from s = 0 to s = su. For bonded length of 300 or 350 mm, the number of curves per each test is equal to six because between the seven strain gauges applied there were six intervals and a curve was constructed for each interval. For the 200 and 250 mm bonded lengths, the total number of curves per
¯ th (s )
It is important to mention that the expression for the ascending branch is the same as that introduced in [38]. In Eq. (7) and the two parameters that govern the shape of the ascending and descending branch, respectively. The parameter can only assume a value between 0 and 1. The set of unknown parameters in the latter case is { max, smax , , } . The fourth model was proposed by Dai et al. [41] where a single expression is presented for both branches as indicated by Eq. (8).
(s ) = 4
1) +
N
(7a)
su
n
1 and K > 1 are shape defining parameters In Eqs. (10) 0 while the complete set of unknown parameters is { max, smax , , K } . Thus, the proposed model has the same number of unknown parameters as the Cosenza et al. and one of Dai et al. models. The set of unknown parameters for the six interface models can be calibrated by applying the Mean Squared Error (MSE) minimization technique, which is expressed as Eq. (11).
The set of unknown parameters for the calibration of this model is smax , su} . The third model considered was proposed by Dai et al. [40], and its ascending and descending branches are represented by Eqs. (7a) and (7b), respectively.
{
s smax (n
In Eq. (9) n is a parameter mainly governing the softening branch and must be greater than 2. In this case, the set of unknown parameters is { max, smax , n} . The last model considered is proposed by the authors. It treats both the ascending and descending branches to be non-linear and are characterized by Eqs. (10a) and (10b), respectively.
(6a)
s
max
(8) 224
¯ exp (s )
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2.50
2.00
1.50
1.00
0.50
0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.35 s [mm]
0.40
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
a) grinded surface specimens 2.50
2.00
1.50
1.00
0.50
0.00 0.00
0.05
0.10
0.15
0.20
0.25 0.30 s [mm]
0.35
0.40
b) bush hammered surface specimens Fig. 4. Experimental bond-slip curve for SRP-to-concrete.
each test is three and four, respectively. Figs. 4(a) and 4(b) show the experimental shear stress-slip relationships for the specimens with grinded and bush-hammered surface finish, respectively. In these graphs, the precipitous drop in the shear stress after the peak stress can be observed, especially for specimens with high interfacial shear strength. As mentioned earlier, the proposed model was motivated by the latter observation. The total number of ( , s ) points used were 15,986 and 13,239 in the case of grinded and bush-hammered surface treatment, respectively. To reduce the number of unknown parameters, the following constraint was imposed: the average experimental interface fracture enav ergy, Gexp , is equal to the theoretically computed fracture energy, Gth , i.e. av Gexp = Gth
Equality of the two fracture energy values in Eq. (12) is assumed if they differ by no more than 0.2%. The theoretical fracture energy is computed using
Gth =
(s ) ds
0
(13)
After some algebraic manipulations, it can be verified that Eq. (13) leads to the following expressions for the six interface constitutive laws described earlier where the normalization with respect to the square root of fcm,28 has been introduced
(12) 225
th G¯Cosenza et
al
th G¯Monti et
=
al
= ¯max
¯max su 2
smax s + max 1+ 2p
(14) (15)
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Table 5 Minimization process: values of unknown parameters. Authors
Parameters
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Note: ¯max =
max /
Surface
{smax , , p},{¯max } {¯max , smax },{su} {smax , , },{¯max } {B},{¯max } {¯max , n},{smax } {smax , , K },{¯max }
Grinded
Bush Hammered
{0.024, 1.000, 0.056},{0.475} {0.475, 0.024},{0.444} {0.027, 1.000, 5.684},{0.555} {10.678},{0.562} {0.563, 2.595},{0.044} {0.045, 0.907, 1.377},{0.735}
{0.035, 1.000, 0.076},{0.616} {0.616, 0.035},{0.494} {0.038, 1.000, 4.868},{0.678} {8.616},{0.655} {0.664, 2.643},{0.057} {0.052, 1.000, 1.348},{0.868}
fcm,28 with fcm,28 in MPa; smax and su in mm.
1.00 0.90 0.80 0.70
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Ascione et al. (pp)
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
a) grinded surface specimens 1.00 0.90 0.80 Cosenza et al. [38]
0.70
Monti et al. [39]
0.60
Dai et al. [40] Dai et al. [41]
0.50
Ferracuti et al. [42]
0.40
Ascione et al. (pp)
0.30 0.20
0.10 0.00 0.00
0.05
0.10
0.15
0.20
0.25 0.30 s [mm]
0.35
0.40
0.45
0.50
b) bush hammered surface specimens Fig. 5. Bond-slip curve for SRP-to-concrete based on MSE minimization technique.
226
0.55
0.60
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1.00 0.90 0.80 0.70
fcm 19 MPa fcm 22 MPa fcm 26 MPa
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
a) grinded surface specimens 1.00 0.90 0.80 0.70
fcm 13 MPa fcm 22 MPa fcm 26 MPa
0.60 0.50 0.40
0.30 0.20 0.10 0.00
0
0.05
0.1
0.15
0.2
0.25
0.3 0.35 s [mm]
0.4
0.45
0.5
0.55
0.6
b) bush hammered surface specimens Fig. 6. Effect of concrete strength on interfacial bond-slip relationship according to the proposed model. Table 6 Values of the proposed model parameters as function of concrete strength. fcm
13 19 22 26
Parameters
{smax , , K },{¯max }
Note: ¯max in
Surface Grinded
Bush Hammered
–
{0.057, 0.724, 1.986},{0.832} –
{0.054, 0.881, 1.832},{0.777} {0.048, 0.869, 1.496},{0.730} {0.042, 0.819, 1.155},{0.702}
th G¯Dai et
al [26]
= ¯max
th G¯Dai et
al [27]
=
smax 1 + 1+
2 ¯max B
th G¯Ferracuti et al = gf (n) ¯max smax ,
{0.058, 0.984, 1.772},{0.882} {0.042, 1.000, 0.984},{0.799}
th G¯ proposed = ¯max
N / mm , fcm,28 in MPa, smax in mm.
227
(16) (17)
gf (n) =
smax s + max 1+ K 1
2 1 n
1 n
1
sin
2 n
1
(18) (19)
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1.00 0.90 0.80 0.70
Grinded
0.60
Bush Hammered
0.50 0.40 0.30 0.20 0.10 0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.35 s [mm]
0.40
0.45
0.50
0.55
0.60
Fig. 7. Bond-slip curves for SRP concrete interfaces based on the proposed model.
5. Discussion and comparisons
Table 7 Overall mean square error of the calibrated models. Authors
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Set of parameters
{¯max , {¯max , {¯max , {¯max , {¯max , {¯max ,
smax , smax , smax , smax , smax , smax ,
, p,} su} , } B} n} , K}
The bond-slip curves obtained via the minimization technique for all the above constitutive models are depicted in Fig. 5. Besides the general shape, the magnitudes of the three key parameters; namely, the maximum bond stress, the corresponding slip and the ultimate slip, are used to assess the relative accuracy of each model. It is important to note that the six models differ substantially with respect to the predicted value of maximum shear stress and the shape of the descending branch. For the minimization technique, assuming α equal to one, the partially nonlinear model of Cosenza et al. [38] essentially reverts to the bilinear model of Monti et al. [39]. Among all the models, the latter two models furnish the lowest value of ¯max , viz. 0.47 in the case of grinded specimens; 0.62 in the case of bush hammered specimens. On the contrary, the non-linear model proposed by the authors furnishes the highest value of ¯max , viz. 0.73 in the case of grinded specimens; 0.87 in the case of bush hammered specimens. All the other models investigated here give values of ¯max ranging between the preceding maximum and minimum values. Furthermore, the Cosenza et al., Monti et al. and Dai et al. [40] models are characterized by the same ascending linear elastic relationship for both surface treatments. The model proposed by the authors is characterized by a nearly linear ascending branch but exhibiting a lower stiffness in the case of grinded surface. Considering the descending branch, the models by Cosenza et al. and Monti et al. furnish the highest value for the experimental (s ) among all the models considered while the nonlinear model proposed by the authors gives the lowest value. The precipitous drop in the shear stress after the peak stress can be observed, especially for specimens with grinded surface treatment. The present model, through the parameter K, captures the effect of concrete strength on the descending branch shape as illustrated in Fig. 6. The values of the parameters associated with the curves in Fig. 6 are reported in Table 6. For the grinded surface treatment, the 13 MPa prisms results were neglected because as shown in Table 3, only two such specimens were tested. For the same reason, the results for the 19 MPa prisms with bush-hammered surface treatment were also neglected. It may be noticed in Fig. 6 that increasing the concrete
Surface Grinded
Bush Hammered
[%]
[%]
2.68 2.68 2.57 2.80 2.61 2.74
2.72 2.72 2.66 2.87 2.71 2.82
Table 8 Comparison with bond-slip law proposed in [46]. Authors
Grinded max
Cosenza et al. [38] Monti et al. [39] Dai et al. [40] Dai et al. [41] Ferracuti et al. [42] Proposed
Bush Hammered max
[MPa]
[MPa]
2.769 2.769 3.236 3.276 3.282 4.285
3.591 3.591 3.819 3.819 3.871 5.061
In Eqs. (14)–(19) G¯ th and ¯max represent the interfacial theoretical fracture energy and the maximum shear stress normalized with respect to the square root of fcm,28 , respectively. The results of the minimization process are summarized in Table 5, where for each bond-slip model considered and for both types of surface treatment, the values of the unknown set of independent parameters (see first bracket in column 2) and the dependent parameter (see second bracket in column 2) are reported in the first and second bracket, respectively, in columns 3 and 4.
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strength, increases the (s ) values on the descending branch. In order to predict the effect of concrete surface finish on the shear stress-slip response of the SRP-concrete interface, in Fig. 7 the normalized shear stress-slip curves of the two types of interfaces, predicted by the proposed model (see last row in Table 5), are plotted. It can be seen that the bush hammered treatment has little effect on the shape of the curve, but it leads to higher shear strength and higher slip at failure; or overall to higher fracture energy. To determine the relative accuracy of the calibrated models, with reference to Eq. (11), the corresponding error norms using Eq. (20) are computed for each model and the results are shown in Table 7.
finish. 7. The proposed model can be used to capture the effect of the concrete strength on the shape of the descending branch of the shear stressslip curve while none of the existing models can capture this effect. 8. All factors considered, the proposed model predicts the interface shear stress-slip response for SRP retrofitted concrete substrates better than the existing models. References [1] Teng JG, Chen JF, Smith ST, Lam L. FRP-strengthened RC structures. UK: John Wiley & Sons; 2002. [2] Yao J, Teng JG, Chen JF. Experimental study on FRP-to-concrete bonded joints. Compos B Eng 2005;36:99–113. [3] De Lorenzis L, Miller B, Nanni A. Bond of fiber-reinforced polymer laminates to concrete. ACI Mater J 2001;98:256–64. [4] Nakaba K, Toshiyuki K, Tomoki F, Hiroyuki Y. Bond behavior between fiber-reinforced polymer laminates and concrete. ACI Struct J 2001;98:359–67. [5] Yuan H, Teng JG, Seracino R, Wu ZS, Yao J. Full-range behavior of FRP-to-concrete bonded joints. 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N
MSE =
1 min (¯ th (s ) N k=1
¯ exp (s )) 2
(20)
With reference to the values in Table 7, the overall or average mean square error of all the models is practically the same. This indicates that the minimization process used to derive the values of the unknown parameters for each model achieves practically the same variance and bias. Finally, the proposed bond-slip model is compared with the one proposed in [46]. In the latter, the authors recalibrated four bi-linear bond–slip models for FRP strengthened interfaces to evaluate the interfacial behavior of RC beams externally strengthened with SRP. For beams with concrete strength of 34 MPa, they found an optimized value of 5 MPa for the maximum shear stress. For the sake of comparison, the maximum shear stresses were evaluated by multiplying the normalized ones, reported in Table 5 for both surface treatments, for the compressive concrete strength equal to 34 MPa. The obtained values are collected in Table 8. Note, the type of surface treatment was neither considered nor reported in [46]. It can be observed in the latter table that the maximum shear stress values predicted by the proposed model are closest to the 5 MPa reported in [46]. Based on this comparison and the more accurate prediction of the descending branch of the shear-slip curve mentioned earlier, it is believed that the model herein proposed is more suitable for application to SRP retrofitted concrete interfaces than the existing models. 6. Conclusions Results of a comprehensive experimental program are used to investigate the applicability and accuracy of five interfacial shear-slip models, previously proposed for FRP-concrete interface, to SRP-concrete interface. In addition, a new model is proposed for SRP-concrete interface as part of this study. All the models are characterized by shear stress-slip curves with an ascending and a descending branch. The mean squared error technique is used to calibrate each model’s parameters. The results support the following conclusions: 1. All the models predict well the slope of the ascending branch, but they give different values for the maximum shear stress resisted by the interface. 2. The descending branch of the curves provided by the six models differ substantially from each other. 3. Among all the models, the model proposed in this study provides a more accurate representation of the descending branch. 4. Among all the models, the proposed model predicts the highest shear stress resisted by the interface, which compares better with the corresponding experimental values than the values predicted by the other models. 5. Based on the percent error provided by the minimization technique, all the models provide practically the same overall level of accuracy in terms of fracture energy. 6. The accuracy of the models is not affected by the concrete substrate surface finish, but the bush-hammered finish gives higher shear strength and higher slip at failure compared to the grinded surface 229
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