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Near free-edge stresses in FRP-to-concrete bonded joint due to 1
mechanical and thermal loads 2
3
H. Samadvanda, and M. Dehestani 4
Corresponding author, Associate Professor, Faculty of Civil Engineering, Babol Noshirvani University of 5
Technology, Babol, Iran 6
Postal Box: 484, Babol, 7
Postal Code: 47148-71167, I.R. Iran 8
Tel: +989113136113 9
Fax: +981132331707 10
Email: [email protected] 11
12
a Graduate Student, Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran 13
Postal Box: 484, Babol, 14
Postal Code: 47148-71167, I.R. Iran 15
Tel: 16
Email: [email protected] 17
18
Abstract 19
Over the last few decades, a considerable amount of theoretical and experimental investigations have been 20
conducted on the mechanical strength of composite bonded joints. Nevertheless, many issues regarding the 21
debonding behavior of such joints still remain uncertain. The high near free-edge stress fields in most of these 22
joints are the cause of their debonding failure. In this study, the performance of an externally bonded fiber-23
reinforced polymer (FRP) fibrous composite to a concrete substrate prism joint subjected to mechanical and 24
thermomechanical loadings is evaluated through employing the principles of lamination theory. An inclusive 25
Matlab code is generated to perform the computations. The bond strength is estimated to take place in a region- 26
2
also termed the boundary layer- where the peak interfacial shearing and transverse peeling stresses occur; whereas 27
the preceding stress field is observed to be the main failure mode of the joint. The proposed features are validated 28
through the existing experimental data points as well as the commercial finite element (FE) modeling software, 29
Abaqus. Comparison between the calculated and experimental results demonstrates favorable accord, producing 30
quite a high average ratio. The current approach is advantageous to failure modeling analysis, optimal design of 31
bonded joints, and scaling analyses among others. 32
Keywords: FRP-to-concrete bonded joint; thermomechanical load; free-edge interfacial stress; boundary 33
layer; debonding failure; finite element method 34
35
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1. Introduction 36
Bonded joints are significantly desired in comparison to mechanical bolts, rivets, and welds due to 37
their compact design, low manufacturing cost, enhanced mechanical durability, sound noise 38
suppression, and obvious reduction of weight. Accordingly, they have been extensively adopted and 39
increasingly used in order to transfer load in structural elements and connections in civil structures 40
[1]. Furthermore, the problem of function degradation in such structures has gained exceptional 41
concern over the past several years [2, 3]. Meanwhile, thermal stresses play a very important role in 42
the study of bonded joints for a variety of reasons. On the one hand, these joints have considerable 43
residual thermal stresses from the fabrication process. On the other hand, the materials are to be 44
selected in order to behave well at elevated or dropped temperatures. The adherends shall thus retain 45
their properties at such temperatures; this is particularly true about the composite materials. They can 46
be chosen in that the coefficient of thermal expansion (CTE) is a specified value, possibly zero [4]. 47
As a historical point of view, Timoshenko conceived the concept of free-edge stress, who first 48
considered the deflection of bimetal thermostats subjected to uniform change of temperature [5]. 49
Later, continuous shear springs were practiced in bonded joints, in order to find the maximum 50
singular stresses at the edges [6]. This joint model had the advantage of simplicity, though ignored the 51
role of normal stresses in the failure criterion. However, the aforementioned joint model and the 52
associated interfacial stress solution has been adopted in many popular textbooks for design and 53
strength analysis of these types of joints. Following the asserted classic models, there have been a 54
significant number of theoretical complements. To mention a few, an efficient adhesively-bonded-55
joint model based on the elementary Euler-Bernoulli beam theory was formulated [7], to take into 56
account all mechanical properties and geometries of the adhesively bonded joints, in which the 57
adherends were treated as Euler-Bernoulli beams and the adhesive layer was ignored due to its 58
negligible deformation across the thickness. So far, with the fast development of microelectronics 59
since the 1970s, thermal stress-induced debonding failure in the so-called packaging has become one 60
of the main technical concerns and has attracted exceptional research over the last few decades. For 61
instance, an innovative bonded joint model was developed within the framework of 2D elasticity [8]. 62
4
This model was capable of predicting the reasonable location of the peak interfacial shear stress, 63
being located at a distance about 20% from the free end of adherend, still disregarding the normal 64
stress field. An analytical solution of stress-strain relation in a bonded joint was developed [9], from 65
which a parametric study of the overlap length and thickness of layers was demonstrated. 66
Since the 1990s, several improved methods for stress analysis of bonded joints have been reported 67
in the literature. For instance, engineering solutions to the singular stress field near interface edge of 68
bimaterials were proposed [10]. Furthermore, an analytical elastoplastic model of bonded joints was 69
presented, which was then verified by finite element method (FEM) [11]; they also conducted an 70
analytic study and concluded that the strength of adhesively bonded joints can be increased via 71
introducing proper residual stresses. A model of layerwise adhesively bonded joints was formulated, 72
from which the divided sub-layers were treated as field variables and solved through a set of 73
governing ordinary differential equations (ODEs) by evoking the theorem of minimum strain energy 74
[12]. An experimental investigation [13] was done on the suitability of carbon fiber-reinforced 75
polymer (CFRP) in strengthening of concrete filled steel tubular members (CFST) members under 76
flexure. They showed that the presence of CFRP in the outer limits significantly increase the 77
performance of beams. The efficiency of near-surface-mounted (NSM) method for both flexural and 78
shear strengthening of reinforced concrete (RC) beams was examined by applying an innovative 79
manually-made CFRP bar through experimental and numerical investigations [14]. They defined 8-80
noded solid elements to represent concrete beams, and link elements for modeling FRP in Ansys 81
software. It was indicated that using the proposed bars significantly improve the shear capacity of 82
deficient concrete beams. In a peer study, the flexural performance of FRP-strengthened RC beams 83
under cyclic loads was examined to assess its flexural behavior [15]. They tested eight RC beams 84
under monotonic and cyclic loadings and presented the results in the form of load-deflection curves, 85
reporting initiation and propagation of the interfacial debonding at mid-span for specimens with plate 86
end anchoring. In a partly similar experimental study, debonding mechanism in FRP-strengthened RC 87
beams within the framework of FE was studied [16]. Investigating the effects of length and width of 88
the strengthening sheet on beam's behavior and its failure mechanism, they proved that longer FRP 89
5
sheets increase the load-carrying capacity. Moreover, analysis of a membrane element of a FRP-90
strengthened concrete beam-column joint was proposed [17]. They considered the bond effect 91
between steel, FRP and concrete, which was able of predicting the shear strength. It was stated that 92
even a low quantity of FRP can enhance the shear capacity of joint through increasing the 93
confinement. In addition, shear-torsion interaction and cracking load of RC beams via FRP sheets 94
were investigated [18]. Lower-strengthened specimens were discovered to undergo higher capacity 95
increase. Furthermore, the flexural and impact behavior of glass fiber-reinforced polymer (GFRP) 96
laminates exposed to elevated temperatures were investigated [19]. It was pointed that the flexural 97
and impact properties of laminates decrease by increasing temperature. They also observed that 98
laminates with unidirectional fibers have the best performance subjected to elevated temperatures. A 99
micro-modeled computational framework for simulating tensile response and tension-stiffening 100
behavior of FRP–strengthened RC elements was presented [20]. The bond stresses at the FRP-to-101
concrete interface were then computed based on the local stress transfer mechanisms. A FE model to 102
characterize the interfacial shear strength between polymer matrices and single filaments was 103
developed [21]. The statistical study was then followed to evaluate the influence of key geometrical 104
test parameters on the variability of interfacial shear strength values. The RC beam specimens 105
strengthened with bonded CFRP strips with fan-shaped end anchorages to tension region were 106
examined [22]. They indicated that the numbers of anchorages applied to ends of CFRP strips were 107
more effective than the width of CFRP strips, concerning strength and stiffness of the specimens. A 108
numerical solution to calculate the interfacial stress distribution in beams strengthened with FRP plate 109
with tapered ends subjected to thermal loading was developed by applying finite difference method 110
(FDM) [23]. They investigated different profiles to consider the effect on stress concentration 111
reduction. Yet, simplifying assumptions of parabolic thru-thickness shear stress, as well as correction 112
factors were implemented in the methodology. 113
Some researchers conducted a thermal analysis on the hybrid GFRP-concrete deck [24]. They 114
theoretically investigated the CTE of GFRP laminates on micro scale. Based upon this analysis, 115
thermal behavior of composite girders with hybrid GFRP-concrete deck were studied on macro scale, 116
6
from which the results were more conservative than those of FEM outputs. Others, developed a 117
combined experimental-numerical approach to characterize the effect of cyclic-temperature 118
environment on bonded joints [3]. Experimental tests were performed on single-lap joints with CFRP 119
and steel adherends in a cyclic-temperature environment. It was then confirmed that adhesively 120
bonded joints permit to have more uniform stress distributions, join complex shapes, and reduce the 121
weight of structures. An externally-bonded CFRP plate to reinforce the concrete beam was utilized in 122
order to predict the interfacial stresses in the adhesive layer accounting for various effects of Poisson's 123
ratio and Young's modulus of the layer [25]. However, the adherend shear deformations were assumed 124
to be a parabolic shear stress through thickness of the layers. Furthermore, a higher-order method was 125
used [26] to study thick composite tubes under a combination of loads. In their analytical approach, a 126
displacement field of elasticity for a laminated composite tube was obtained to calculate stresses. In 127
addition, a higher-order theory for hygrothermal analysis of laminated composite plates [27], 128
satisfying the inter-laminar shear stress continuity at the interfaces and zero transverse shear stress 129
conditions at the top and bottom of plate was used. They applied a 9-noded C0 continuous 130
isoparametric element to simulate the FE model. 131
In spite of the many works done, habitually obtained by regression analyses and fittings, 132
outstanding issues still exist when applying the above adhesively bonded joint models for determining 133
the stress field in composite bonded joints, in which stress singularities occur at the end of binding 134
lines and laminate interfaces. As a matter of fact, transverse deformation plays a critical role in 135
delamination failure of composite joints, in which the transverse modulus of composite laminates is 136
very low. 137
1.1 Research attributes 138
The main objective of this work is to establish two interfacial shear and transversely normal stress 139
functions via integrating the geometric, mechanical and thermal parameters of the joint under 140
consideration, in order to accurately determine the stress field distributions along the interface of the 141
CFRP cover and concrete substrate layers. The salient feature is that the approach exactly satisfies all 142
the boundary conditions (BCs) in the joint as necessitated by the theoretical framework, though 143
7
ignored by many of its preceding works, even those renowned ones. Furthermore, it expresses two 144
simple and explicit equations for defining the interfacial shear and transversely normal stress fields at 145
any point of the bonding line. In addition, the approach can be utilized in extending the scopes of 146
failure modeling analysis, optimal design of bonded joints, and scaling analysis through altering the 147
type of loading, joint configurations, material type, etc. Moreover, the approach has cast a light over 148
the problem without any conventional oversimplifications leading to a closed form determination of 149
the cited stress fields. 150
2. Stress analysis through thermo-elastic lamination theory 151
In this section, the determination of interfacial shear and normal stresses of the externally-bonded 152
FRP-to-concrete joint subjected to mechanical and thermomechanical loads is discussed in detail. The 153
stress analysis is based on the mechanics of composite materials approach incorporated with the 154
efficient algorithms offered by Matlab software. The generated computational code embraces all the 155
mechanical, thermal and geometric properties of layers, in addition to the external loadings of the 156
joint. 157
In order to develop the stress-strain relationship for the layers of the joint, the 2D constitutive 158
equation for each layer has to be established as Equation 1, 159
11 12 16
21 22 26
61 62 66
T
x x x
T
y y y
T
xy xy xy
Q Q Q
Q Q Q
Q Q Q
(1) 160
where Q̅ij are the transformed reduced stiffness coefficients. The [Q̅] matrix can be expressed by 161
Equation 2, 162
1 1
Q T Q R T R
(2) 163
where [𝑇] is termed the transformation matrix and is defined by Equation 3, 164
2 2
2 2
2 2
cos sin 2sin cos
sin cos 2sin cos
sin cos sin cos cos sin
T
(3) 165
8
and [R] is the Reuter matrix. The inverse of compliance matrix is [Q], the stiffness matrix, i.e. 166
[Q] = [S]-1. The compliance matrix, [S], can be expressed as Equation 4 in terms of the engineering 167
constants of the material which writes, 168
121
111 12 16
1221 22 26 2
1
61 62 66
12
1/ 0
1/ 0
10 0
EE
S S S
S S S S EE
S S S
G
(4) 169
To determine the strains in Equation 1, Equation 5 reads, 170
0
0
0
T
x x x x
T
y y y y
T
xy xy xy xy
z
(5) 171
where 휀0, 𝜅 and 휀𝑇 terms are the midplane strains, curvatures and thermal strains, respectively; 172
Equation 5 is considered to be a fundamental equation of the lamination theory. The total strains are 173
the superposition of the midplane strains, the strains associated with curvature and strains originated 174
by thermal effects. It is worth noticing that this result was achieved without specification of the type 175
of material or any statement as to whether or not the layer is composed of more than one ply. It is then 176
the result of Kirchhoff assumptions on the displacements, citing that the normals to the midplane 177
remain straight and normal to the deformed midplane after deformation; normals to the midplane do 178
not change length; and the displacements are independent of material considerations [4]. 179
Now only if the midplane and thermal strains and curvatures are known, the stress state can be 180
determined from the constitutive Equation 1. The z in Equation 5, is the point of interest at which the 181
stresses and strains are calculated, measured from the midplane. The midplane strains and curvatures 182
can be considered using Equation 6 as it writes, 183
9
0
0
0
T
x xx
T
y yy
T
xy xyxy
T
x x x
T
y y y
T
xy xy xy
N N
N N
N NA B
B D M M
M M
M M
(6) 184
where, [𝐴], [𝐵] and [𝐷] matrices respectively denote extensional, coupling and bending 185
compliance matrices, and are defined by Equation 7, 186
1
2
1
3
1
1, 2,6
1/ 2 1, 2,6
1/ 3 1, 2,6
n
ij ij kkk
n
ij ij kkk
n
ij ij kkk
A Q t i
B Q t i
D Q t i
(7) 187
where tk is the thickness of kth layer. The resulting forces and moments are expressed as follows. Nx 188
and Ny are normal forces, and Nxy is the shear force per unit length. Likewise, Mx and My are bending 189
moments, and Mxy is the twisting moment per unit length. Hence, the only remaining consideration for 190
thermo-elastic lamination theory is the utilization of the equivalent thermal force, NT, and the 191
equivalent thermal moment MT. These quantities are respectively titled equivalent forces and 192
equivalent moments since they have units of force and moment per unit length, and in themselves, are 193
not physically applied forces and moments. 194
3. Problem modeling 195
Configuration of the composite bonded joint comprising a straight CFRP cover layer and a 196
concrete substrate bar, together with the resultant interfacial shear (τ) and normal (σ) stresses are 197
illustrated in Figs. 1(a)-(b), where the bonding line is denoted as L. The beam is assumed to be 198
subjected to the action of uniform axial tension, P0, which is applied at both ends of the tension bar, 199
namely the concrete substrate layer. The subscripts 1 and 2 are designated to the cover layer and 200
substrate bar, respectively. The layers have the mechanical properties of Young’s modulus Ei (i=1,2), 201
Poisson’s ratio νi (i=1,2); the geometric parameters of thickness hi (i=1,2) and CTE αi (i=1,2). The 202
10
width is represented with b. The x-coordinate in a 2D model is taken from the mid-span of the joint; 203
the z-coordinate is selected perpendicular to the x-coordinate and along the layer edge. 204
The adherends of bonded joints are in a complicated three-dimensional (3D) stress state. This is 205
mainly owing to the varying Poisson’s ratios across the adherends’ interface. To facilitate the analysis 206
process, the joint is considered to be in plane-stress state and subjected to a uniform temperature 207
change ΔT relative to the reference temperature. It can be expected that interfacial shear and normal 208
stresses are caused by the mismatch of material properties along the bonding line; lateral deflection is 209
probable to be induced since the joint is not laterally symmetric. The joint layers are treated as 210
isotropic for the concrete substrate bar and transversely isotropic for the CFRP cover layer, and 211
linearly thermoelastic materials. 212
Thereafter, FEM based on the commercial finite element analysis (FEA) software package Abaqus 213
is used to further authenticate the proposed composite approach. During FEM of the joint, the 8-noded 214
linear elements (C3D8R) and exclusively hexahedral meshes are employed. A nonuniform biased 215
distribution of elements along the bondline is used to exhibit the trend of singular stresses along the 216
bondline as shown in Fig. 2. In addition, as illustrated in Fig. 3, two of the substrate ending edges are 217
used to constrain the movement of the selected degrees of freedom to zero so as to simulate the BCs. 218
The whole solving process is executed by programing a robust Matlab code which encompasses all 219
the mechanical properties, geometries and external thermal load of the joint. Consequently, the entire 220
stress field of the bonded joint can be determined accurately, which is validated by the existing 221
models in the literature. The detailed workflow of the stress coefficients derivation is demonstrated in 222
Fig. 4. It characterizes the incremental analysis procedure flowchart for the FRP-to-concrete 223
composite bonded joint model under consideration. 224
4. Governing equation of interfacial stress functions 225
Figs. 5(a)-(b) illustrate typical representative segmental elements of the reinforcing CFRP-laminate 226
and the concrete substrate layer together with the stress resultants, which conform to the conventional 227
sign standards [28]. Therefore, as in the representative segmental element of the reinforcing patch 228
(Fig. 5(a)), the two of static equations of equilibrium are expressed by Equations 8 and 9, 229
11
1dQ
bdx
(8) 230
1 1
12
dM h bQ
dx
(9) 231
Meanwhile, the equivalent equilibrium equations for the representative segmental element of the 232
concrete substrate layer are formulated as Equations 10 and 11, 233
2dQ
bdx
(10) 234
2 2
22
dM h bQ
dx
(11) 235
Two unknown stress functions are adopted to represent the interfacial shear and normal stresses as 236
𝜏(𝑥) and 𝜎(𝑥). Since the joint is symmetric with respect to y-axis, the shear stress function 𝜏(𝑥) and 237
the normal stress function 𝜎(𝑥) are to be odd and even functions, respectively. 238
Furthermore, the x-axis is aligned horizontally, so that the two unknowns are expressed as functions 239
of variable 𝑥. The shear stress at the free edges of the bonding line, 𝑥 = −𝐿 2⁄ and 𝑥 = +𝐿 2⁄ 240
imposes the interfacial shear stress function be zero. 241
Integration of Equations 8 and 9 with respect to 𝑥 from 𝑥 = −𝐿 2⁄ along with the applying 242
associated boundary conditions, i.e. 𝑄1(−𝐿/2) = 0, and 𝑀1(−𝐿/2) = 0, respectively give 243
Equations 12 and 13, 244
1
/2
x
L
Q x b d
(12) 245
1
1
/2 /2 /22
x x
L L L
h bM x b d d d
(13) 246
The same trend is used for the stress resultants of the concrete substrate by integration of 247
Equations 10 and 11 with respect to 𝑥 from 𝑥 = −𝐿 2⁄ and taking into account the relevant 248
boundary conditions, i.e. 𝑄2(−𝐿/2) = 0 and 𝑀2(−𝐿/2) = 0, for the shear and moment, 249
respectively. The shear force 𝑄2(𝑥) and bending moment 𝑀2(𝑥), can be simply determined as 250
Equations 14 and 15, 251
12
2
/2
x
L
Q x b d
(14) 252
2
2
/2 /2 /22
x x
L L L
h bM x b d d d
(15) 253
In order to correlate the interfacial shear 𝜏(𝑥) and normal 𝜎(𝑥) stress functions along the interface 254
as well as to simplify the calculations, the deformation compatibility represented by Equation 16 is 255
used in a way that the radii of curvature of CFRP-composite laminate and the concrete substrate layer 256
are assumed to be approximately the same, 257
1 2
1 2 1 1 2 2
1 1 M M
E I E I (16) 258
where 𝐼1 = 1 12⁄ 𝑏ℎ13 and 𝐼2 = 1 12⁄ 𝑏ℎ2
3 are moments of inertia of the CFRP-laminate 259
composite and the concrete substrate layer, respectively. 260
Hence, substituting from Equations 13 and 15 into Equation 16, it becomes, 261
3
2 2 1 2
1 1
2 2 2 2 2 2
2 2
x x x x
L L L L L L
E h h hd d d d d d
E h
(17) 262
By differentiating at both sides of Equation 17 and defining the parameters as 𝑒21 = 𝐸2 𝐸1⁄ , 263
ℎ21 = ℎ2 ℎ1⁄ and 𝜂0 = 𝑒21ℎ213 + 1 (𝑒21ℎ21
3 − 1)⁄ , Equation 17 can be reduced to Equation 18, 264
0
/22
2
x
L
x dh
(18) 265
Then the entire strain energy of the CFRP-to-concrete joint can be determined by integrating the 266
strain energy density (per unit length) with respect to the 𝑥 from 𝑥 = −𝐿 2⁄ to 𝑥 = +𝐿 2⁄ . This 267
can be seen to be given by Equation 19, 268
13
1
1
1
2
2
2
/222
1 1 1 1 11
1
/2 1
2
/222
2 2 2 2 22
2
/2 2
2
11
2
11
2
L
h
xx xx yy yy y x
L h
L
h
xx xx yy yy y x
L h
U b dxdyE
b dxdyE
(19) 269
The aforementioned complementary strain energy is a functional with respect to the unknown 270
interfacial stress function 𝜏(𝑥). 271
Based on the theorem of minimum complementary strain energy, the complementary strain energy of 272
the joint with respect to the interfacial stress function 𝜏(𝑥), reaches a stationary point at static 273
equilibrium such that 𝛿𝑈 = 0, where 𝛿 is the mathematical variational operator with respect to the 274
unknown interfacial shear stress 𝜏(𝑥). By setting it equal to zero, yields a 4th-order ODE with respect 275
to the interfacial shear stress 𝜏(𝑥) as Equation 20, 276
22 0( ) ( ) ( )IV IIp q t (20) 277
In Equation 9, 278
11 0 /2
( ) (1
)
x
h L
xdx
h Px
(21) 279
is a dimensionless stress function; the coefficients 𝑝 = 𝐵 −2𝐴⁄ , 𝑞 = √𝐶 𝐴⁄ and 𝑡 = 𝐷 𝐴⁄ are 280
related to the properties of the joint. Moreover, each of the coefficients A, B, C and D are in turn 281
solved by using an efficient code generated by the authors in Matlab elaborated in Fig. 4. 282
Mathematically, in the case of 𝑞 > 𝑝, the solution to Equation 9 gives, 283
2
1 2cosh cos sinh sin /( ) C C t q (22) 284
Where = √(𝑞 − 𝑝) 2⁄ ; 𝜇 = √(𝑝 + 𝑞) 2⁄ and C1 and C2 are constants. By applying the shear-285
free and axial traction-free conditions at 𝑥 = ±𝐿 2⁄ , C1 and C2 can be determined and 𝜏(𝑥) can be 286
expressed as Equation 23, 287
14
1 2 1 1
0 1 0
1 2 1 1
sinh / /
cosh / sin /
C C x h cos x hdx P h P
dx C C x h x h
(23) 288
In addition, if 𝑝 > 𝑞, the solution to Equation 9 yields, 289
2
1 2cosh cosh /C C t q (24) 290
where = √𝑝 − √𝑝2 − 𝑞2 ; 𝜇 = √𝑝 + √𝑝2 − 𝑞2and C1 and C2 are two constants. By plugging 291
the shear-free and axial traction-free conditions at 𝑥 = ±𝐿 2⁄ , C1 and C2 can be determined and 𝜏(𝑥) 292
can be expressed as Equation 25, 293
0 1 0 1 1 2 1sinh / ωsinh /d
x P h P C x h C x hdx
(25) 294
Correspondingly, the interfacial normal stress 𝜎(𝑥), which was related to 𝜏(𝑥) through Equation 18 295
can be defined as Equation 26, 296
2
0
'2
hx x
(26) 297
5. Validating process 298
In order to validate the present model, herein, the thermomechanical stress analysis of a bimaterial 299
thermostat subjected to a uniform temperature change [29, 30 and 31] is considered. Consistent with 300
the coordinate system depicted in Fig. 1, the parameters are: h1 = 2.5 mm, E1 = 70 GPa, ν1 = 0.345, α1 301
= 23.6e-6 1/C, h2 = 2.5 mm, E2 = 325 GPa, ν2 = 0.293, α2 = 4.9e-6 1/C, L = 50.8 mm and ΔT = 240 °C; 302
where αi is the CTE with respect to principal material directions. 303
By implementing the aforementioned values into the Equations 25 and 26, the determination of 304
interfacial shear and normal stresses of the joint subjected to mechanical and thermomechanical loads 305
for a joint model with a given set of parameters, the analysis includes each of the resulting interfacial 306
shear and normal stress fields, and can be expressed explicitly through simple but straightforward 307
equations, i.e. by Equations 27 and 28, respectively. 308
It reads, 309
15
015 009
= 4.2328 10 sinh(1.5631 ) 2.2095 10 sinh(1.0448 )xz
x x
(27) 310
where 𝜏𝑥𝑧 is the interfacial thermal shear stress and 311
009 015
= 1.8629 10 cosh(1.0448 ) 5.3392 10 cosh(1.5631 )zz
x x
(28) 312
where 𝜎𝑧𝑧 is the interfacial thermal normal stress along the bonding line. 313
Figs. 6 (a)-(b) show the distribution of the interfacial thermal shear and normal stress fields of the 314
tested joint subjected to a uniform change of temperature ΔT, predicted by the present lamination 315
theory approach, and compared with those obtained by FEM (Abaqus) and those of [32] along the 316
interface. It can be observed that the shearing and peeling stresses are highly localized at the near 317
free-end of the bondline. Obviously, the interfacial shear stresses predicted by the present lamination 318
theory approach exactly satisfy the shear-free BCs in the joint as required by the theoretical 319
formulation. 320
Even though the normal and shear stresses adopt the very similar varying trends for the proposed 321
approach, FEM and [32], shear stress results obtained by FEM do not satisfy the stress free BCs at the 322
very ending edges of the cover layer because of the singular state of such stresses. However, the peak 323
values happen to take place on the same point as those of lamination theory approach. The interfacial 324
shear stress values are 25 and 35% lower at the adherend end than those predicted by [32] and FEM, 325
respectively. This can be due to the lower-order nature of the approach in the current work. Moreover, 326
the peak values of normal stresses predicted by the formulations in the present study are 30% higher 327
and 20% lower than [32] and FEM with the same geometries, material properties, and temperature 328
change. While employing the temperature change to the joint, the cover layer illustrates a great 329
thermal expansion and tends to endure larger deformations due to its large CTE in comparison to the 330
substrate layer with gross ratio of 5 to 1. 331
6. Scaling analysis 332
In the view of scaling analysis, the present solution to the interfacial stresses in bonded joints has 333
advantages over the conventional approaches on account of its simplicity and explicit expressions of 334
the stress components. A compact and efficient computational Matlab code was designed to be 335
16
implemented to examine the dependencies of the interfacial shear and normal stresses upon the 336
temperature parameter, as well as other mechanical variables. Table 1 depicts the mechanical and 337
geometric properties of concrete substrate in the scaling analysis. Likewise, Table 2 displays the 338
mechanical and geometric properties of CFRP composite cover layer through this Section. 339
To carry out the scaling analysis of thermomechanical stresses in the joint, four values for the 340
positive and four for the negative temperatures, ranging from ΔT= 25 °C to ΔT= 100 °C and ΔT= -25 341
°C to ΔT= -100 °C at every span of 25 are adopted. Figs. 7(a)-(b), as well as Figs. 8(a)-(b) illustrate 342
the variations of interfacial shear and normal stresses with distance x from the left end of the 343
reinforcing CFRP cover layer, respectively for positive and negative thermal loads. 344
It is perceived that shear and normal stresses are symmetric with respect to mid-span of the joint. It 345
is also comprehended from Figs. 7(a)-(b) that as the positive values of temperature increase, the 346
interfacial shear stress field increases at the near free-edge zone. Specifically, the peak values of 347
interfacial shear stress appear at a distance close to free ends, also termed the boundary layer region. 348
In this research, the distance is measured to be the L/8 from the free edges. The shear stress is equal to 349
zero at free edges, though on the contrary, the peak values of the positive normal stress happen 350
exactly at free ends. In Figs. 8(a)-(b), the increasing trend of stresses is the same, i.e. the absolute 351
increase in temperature causes a growth in shear and normal stresses. It is then observed that the 352
normal stresses take negative values for negative thermal loads. Due to the main contribution of shear 353
stresses, the peak values of interfacial shear stresses are much larger compared to the normal stresses 354
in all the cases under examination, implying that it’s the dominant mode of debonding failure in the 355
joint. 356
In order to evaluate the mechanical loading effect, four positive values of P0= 1, 10, 20 and 30 357
MPa are adopted in the study. Table 3 illustrates the joint material properties through the analysis. 358
The distribution of the shear and normal stress fields along the interface with respect to different 359
values of P0 are depicted in Figs. 9(a)-(b). It is noticed that the stress fields are symmetric with respect 360
to mid-span of the joint. The stress distribution is identical to that of the thermal loading effect, 361
though much intense. In both Figs. 9(a)-(b), the stresses increase as P0 escalates. Additionally, four 362
17
negative values of P0= -1, -10, -20 and -30 MPa are assumed. Material properties are represented in 363
Table 3, and Figs. 10(a)-(b) plot variations of the interfacial shear and normal stresses along the 364
bondline with respect to negative traction loads, respectively. As P0 increases in absolute value, i.e. 365
from P0= -1 to -30 MPa, the interfacial stresses tend to decrease up to a certain load which is P0= -19 366
MPa, whereas the curvature at this load level changes so that the peak interfacial normal stresses 367
appear to be negative after this loading point, having an increasing trend. 368
Finally, the distribution of interfacial shear and normal stress fields along varying temperature with 369
respect to different values of P0 are illustrated in Figs. 11(a)-(b), respectively. As the thermal load, ΔT 370
increases from ΔT= -100 °C to ΔT= 100 °C, the shear stress decreases. Thus, for a particular thermal 371
load, shear stress decreases with the increase of mechanical traction load. This tendency is inverse 372
about the normal stresses, in a sense that the increase in traction load for a specific thermal load leads 373
to an increase in the normal stress field. It is vivid in all figures that the stress values are zero at points 374
without any thermal or mechanical loads. 375
As it is acknowledged thru validating process (Section 4), the obtained results in the present study 376
show the same quantitative features, and are in sensibly acceptable agreement with the previously 377
performed investigations. Yet, the formalism can be readily extended to an assortment of bonded 378
structures via implementing the robust computational code which is based on the presented theoretical 379
approach by introducing an additional ply as the adhesive layer and plugging the mechanical and 380
geometrical properties into the algorithm in order to take into account the effects of the adhesive layer 381
on the stress distribution. As a result it has the capability to provide a framework for future studies. 382
7. Concluding remarks 383
The performance of an externally-bonded FRP-to-concrete joint subjected to mechanical and 384
thermal loadings was evaluated. A theoretically self-consistent lower-order lamination-theory 385
approach was proposed to establish two interfacial shear and transversely normal stress functions in 386
terms of two simple and explicit equations via integrating the geometric, mechanical and thermal 387
parameters of the joint. An improved understanding of the scaling behavior of stress variations was 388
offered thru devising a compact and efficient computational Matlab code to examine the dependencies 389
18
of the interfacial shear and normal stresses upon temperature, as well as other mechanical variables. 390
The following conclusions were thus drawn: 391
1- Normal and shear stresses adopt the very similar varying trends for the proposed approach as to 392
FEM, in that the peak values happen to take place at the same point, though shear stresses obtained 393
via FEM do not satisfy the stress-free BCs at the very ending edges of the bondline due to singular 394
nature of such stresses. 395
2- Compared to FEM, the proposed approach is more competent for stress analysis of bonded 396
joints since it only calls for the input of basic dimensions and material properties. 397
3- Increase in temperature, escalates both shear and normal stresses. 398
4- The stress distribution subjected to mechanical loading is identical to that of the thermal loading 399
effect, yet much intense. 400
5- Shear stress is equal to zero at free edges, though on the contrary, the absolute peak value of 401
normal stress happens precisely at free ends. 402
6- Peak values of interfacial shear stress appear at a distance close to free ends, termed the 403
boundary layer region. In this research, the length of this region was measured to be L/8 from the free 404
edges. 405
7- Peak values of interfacial shear stresses are much larger compared to normal stresses in all the 406
cases under examination, implying that it’s the dominant mode of debonding failure in the joint. 407
408
No funding source was provided in the conduction or preparation of the current research. 409
410
References 411
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65 (2002). 414
19
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20
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beams strengthened by bonded CFRP laminates under monotonic and cyclic loads”, Scientia 445
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flexural and impact behavior of GFRP laminates after exposure to elevated temperatures”, J. 455
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2596 (2018). 459
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stress concentration in strengthened beams under thermal load”, Int. J. Steel and Composite 466
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21
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joints under mechanical and thermal loads”, Int. J. Engineering Science, 49, pp. 279-294 (2011). 487
488
489
22
Tables’ captions 490
Table 1 Mechanical and geometric properties of concrete substrate prism 491
Table 2 Mechanical and geometric properties of CFRP-composite cover layer 492
Table 3 Mechanical and geometric properties of the joint layers 493
494
495
23
496
Table 1 Mechanical and geometric properties of concrete substrate prism 497
Axial
modulus
E2 (GPa)
Shear
modulus
G12
(GPa)
Poisson’s
ratio ν2
Concrete
cylinder
strength
(MPa)
Axial
CTE α1
(μ/°C)
Thickness
h2 (mm)
Joint
width b
(mm)
Bondline
L (mm)
Traction
load P0
(MPa)
33.8 14.08 0.2 50 10 50 50 100 +1
498
499
500
501
Table 2 Mechanical and geometric properties of CFRP-composite cover layer 502
CFRP Axial
modulus
E1 (GPa)
Shear
modulus
G12 (GPa)
Poisson’s
ratio ν1
Axial
CTE α1
(μ/°C)
Thickness
h1 (mm)
Joint
width b
(mm)
Bondline
L (mm)
Traction
load P0
(MPa)
T300/5208 132 5.65 0.24 -0.77 5 50 100 +1
503
504
505
Table 3 Mechanical and geometric properties of the joint layers 506
CFRP
axial
modulus
E1 (GPa)
Concrete
axial
modulus
E2 (GPa)
CFRP
thickness
h1 (mm)
Concrete
thickness
h2 (mm)
CFRP
Poisson’s
ratio ν1
Concrete
Poisson’s
ratio ν2
Joint
width b
(mm)
Bonding
length L
(mm)
Thermal
load ΔT
(°C)
132 33.8 5 50 0.24 0.2 50 100 100
507
508
cf
24
Figures’ captions 509
Fig. 1 Configuration of the joint consisting of a straight CFRP cover and concrete substrate with distribution of 510
near free-edge interfacial stresses (a) bonded joint (b) distribution of interfacial shear (τ) and normal (σ) stresses 511
Fig. 2 Representative of the biased mesh grid density of bonded joint 512
Fig. 3 Loading and BCs applied to ending edges of the concrete substrate 513
Fig. 4 Incremental procedure to determine interfacial stress coefficients of the bonded joint 514
Fig.5 Free-body diagrams of representative segmental elements of the adherends: (a) CFRP-composite cover 515
and (b) concrete substrate layer 516
Fig. 6 Comparison of stresses proposed by the present model with those by Wu and Jenson’s and FEM due to a 517
uniform change of temperature ΔT (a) interfacial thermal shear stress(τ) (b) interfacial thermal normal stress 518
σ 519
Fig. 7 Variations of the interfacial shear and normal stresses with respect to positive thermal loads (a) interfacial 520
shear stress (τ) (b) interfacial normal stress σ 521
Fig. 8 Variations of the interfacial shear and normal stresses with respect to negative thermal loads (a) interfacial 522
shear stress (τ) (b) interfacial normal stress σ 523
Fig. 9 Variations of the interfacial shear and normal stresses with respect to positive mechanical loads (a) 524
interfacial shear stress (τ) (b) interfacial normal stress σ 525
Fig. 10 Variations of the interfacial shear and normal stresses with respect to negative mechanical loads (a) 526
interfacial shear stress (τ) (b) interfacial normal stress σ 527
Fig. 11 Stress field distribution along varying temperature with respect to different values of P0 (a) interfacial 528
shear stress (τ) (b) interfacial normal stress σ 529
530
531
25
532
533
534
Fig. 1 Configuration of the joint consisting of a straight CFRP cover and concrete substrate with distribution of 535
near free-edge interfacial stresses (a) bonded joint (b) distribution of interfacial shear (τ) and normal (σ) stresses 536
537
26
538
539
540
541
542
Fig. 2 Representative of the biased mesh grid density of bonded joint
27
543
Fig. 3 Loading and BCs applied to ending edges of the concrete substrate 544
545
546
28
547
Fig. 4 Incremental procedure to determine interfacial stress coefficients of the bonded joint 548
Yes
No
Yes
No
Read thermomechanical parameters
Calculate Si and Qi, the compliance and stiffness matrices for each ply
Is i ≤ the number
of used materials?
Read the angle of orientation
Calculate Ti and 𝑄 𝑖, the transformation and reduced
stiffness matrices for each ply, Eqs. (3) and (2)
Is i ≤ the number
of layers?
Read each ply thickness
Calculate A, B and D, the extensional, coupling and
bending compliance matrices, Eq. (7)
Is i ≤ the number
of layers-1?
Read loadings
Calculate midplane strains and curvatures, Eq. (6)
Read point coordinates in which strains will be calculated
Calculate strains and stresses Eqs. (5) and (1)
i=i+1
i=1
i=i+1
i=1
i=i+1
i=1
Yes
No
29
549 550
551
552
553
Fig. 5 Free-body diagrams of representative segmental elements of the adherends:
(a) CFRP-composite cover and (b) concrete substrate layer
30
554
555
(a) 556
557
(b) 558
Fig. 6 Comparison of stresses proposed by the present model with those by Wu and Jenson’s and FEM due to a 559
uniform change of temperature ΔT (a) interfacial thermal shear stress (τ) (b) interfacial thermal normal 560
stress σ 561
562
31
563
(a) 564
565
(b) 566
Fig. 7 Variations of the interfacial shear and normal stresses with respect to positive thermal loads (a) interfacial 567
shear stress (τ) (b) interfacial normal stress σ 568
569
32
570
(a) 571
572
(b) 573
Fig. 8 Variations of the interfacial shear and normal stresses with respect to negative thermal loads (a) interfacial 574
shear stress (τ) (b) interfacial normal stress σ 575
576
33
577
(a) 578
579
(b) 580
Fig. 9 Variations of the interfacial shear and normal stresses with respect to positive mechanical loads (a) 581
interfacial shear stress (τ) (b) interfacial normal stress σ 582
583
34
584
(a) 585
586
(b) 587
588
Fig. 10 Variations of the interfacial shear and normal stresses with respect to negative mechanical loads (a) 589
interfacial shear stress (τ) (b) interfacial normal stress σ 590
591
35
592
593
(a) 594
595
(b) 596
Fig. 11 Stress field distribution along varying temperature with respect to different values of P0 (a) interfacial 597
shear stress (τ) (b) interfacial normal stress σ 598
Authors’ Technical Biographies 599
Dr. Dehestani is currently an associate professor of Civil Engineering at Babol Noshirvani University of 600
Technology, Babol, Iran. He is also a faculty member of the Structural and Earthquake Engineering at Babol 601
Noshirvani University of Technology. He received his Ph.D. degree in Structural and Earthquake Engineering 602
from Sharif University of Technology, Tehran, Iran. His research expertise lie in the area of reinforced concrete 603
structures, specifically the application of fiber-reinforced polymer composites and the fracture mechanics of a 604
variety of reinforced concrete components. He has already issued over 80 research papers with the most 605
prominent journals and conferences in the field of Civil Engineering. 606
607
Hojjat Samadvand received his M.Sc. in Structural Engineering at Babol Noshirvani University of Technology, 608
and is currently a Ph.D. candidate in Structural/Earthquake Engineering at Urmia University. His major areas 609
36
of interest are the mechanics of reinforced concrete structures and composite materials, as well as numerical 610
modeling. He has already published several articles with the leading international and national conferences on 611
Civil Engineering including ICCE and NCCE. 612
613
614
615
