Jian Et Al 2018 - FE Modeling of FRP-strengthened RC Beam Under Sustained Load-1
Jian Et Al 2018 - FE Modeling of FRP-strengthened RC Beam Under Sustained Load-1
Jian Et Al 2018 - FE Modeling of FRP-strengthened RC Beam Under Sustained Load-1
Research Article
Finite Element Modeling of FRP-Strengthened RC Beam under
Sustained Load
Copyright © 2018 Shiyong Jiang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
External bonding of FRP laminates to the tension soffit of concrete members has become a popular method for flexural
strengthening. However, the long-term field performance of FRP-strengthened RC members under service conditions is still
a concern, and more work needs to be done. Based on concrete smeared-crack approach, this paper presents a finite-element (FE)
model for predicting long-term behavior of FRP-strengthened RC beam, which considers the time-dependent properties of all
components including the aging effect of concrete. According to the comparison between theoretical predictions and test results,
the validity of the FE model is verified. The interfacial edge stresses in adhesive layer were determined through appropriate mesh
refinement near the plate end, and their time-dependent characteristics were investigated. The results show that creep of concrete
and epoxy resin cause significant variations of the edge stresses with time. According to the research in this paper, the FE approach
is found to be able to properly simulate the long-term behavior of the FRP-strengthened beam and help us better understand the
complex changes in the stress state occurring over time.
model such tension-stiffening effect appropriately and ac- interfacial stresses were accurately determined by the same
curately. Thus, it can be noted that the above influential FE model, with appropriate mesh refinement near the end of
factors are usually roughly approximated and even neglected CFRP plate. Additional numerical investigations were car-
[11, 16, 19]. On the other hand, in order to improve the ried out to study the changes of edge stresses over time
efficiency of solving process or reduce the difficulty in induced by the creep of materials.
mathematics, simplifications have been ordinarily in-
troduced in analytical method. Based on axial strain re- 2. Long-Term Deformation Monitoring
duction coefficient and curvature reduction coefficient,
Hong and Park [11] calculated the time-dependent change of Time-dependent deformation of a FRP-strengthened RC
the concrete stress in a simple and empirical way. Marı́ et al. beam under sustained load was monitored for 200 d. The
[16] assumed that the stresses of tension bars and FRP sectional dimension was 250 × 400 mm with a 30 mm
laminate remain constant over time. Although, reasonable thickness. The total and effective span was 3300 mm and
theoretical results are still obtained in their studies [11, 16], 2900 mm, respectively. D28 reinforcement (diameter is
there is no evidence to show those simplifications are 28 mm) was used for reinforcement on the compression side,
suitable for other cases, where the creep responses of ma- while D14 reinforcement (diameter is 14 mm) was used for
terials, the geometry of member, and reinforcement ratio reinforcement on the tension side. Under service state,
(for steel and FRP) are different. It is therefore implied that concrete stress is not high, which presents linear creep. In
the universality of the analytical method applying specific this study, adequate compression bars were used to control
simplification is not favorable enough. Additionally, it instantaneous stress in compressive concrete at lower level
should be noted that despite simplifications have been made, simulating the service condition. CFRP plate with length and
large amount of iterative calculations still may be needed to width of 2600 and 200 mm was bonded on the tension soffit
obtain the final results [11]. Moreover, the stresses at the of beam. D8 (diameter is 8 mm) reinforcement at interval of
adhesive interfaces near the bonding edges are responsible 200 mm was used for the stirrups. To avoid premature
for the debonding failures, and their redistribution with time debonding near the end of CFRP plate, CFRP sheets were
is an issue worth studying. Nevertheless, when determining used to wrap around the test beam. Figure 1 provides the
and investigating the time-dependent interfacial stresses, detailed information of the test beam.
cross section analyzing is no longer competent, and a quite The test beam was cast by commercial concrete. The 28-
different analysis procedure (particularly to interfacial day cylinder compressive strength of the concrete was
forces) needs to be carried out [22, 23]. In the related studies, 34.0 MPa. Other parameters of concrete are listed in Table 1.
it can be found that although precise and complicated Table 2 shows the properties of steel reinforcement. Table 3
derivations had been conducted, assumption not fully gives the properties of epoxy resin (applied as adhesive),
corresponding to the fact was still made in order to make the CFRP sheet (for avoiding premature debonding), and CFRP
mathematical process more manageable [22, 23]. The plate (for flexural strengthening). The reported material
stresses within the adhesive layer were assumed to be strengths above are measured mean values.
constant through the thickness of the adhesive layer in both The test beam was simply supported and loaded in four
studies of Benyoucef et al. [22] and Zhang and Wang [23], point bending. The sustained load F was 75% of the the-
which in fact vary strongly across the layer thickness [24]. oretical ultimate capacity Fu (on the basis of cross section
To eliminate the aforementioned deficiencies, finite el- analysis) of the same dimensional RC beam without
ement (FE) modeling may be a feasible scheme. Comparing strengthening. The beam was cured for 7 d, then placed
to the analytical method, to study the behavior of FRP- indoors for another 69 d before loading. In the test,
strengthened RC member under sustained load, the FE a dedicated experimental system, made up of vertical re-
method may provide a more powerful tool, which calculates action frame, screw jack, and steel distribution beam, was
numerically and does not operate on the basis of cross designed to apply and sustain the external load, as illus-
section analysis. Those specific simplifications mentioned trated in Figure 2. Load was applied to the beam through
above may also be avoided, which makes the FE method slowly lifting up the screw jack, meanwhile, its value was
more generally applicable. However, to date, studying the monitored by the force sensor at the top of the jack. As the
time-dependent performance by FE approach has not been value reached to 0.75 Fu (117.71 kN), screw jack was locked
extensively conducted. One of the likely reasons should be to prevent lifting or declining, which sustained the load.
that reasonably developing time-dependent constitutive When the deformation of test beam increased due to creep
models of materials can be a challenging task. behavior, the applied load might decrease, then the Jack
The major thrust of this paper is to propose a finite was unlocked and further lifted up to replenish the force to
element model to investigate the long-term response of FRP- the designed level (0.75Fu). During the long-term test, the
strengthened RC beam under sustained load. The time- error of applied load was controlled within ±1% of the
dependent characteristics of all components including the planned value.
evolution of concrete property along with time were As shown in Figure 1, in the midspan, resistance strain
modeled as user-defined subroutines in ABAQUS 6.12 [25]. gauges were bonded to the tension, compression steel, and
Besides, load sustaining experiment of a strengthened beam CFRP plate to measure the instantaneous strains only, since
was also conducted. By comparing the experiment and FE their long-term working performance is unreliable. Vi-
results, the validity of the FE model is demonstrated. Also, brating wire gauges were especially used to obtain both
Advances in Materials Science and Engineering 3
Compression
WG2 bars: 3-D28
The wrapped Steel distribution
CFRP sheet beam Tension
bars: 3-D14
TS1 TS3
TS2 Adhesive layer
CFRP plate
Front view of the test beam Setup of strain gauges at mid-span section
CFRP plate
FP1
FP2
Bottom view of the test beam
WG 1, 2: vibrating wire strain gauges
CS 1, 2, 3: compression steel gauges (electric resistance gauge)
TS 1, 2, 3: tension steel gauges (electric resistance gauge)
FP 1, 2: FRP plate gauges (electric resistance gauge)
Figure 1: Test beam details.
Table 2: Materials’ properties of steel reinforcement. The experiment was conducted indoors. Figure 3 shows
the recorded temperature and humidity, which presented
Yield strength Ultimate strength unavoidable variations with time. It should be noted that the
Reinforcement type
(MPa) (MPa) observed creep might be affected by the change of envi-
Tension, D14 461 625 ronmental conditions in the laboratory.
Compression, D28 420 570
Shear, D8 435 610
3. FE Modeling
Based on ABAQUS 6.12 [25], nonlinear numerical analysis
Table 3: Adhesive and CFRP properties. was performed to study the time-dependent characteristics
Ultimate of concrete beam strengthened with external FRP re-
Elastic inforcement. The FE model used herein is on the basis of the
strength in Thickness
modulus
tension (mm) one developed by Jiang et al. [26]; however, the current FE
(GPa)
(MPa) model in this paper was further updated and is more ad-
Epoxy resin vanced: the aging effect of concrete was taken into account
39.6 3.468 1.4
layer and the corresponding calculating method was presented,
CFRP sheet 3510 241 0.15 and also, the method for analyzing the time-dependent
CFRP plate 2482 174 1.4 interfacial stress was developed.
instantaneous and long-term strains in some places because 3.1. Modeling of Instantaneous Material Property. In this
of their reliable performance in long period of monitoring. study, concrete cracking was simulated by using the smeared-
The 1# wire strain gauge (WG1) was bonded on the top cracked approach. Unlike the discrete-crack approach, there is
surface of the beam. The 2# wire strain gauge (WG2) tied no need to predefine the crack paths in the smeared-cracked
with the middle compression bar was buried in the specimen approach. In addition, aiming to overcome the mesh-
when casting concrete. One dial gauge was installed at the nonobjectivity problem in the conventional smeared-crack
bottom midspan point and other two dial gauges installed at approach, the crack band model was used [27].
pedestals were used mainly to check the symmetrical nature The concrete was modeled within the framework
of the loaded beam. of the concrete-damaged plasticity model provided by
4 Advances in Materials Science and Engineering
Reaction frame
Force sensor
Screw jack
Steel distribution beam
Dial gauge
40
Highest temperature: 36.1°C 90
Highest relative humidity: 88%
80
Relative humidity (%)
30
Temperature (°C)
Average: 24.4°C 70
20 Average: 73%
60
50
Lowest temperature: 13.8°C Lowest relative humidity: 50%
10
0 50 100 150 200 0 50 100 150 200
Time (days) Time (days)
(a) (b)
Figure 3: Environmental conditions during test. (a) Temperature indoors. (b) Relative humidity indoors.
E0
ε
~p
ε εde1
E0
εp εe1 = σt/E0
Compression
fc
Based on Bazant and Planas [32], mesh sensitivity wrapped CFRP sheets for avoiding premature debonding
problem can be effectively overcome through applying the were reasonably ignored in the FE model based on the
crack band model [33], in which the displacement of crack relatively low stress level [20].
opening w equals to the cracking strain εcr accumulating
over the width of the crack band hc .
3.2. Modeling of Time-Dependent Material Property.
w εcr dh.
Under sustained load, concrete presents significant creep
(6) behavior, and additional stress-independent shrinkage de-
hc
formation is also considerable. The different time-dependent
According to Rots [34], when meshing concrete with characteristics of materials induce complicated stress re-
four-node plane stress element with √ integration points,
four distribution in structure along with time.
the crack band width is taken to be 2e, where e is the side The loading age is written as t0 , and the load sustaining
length of the element [27]. Thus, the tensile stress-crack period is from t0 to tn (tn > t0 ). The total time period t0 to tn
opening displacement relationship given by Equation (2) is divided by a number of discrete periods: t0 to t1 , t1 to
can be transformed into a stress-strain curve needed by FE t2 ,. . ., tn−2 to tn−1 , tn−1 to tn . This paper presents a quasielastic
program through Equation (6). constitutive model of concrete considering creep and
The damage evolution was also considered to better shrinkage based on age-adjusted effective modulus method
describe the nonlinearity of concrete. The relationship (Equations (8)–(10)). The model is in recursion form, which
between the damage factor d and plastic strain εp , which comforts codes programming and improves calculating
does not take stiffness degradation into account was de- efficiency. The detailed derivation process of Equations
fined. Damage factor can be determined as Equation (7) (8)–(10) can be found in Jiang et al. [26], which is not
according to Tao and Chen [35] for both uniaxial tension presented herein for brevity.
E tn−1
and compression. The patterns of damage are illustrated in
Figure 4. σ ∗ tn
1 + χ tn , tn−1 m
j1 bj tn−1 1 − e
−λj ·Δtn
(8)
(1 − σ/f)εp
· εn − Δε,
d , (7)
(1 − σ/f)εp + σ/E0
where σ is the uniaxial tensile or compressive stress and f is m
uniaxial tensile or compressive strength. Δε εsh tn − εsh tn−1 + A∗n,j · 1 − e−λj ·Δtn , (9)
In this study, the response of FRP-strengthened RC j1
beam under service state is investigated, therefore, the in-
stantaneous material properties of epoxy adhesive, steel A∗n,j A∗n − 1,j · e−λj ·Δtn − 1 + bj tn−2 · e−λj ·Δtn − 1
reinforcements, and CFRP plate were modeled as linear
χ tn , tn−2 ∗ (10)
elastic. Also, perfect bonds were assumed between adhesive- · σ tn−1 for n > 1.
E tn−2
·
concrete, adhesive-CFRP plate, and concrete-steel. The
6 Advances in Materials Science and Engineering
When n 1: 50 40
σ
A∗n1,j bj t0 · 0 ,
1 1
χ(t, τ) − . (13) that epoxy adhesive presented evident viscoelastic manner
1 − e−φ(t,τ) φ(t, τ)
[20, 21, 38]. As a result, the relaxation of interfacial stresses
Due to the progress of hydration, concrete properties caused by adhesive creep affects the strengthening mecha-
including strength and elastic modulus develop with time. nism, which therefore influences the long-term response of
The above time-dependent elastic modulus of concrete strengthened beam. In this research, to describe the decreased
E(ti )(i 0, 1, 2, . . . , n − 2, n − 1, n) was determined by the shear relaxation modulus of epoxy Ga (t) with time, Maxwell
time-dependent concrete strength fc (ti ) (concrete
strength chain was used [20, 21].
at time ti ) according to ACI 318 [31] (Ec 4730 fc ). The NG
time interval (t0 to t1 , t1 to t2 , . . ., ti to ti+1 , . . ., tn−2 to tn−1 ) Ga (t) Gu + Gi · e−t/τGi , (15)
was set small (less than 5 d) to obtain accurate enough result. i1
The elastic modulus of concrete was assumed to be constant
in each time interval, but renewed and updated when cal- where Gu is the shear modulus at infinite time, NG is number
culation of the latest time interval was finished. In this way, of Maxwell elements, Gi refers to the relaxed shear modulus
the aging effect of concrete was considered. This study of the ith Maxwell element, and τ Gi is material constant.
applied the evolution law of concrete strength proposed by Following Choi [20], NG 1, τ G1 2 d, and Gu G1 /5. In
ACI 209 [37]. addition, the creep of epoxy in axial direction was also taken
into account, since near the midspan, where section moment
fc ti fc28 · ,
ti is large and the instantaneous stress of epoxy resin in axial
(14)
a + b · ti direction usually reaches a noticeable value, which causes
nonignorable creep response. It is assumed that Poisson’s
where fc28 is the concrete mean compressive strength at ratio of adhesive ]a remains constant with time [39, 40], and
28 d in MPa and ti is the age of concrete in days. a 4.0 and so, the relaxation modulus in axial direction Ea (t) is derived
b 0.85 are material constant for concrete using ordinary from that in the shear direction or vice versa: Ea (t) 2(1 +
Portland cement. In this study, the test beam was loaded at ]a )Ga (t) [14, 15].
the age of 76 d after casting concrete. Thus, the calculated The viscous property of CFRP was determined as follows
76 d strength 37.7 MPa and its corresponding elastic [41]:
modulus 29.0 GPa were used to determine the instantaneous
σ σ
response of the test beam under loading. The evolution of εCFRP (t) ε0 · sinh + m · sinh · tn1, (16)
concrete property along with time is shown in Figure 5. σm σm
The above time-dependent property of concrete was
where εCFRP (t) time-dependent strain; t time after
programmed as FORTRAN code in user-defined subroutine
loading, in hour; and σ applied stress. The ε0 0.379; σ m
UEXPAN and USFLD in ABAQUS. Concrete creep co-
68950 MPa; m 0.0011; n1 0.123 are material constants
efficient and shrinkage strain was calculated using ACI 209
[41]. Similar to concrete, the above mechanical properties of
code [37], and during the calculation, the measured average
adhesive and CFRP related to time were also programmed in
humidity (73%) was used.
user subroutines UEXPAN and USFLD.
In this study, both shear and axial creep of adhesive resin
were modeled in the FE approach. For FRP-strengthened
system, the performance of adhesive is of crucial importance 3.3. Solution Strategy. In the FE model, only a half of the
in providing effective stress transfer. It has been demonstrated beam was included by taking advantage of symmetry about
Advances in Materials Science and Engineering 7
midspan plane, which effectively reduces calculating time. with the experimental results. The predicted time-dependent
The concrete, epoxy adhesive, and CFRP plate were modeled deflection at the 50, 99, 155, and 200 d after sustaining load is
by plane stress element CPS4. The steel reinforcement was 4.97, 5.24, 5.42, and 5.52 mm, respectively, which is close to
modeled using truss elements. the corresponding test values of 5.00, 5.43, 5.52, and
During the FE analyses, step-by-step procedures were 5.59 mm, with differences of only 0.6%, 3.5%, 1.8%, and
applied for calculations. Instantaneous behavior of the beam 1.3%, respectively. The time-dependent strains of top con-
was calculated first, which was set as the initial condition of crete, concrete near compression bars, are also predicted
the calculation of long-term response afterwards. Then, with good accuracy as shown in Figure 6(b). Besides, The
based on the time-dependent properties of materials, nu- predicted results (FE2) without considering the further
merical simulation of long-term behavior of the test beam evolution of concrete property are also displayed in Figure 6.
was carried out. In this study, if not otherwise stated, square As expected, neglecting the evolution of concrete property
element with an element size of 10 mm was used to model leads to overestimated results, however, with slight effect
concrete. Matching element sizes were chosen to represent only. The reason is that the maturity of the used commercial
the steel bars, the epoxy layer, and the FRP plate. The epoxy concrete has been good enough after 76 d of aging (the time
layer and the FRP plate were each modeled with two layers of that the long-term test commenced), as shown in Figure 5.
elements in their thicknesses. Nevertheless, when discussing The result of FE1 and FE2, respectively, refers to the case
the time-dependent interfacial stresses at the bonding edge, with and without considering the evolution of concrete
mesh refinement is needed, which is particularly mentioned property. The additional deflection (Δf) is defined as the
in the later section. delayed deflection occurring in load sustaining period. The
total additional deflection (Δftotal ) is the delayed deflection
4. Results and Discussion after 200 d of sustaining load. The ration between Δf and
Δftotal represents the creep progress of the test beam.
4.1. Verification of the Recommended FE Model. Under in- As shown in Figure 6, in this study, the test results of
stantaneous loading, comparison of FE-predicted results and both strain and deflection do not present continuing growth
experimental results of the specimen is shown in Table 4. As along with time, which is likely to be caused by the fluc-
shown in Figure 1, the tension steel gauge, compression steel tuations of environmental humidity and temperature [11].
gauge, and FRP plate gauge are labeled as TS, CS, and FP. However, effects of climate changing were currently not
Several strain gauges were employed at the place of midspan: considered during FE modeling; therefore, the predicted
TS1, TS2, and TS3 were bonded to each of the tension bar; long-term deformation increases continuously and
CS1, CS2, and CS3 were bonded to each of the compression smoothly. Whereas, from the comparison above, the ac-
bar; and FP1 and FP2 were adjacently bonded to the CFRP curacies of the FE results are favorable enough.
plate. In Table 4, the strain of tension bar, compression bar, The proposed FE model rationally predicts the cracking
and CFRP plate is the average value of the measured data by behavior of concrete. Figure 7(a) shows the experimentally
TS1, TS2, TS3; CS1, CS2, CS3; FP1, and FP2, respectively. observed cracking pattern of test beam under instantaneous
The strain of the top concrete of test beam was obtained by loading. In this study, because adequate stirrups were
the 1# wire strain gauge (WG1). Table 4 shows that the test deployed to prevent the occurrence of shear failure, in the
results are reasonably predicted by the FE model. earlier stage of loading, the observed cracks were flexural
From Table 4, the compressive strain of top concrete is cracks in vertical direction mainly. With the increasing of
277 με under instantaneous loading, indicating the stress load, new cracks gradually emerged in the bending-shear
level does not exceed 40 percent of strength, thus, applying section, propagating vertically first but then obliquely af-
the linear creep theory for concrete is appropriate in this terwards. As shown in Figure 8, the FE results appropriately
paper. The time-dependent midspan deflection is shown in simulate this phenomenon. The effect of tension stiffening,
Figure 6(a). The time-dependent strains of top concrete which causes the concrete between adjacent cracks to suffer
and the concrete near compression bars are presented in tensile stress, is also simulated by the FE model, as presented
Figure 6(b), which were experimentally obtained by the in Figure 8(c). During the test, further development of in-
vibrating wire gauges WG1 and WG2, respectively. Distinct stantaneous cracks during load sustaining period was clearly
additional long-term deformation was observed during load observed (Figure 7(b)). The combined effect of materials’
sustaining period. After time passage of 200 d, the deflection, creep behaviors and decreasing of section stiffness induced
strain of top concrete, and strain of concrete near com- by further formation of cracks causes the additional long-
pression bars were respectively 56.7%, 98.6%, and 59.2% term deformation of test beam.
higher than their initial value under instantaneous loading. The instantaneous cracks are marked by color grey; the
The increasing of deformation corresponds to the typical law further development of cracks in long-term are marked by
of RC member under sustained load: quick in early period color red.
and slow in later time. From Figure 6(a), it is shown that The FE model in this paper has been further validated
after sustaining load for 78 d (39% of the total time period), against the test results in other literatures [42, 43]. Totally,
most part of additional long-term deflection (87.2%) is seven specimens are selected herein: four of them are from
finished. Also, as shown in Figure 6(b), there are no clear Sobuz et al. [42] including one control beam (not
increases for the strains of concrete after 78 d of sustaining strengthened) and three CFRP-strengthened beams and the
load. The results (FE1) predicted by the FE model agree well other three are from Tan and Saha [43]. Sobuz et al. [42]
8 Advances in Materials Science and Engineering
6
700
600
4
75
400
3
Δ δ Δ 50 300
2
200
25
1 100
78 days after sustaining load
78 days after sustaining load
0 0 0
0 50 100 150 200 0 50 100 150 200
Time (days) Time (days)
Deflection-test Deflection-FE2 Top concrete-test
Deflection-FE1 Δf/Δftotal (test result) Concrete near compression bar-test
Top concrete-FE1
Top concrete-FE2
Concrete near compression bar-FE1
Concrete near compression bar-FE2
(a) (b)
Figure 6: Test and FE results of the specimen. (a) Deflection at the midspan. (b) Concrete strains.
Pure bending section studied the effects of strengthening scheme and level of
Wrapped CFRP sheets
sustained load on long-term response of RC beam
strengthened by CFRP. Four representative specimens are
selected herein to demonstrate the accuracy of the proposed
FE approach, namely, CBC (not strengthened, the
control beam), FBC-1L, FBC-2L, and FBC-3L. Tan and Saha
[43] tested the time-dependent responses of nine GFRP-
The top face of test beam strengthened specimens. Three typical specimens are chosen
herein, namely, GB3-40, GB3-49, and GB3-59. Table 5
(a)
presents the detailed information of these specimens.
Pure bending section According to Sobuz et al. [42] and Tan and Saha [43], the
viscoelasticity of FRP plate was ignored. The aging effect of
concrete was considered based on the aforementioned method
in Section 3.2. For the specimens from Sobuz’s test [42], the
ultimate creep coefficient and ultimate shrinkage strain of
concrete were 1.87 and 418 με, which were determined
through 3.2.1.1 Compressive creep test and 3.2.1.2 Drying
(b)
shrinkage test in Sobuz’s paper [42]. For the specimens from
Tan’s study [43], the specific laws of concrete creep and
Figure 7: Cracking pattern observed experimentally (unfolded- shrinkage have been given in their paper [43], which were,
drawing). (a) At instantaneous loading. (b) After time passage therefore, directly used in the FE simulations. From Figure 9,
(200 d). the proposed FE approach well predicts the test results.
Advances in Materials Science and Engineering 9
(a)
Cloud figure of the first principal strain
E, max. in-plane principal
(avg: 10%)
+4.627e – 02
+9.600e – 04
+8.576e – 04
+7.553e – 04
+6.529e – 04
+5.506e – 04
+4.482e – 04
+3.459e – 04
+2.435e – 04
+1.412e – 04
+3.880e – 05
–6.356e – 05
(c)
Figure 8: FE cracking pattern. (a) Cracking pattern as 50% of instantaneous load applied. (b) Cracking pattern as 100% of instantaneous
load applied. (c) Principle stress presented by vector field (MPa).
4.2. Time-Dependent Interfacial Edge Stresses. The bonding to study the time-dependent edge stresses of adhesive in-
interfaces are critical regions that transfer stresses between terface through the proposed FE approach. The FE model
concrete and FRP. One of the most concerned issues of RC was established on the basis of the test beam in the ex-
beams reinforced by FRP is the premature debonding at the perimental study of this paper. The responses of specimen
plate edges or near cracks, where the stresses (both shear and under different levels of applied load F (23.54 kN, 47.08 kN,
vertical normal) in the adhesive layer are relatively high. and 70.62 kN) were simulated. Since the wrapped CFRP
In this paper, numerical investigations were performed sheets near the end of beam were not modeled, their
10 Advances in Materials Science and Engineering
2.4
12
1.8 10
Deflections (mm)
Deflections (mm)
8
1.2
6
0.6
0.0
0 50 100 150 200 0 100 200 300 400 500 600 700 800
Time (days) Time (days)
CBC-test CBC-FE GB3-59-test GB3-59-FE
FBC-1L-test FBC-1L-FE GB3-49-test GB3-49-FE
FBC-2L-test FBC-2L-FE GB3-40-test GB3-40-FE
FBC-3L-test FBC-3L-FE
(a) (b)
Figure 9: FE versus test deflections at midspan. (a) FE and test results for Sobuz’s [42] specimens. (b) FE and test results for Tan’s [43]
specimens.
influences on the localized interfacial stresses were not layer. A graded and matching mesh was adopted for the
discussed currently. concrete. The stress distribution near the plate end under
As shown in Figure 10, due to the existence of singular instantaneous loading is given in Figures 11 and 12 (applied
points (Point A and B), in order to obtain accurate stresses, load F 70.62 kN), whose characteristics correspond to the
the region near the plate end needs a particularly fine mesh. findings of previous researches [24, 44]. It shows that high
Following the recommendation of Teng et al. [24], in the vertical normal (peeling) and shear stresses exist near the
vertical direction, for the adhesive layer, the top part (near bonding end, which are responsible for the debonding
concrete-adhesive interface) and the bottom part (near failures widely reported in tests. Also, the normal and shear
adhesive-FRP plate interface) were, respectively, meshed by stresses vary strongly across the thickness of the adhesive
two layers of elements (the height of each element is layer in the vicinity of the end of the plate. This aspect cannot
0.1 mm). The rest part of the adhesive layer was evenly be captured by existing approximate analytical solutions
meshed by other three layers of elements. In the horizontal [22, 23] developed with the assumption that stresses are
direction, a graded mesh was used starting with an aspect invariant across the adhesive thickness.
ratio of 1 for the minimum height (0.1 mm) elements. AC interface refers to adhesive-to-concrete interface, PA
The CFRP plate has the same mesh pattern as the adhesive interface refers to plate-to-adhesive interface, and MA refers
Advances in Materials Science and Engineering 11
Loding
Symmetric boundary condition
Support
Concrete
A Epoxy
B FRP plate
0.4 0.5
0.3 0.4
Vertical normal stress (MPa)
0.2
0.3
0.1
0.2
0.0
0.1
–0.1
–0.2 0.0
0 5 10 15 20 0 5 10 15 20
Distance from the plate end (mm) Distance from the plate end (mm)
AC interface AC interface
MA section MA section
PC interface PC interface
(a) (b)
Figure 11: Interfacial Stress distributions under instantaneous loading. (a) Vertical normal stress. (b) Shear stress.
to the horizontal section of the adhesive layer obtained by significant relaxations for both shear and vertical normal
a horizontal cut at midthickness. stresses at the plate end. With the increase of sustained load,
In this research, the time-dependent characteristic of the level of instantaneous stress in epoxy layer is promoted,
interfacial edge stress at point A (in AC interface) is espe- which causes more evident creep response and stress re-
cially discussed, which is one of the most important factors laxation. In Figure 13, the relaxation rate of stress is fast in
causing debonding failures. The changes of interfacial shear the earlier stage, but almost slows down to zero afterwards.
and vertical normal stresses with time are shown in Fig- Such phenomenon is believed to be induced by the shear
ure 13. In this case, only the creep of adhesive in shear creep law of adhesive adopted herein, which presents that
direction is considered (both FRP and concrete are modeled most of creep deformation develops within relatively short
as time-invariant materials). Figures 13(a) and 13(b) show time period compared to the well-known long creep period
the viscous flow of epoxy in shear direction leading to of concrete [21]. For the results shown in Figure 14, the axial
12 Advances in Materials Science and Engineering
S, S22
(avg: 75%) A
0.266
0.242
0.218
0.194
0.170
0.146
0.122
0.098
0.074
0.050
0.026
0.002
–0.022
–0.046
(a)
S, S12
A
(avg: 75%)
0.076
0.102
0.127
0.153
0.179
0.205
0.231
0.257
0.282
0.308
0.334
(b)
Figure 12: Contours of stress near the plate end (MPa). (a) Vertical normal stress. (b) Shear stress.
creep of epoxy is only considered to separately investigate its stresses, however, with a much more inferior effect. The
effect on the edge stress. It is shown that like the shear creep, likely reason is that near the plate end, the section moment is
the axial creep of adhesive also causes relaxations of the edge small, and there is low axial stress in the adhesive layer;
Advances in Materials Science and Engineering 13
0.40
0.40 0.45
0.35 0.35
0.30 0.40
Vertical normal stress (MPa)
0.25
0.30 0.20 0.35
0.15
Figure 13: The separate effect of shear creep of adhesive on the edge stresses. (a) Variation of vertical normal stress with time. (b) Variation
of shear stress with time.
0.40
0.45
0.35
0.40
Vertical normal stress (MPa)
0.30 0.35
Shear stress (MPa)
0.25 0.30
0.25
0.20
0.20
0.15
0.15
0.10 0.10
0.05 0.05
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500
Time (days) Time (days)
Under load as 23.54 kN Under load as 23.54 kN
Under load as 47.08 kN Under load as 47.08 kN
Under load as 70.62 kN Under load as 70.62 kN
(a) (b)
Figure 14: The separate effect of axial creep of adhesive on the edge stresses. (a) Variation of vertical normal stress with time. (b) Variation of
shear stress with time.
hence, the axial creep is not obvious, which does not clearly Therefore, the edge stresses present lower increases with
influence the edge stresses. time, comparing to the case that concrete creep works
The combined effect of concrete creep and adhesive separately. As the relaxation effect is evident enough, which
creep was also investigated, as illustrated in Figure 15. It is is larger than the increasing effect of concrete creep, the
found that the creep of concrete individually leads to in- stresses even show decreases in the earlier stage (curve A, B,
creases of both interfacial shear and vertical normal stresses and C in Figure 15). Following that, the creep of adhesive is
along with time, which may eventually lead to premature almost completed, while the creep of concrete is still sig-
debonding. This corresponds to the findings in previous nificant; thus, there is an ongoing increase for the edge
studies [14, 15, 45], which demonstrates the rationality of stresses afterwards. Besides, since the relaxation effect of
the suggested FE approach. However, as the creep of adhesive creep in axial direction is smaller, its counter-
concrete and adhesive occurs at the same time, the re- action to the increasing effect of concrete creep on edge
laxation effect by adhesive creep counteracts the increasing stress is less evident comparing to that of adhesive creep in
effect by concrete creep on the edge stresses to some extent. shear direction.
14 Advances in Materials Science and Engineering
0.9
0.7 0.8
Vertical normal stress (MPa)
0.3 0.4
Stress at instantaneous loading Applied load F: 47.08 kN
0.3 Stress at instantaneous
0.2
Curve B loading
0.2
0.1
Applied load F: 23.54 kN 0.1 Applied load F: 23.54 kN
0.0
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500
Time (days) Time (days)
Creep in concrete only Creep in concrete only
Creep in concrete and axial creep in adhesive Creep in concrete and axial creep in adhesive
Creep in concrete and shear creep in adhesive Creep in concrete and shear creep in adhesive
Creep in concrete only Creep in concrete only
Creep in concrete and axial creep in adhesive Creep in concrete and axial creep in adhesive
Creep in concrete and shear creep in adhesive Creep in concrete and shear creep in adhesive
Creep in concrete only
Creep in concrete and axial creep in adhesive
Creep in concrete and shear creep in adhesive
(a) (b)
Figure 15: The combined effect of adhesive creep and concrete creep on the edge stresses. (a) Variation of vertical normal stress with time.
(b) Variation of shear stress with time.
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