Flexural Performance of Cracked Reinforced Concrete Beams Strengthened with Prestressed CFRP Sheets under Repeated Loads

Abstract

Because researchers are aiming to restore the deformation and minimize the crack width of existing concrete structures, the strengthening technology of prestressed carbon-fiber-reinforced plastic (CFRP) is currently the focus of many studies and applications. In terms of the strengthening of a prestressed CFRP sheet on the flexural performance of cracked reinforced concrete beams under repeated loads, a four-point bending test of 12 beams was conducted considering the prestress degree reflected by the amount and the prestress force of the CFRP sheet. The longitudinal strengthened CFRP sheet was bonded on the bottom surface of the test beam and fixed by U-jacket CFRP sheets at the ends after tensioning. The strains of concrete, longitudinal tensile steel bars and CFRP sheets were measured at the pure bending segment of test beams, while the cracks, midspan deflection and failure pattern were recorded. The results show that the normal strain on the mid-span section of the strengthened beams by the prestress CFRP sheets was fitted for the assumption of plane section, the cracks and mid-span deflection decreased with the prestress degree of the CFRP sheets to provide better serviceability for the strengthened beams, the load capacity could be increased by 41.0–88.8% at the yield of longitudinal tensile steel bars and increased by 41.9–74.8% at the ultimate state and the ductility at the failure state was sharply reduced by 54.9–186%. The peeling off of broken CFRP sheets played a role in controlling the failure pattern of the strengthened beams under repeated loads. Finally, methods for predicting the bending performance of reinforced concrete beams strengthened by prestressed CFRP sheets were proposed. This study enriches the knowledge about damaged reinforced concrete beams that were strengthened with prestressed CFRP sheets.

Full text

Citation: Wang, H.; Li, C.; Song, S.;
Wang, Y.; Meng, Q.; Li, F. Flexural
Performance of Cracked Reinforced
Concrete Beams Strengthened with
Prestressed CFRP Sheets under
Repeated Loads. Buildings 2023,13,
2115. https://doi.org/10.3390/
buildings13082115
Academic Editors: Denise-Penelope N.
Kontoni and Carmelo Gentile
Received: 26 July 2023
Revised: 6 August 2023
Accepted: 17 August 2023
Published: 21 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
buildings
Article
Flexural Performance of Cracked Reinforced Concrete Beams
Strengthened with Prestressed CFRP Sheets under
Repeated Loads
Huijuan Wang 1, Changyong Li 2, Sihao Song 3, Yao Wang 4, Qingxin Meng 4and Fenglan Li 1,2,3,*
1Low-Carbon Eco-Building Materials Technology Innovation Center of Xuchang City, Civil Engineering and
Architecture School, Zhongyuan Institute of Science and Technology, Zhengzhou 450046, China;
wanghuijuan1@zykjxy.wecom.work
2
International Joint Research Lab for Eco-Building Materials and Engineering of Henan, Collaborative Innovation
Center for Efficient Utilization of Water Resources, North China University of Water Resources and Electric Power,
Zhengzhou 450045, China; lichang@ncwu.edu.cn
3School of Civil Engineering and Communications, North China University of Water Resources and Electric
Power, Zhengzhou 450045, China; z20211030391@stu.ncwu.edu.cn
4
China Construction Seventh Engineering Bureau Co., Ltd., Zhengzhou 450004, China; 0591502@sina.com (Y.W.);
qingxinmeng1995@163.com (Q.M.)
*Correspondence: lifl64@ncwu.edu.cn
Abstract:
Because researchers are aiming to restore the deformation and minimize the crack width
of existing concrete structures, the strengthening technology of prestressed carbon-fiber-reinforced
plastic (CFRP) is currently the focus of many studies and applications. In terms of the strengthening
of a prestressed CFRP sheet on the flexural performance of cracked reinforced concrete beams under
repeated loads, a four-point bending test of 12 beams was conducted considering the prestress degree
reflected by the amount and the prestress force of the CFRP sheet. The longitudinal strengthened
CFRP sheet was bonded on the bottom surface of the test beam and fixed by U-jacket CFRP sheets
at the ends after tensioning. The strains of concrete, longitudinal tensile steel bars and CFRP sheets
were measured at the pure bending segment of test beams, while the cracks, midspan deflection and
failure pattern were recorded. The results show that the normal strain on the mid-span section of the
strengthened beams by the prestress CFRP sheets was fitted for the assumption of plane section, the
cracks and mid-span deflection decreased with the prestress degree of the CFRP sheets to provide
better serviceability for the strengthened beams, the load capacity could be increased by 41.0–88.8%
at the yield of longitudinal tensile steel bars and increased by 41.9–74.8% at the ultimate state and
the ductility at the failure state was sharply reduced by 54.9–186%. The peeling off of broken CFRP
sheets played a role in controlling the failure pattern of the strengthened beams under repeated loads.
Finally, methods for predicting the bending performance of reinforced concrete beams strengthened
by prestressed CFRP sheets were proposed. This study enriches the knowledge about damaged
reinforced concrete beams that were strengthened with prestressed CFRP sheets.
Keywords:
cracked reinforced concrete beams; prestressed CFRP sheet; bending performance;
repeated load; strengthening effect
1. Introduction
Within the service life of building concrete structures, retrofitting and strengthening
are necessary activities due to the deterioration of structural materials and changes in
structural function. This ensures the serviceability of existing concrete structures with
rational loading reliability [
1
,
2
]. With the advantage of being self-lightweight and high-
strength and having the right corrosion resistance, fiber-reinforced plastic (FRP) composites,
including glass FRP (GFRP), basalt FRP (BFRP) and carbon FRP (CFRP), are becoming
the preferred materials for strengthening concrete structures. Normally, FRP sheets are
Buildings 2023,13, 2115. https://doi.org/10.3390/buildings13082115 https://www.mdpi.com/journal/buildings

Buildings 2023,13, 2115 2 of 19
externally bonded along tensile surfaces or vertical sections of concrete flexural members
to enhance their bending capacity or shear resistance [
3
–
6
]. It is easier to bond reasonable
layers and numbers of FRP sheets on the surface of concrete with an adhesive, while rational
measures, including fiber nails, fiber/steel depression strips and fiber/steel stirrups, can be
used to ensure the bond of FRP sheets to the surface of concrete with deformation under
load [
3
,
7
–
11
]. Another approach utilizes near-surface-mounted FRP strips by placing the
FRP strips in the cutting grooves of the concrete surface with epoxy paste, which can help
prevent the FRP from debonding from the concrete and improve the bearing capacity of
strengthened reinforced concrete beams compared with those of externally bonded FRP in
the case of an equal amount of reinforcement being used [
12
–
14
]. However, methods that
utilize directly bonded FRP sheets/strips cannot make use of the advantage of FRP’s high
strength. Therefore, they are more applicable to GFRP and BFRP with tensile strengths
below 1500 MPa rather than CFRP with high tensile strength of over 2300 MPa [
4
]. In
addition to its high mechanical properties, CFRP has the advantage of excellent resistance
to corrosion, fatigue and creep compared to GFRP and BFRP. This also confirms that CFRP
can be used as the main load-bearing component in engineering applications to ensure the
long-term safety and reliability of structures [15,16].
Theoretically, prestressing CFRP can not only help us reduce the total amount of CFRP
required to effectively utilize its high strength but can also help restore deformations, and
the width of existing cracks can be minimized for strengthened structures with restored
serviceability [
17
–
25
]. In regard to prestressed CFRP sheets, effective strength utiliza-
tion certainly relies on the anchorage approach. During the development of prestressing
technology, different anchorage systems have been innovated, including the full-length
self-anchored with adhesive [
19
–
21
], flat anchorage [
22
], steel depression plate [
24
], CFRP
sheet bonded with steel sheet fixed in a steel pocket [
25
], flat anchorage with circular
tooth [
26
,
27
] and a wedge-extrusion bond anchorage system [
28
]. Anchorage efficiency
has become high with the development of new types of anchorage systems. Most research
has indicated that reinforced concrete beams strengthened with prestressed CFRP sheets
presented higher cracking, yielding and ultimate resistances as compared to beams with
directly bonded CFRP sheets and that increasing the prestressing level has a significant
positive influence on loading performance while decreasing the ductility of strengthened
beams. Meanwhile, in one study, the initial prestress loss was around 5%, which was
dominated by slippage of the end anchorage system, and the average prestress loss was
less than 2% in dry and wet environmental conditions under sustained load; however, the
adhesive bonding was weakened by the moisture penetration [
29
]. After a year of exposure
under sustained loading, the strengthened beams were subjected to a four-point bending
test in which the cracking load decreased by up to 23.9%, and there were insignificant
reductions of both the yield and ultimate loads, while the cracking load and stiffness could
be improved through moisture exposure [
30
]. Additionally, reinforced concrete slabs and
beams strengthened using prestressed CFRP sheets behaved according to their expected
performance under fatigue and cyclic actions [
21
,
22
]. This all reveals that using reinforced
concrete beams strengthened with prestressed CFRP sheets is a reliable method. However,
the exposure of the strengthened beams in chloride-containing environments increased
the prestress losses, and the corrosion of metal anchor could induce the anchor pullout
failure of the prestressed CFRP sheets [
31
,
32
]. This reminds us that the durability of pre-
stressed CFRP sheets should be a concern when the reinforced structures are exposed to a
corrosive environment.
Combined with the practical applications, some experimental studies have been
conducted to assess the effect of existing damage before strengthening on the loading
performance of existing reinforced concrete structures strengthened by using prestressed
CFRP. Xie et al. [
33
] reported that the coupling action of corrosion- and cyclic-overload-
induced damage remarkably reduced the flexural stiffness of reinforced concrete beams
but did not greatly impact the ultimate load of the strengthened beams. After exposure
to cyclic overloading, the strengthened beams had a retention of the ultimate load of over

Buildings 2023,13, 2115 3 of 19
80% and a similar failure mode of concrete crushing. Liu et al. [
27
] found that the existing
cracks reduced the flexural stiffness by 27% for a full-scale hollow-section beam taken from
an old bridge, but the cracking resistance, the flexural stiffness and the bearing capacity
of the beam after repairing existing cracks by filling them with an epoxy adhesive could
be strengthened by using the prestressed CFRP plates, while the ultimate failure mode
transferred from under-reinforcement to over-reinforcement with an increasing amount of
CFRP plates.
Generally, most studies have been conducted for strengthened reinforced concrete
beams under static loads, and there has been a lack of studies on the effect of existing
damage on strengthened reinforced concrete beams under repeated loading. Therefore, this
paper carried out an experimental study of cracked reinforced concrete beams strengthened
with different layers and prestress levels of CFRP sheets under repeated loads. A total
of twelve reinforced concrete beams were fabricated and pre-cracked under static load to
simulate the weakened status of engineering beams. Considering one-layer or two-layer
CFRP sheets with different prestress to represent different strengthening conditions, the
flexural behaviors were determined, including the concrete strain along the depth of the
mid-span section, the crack distribution and crack width, the mid-span deflection, the yield
load and the ultimate load. Combined with the theoretical analysis, a method for predicting
the effective prestress of CFRP sheet, the flexural stiffness and the bending capacity of the
strengthened beams is proposed, and the ductility of the strengthened beams is discussed
in view of test factors.
2. Materials and Experiment
2.1. Raw Materials
As planned, the pre-cracked test beams were strengthened by prestressed CFRP sheets.
The CFRP sheets were produced by Toray Joint-stock Company, Japan, with a depth of
0.167 mm and a width of 100 mm. The tested ultimate tensile strength f
fu
= 4060 MPa with
a tensile modulus of elasticity of 2.42
×
10
5
MPa and an elongation of 1.71%. The adhesive
was produced by Dalian Kaihuaxin Tech-Eng Co., Ltd., and was specially used for the
bonding of CFRP. The main properties are presented in Table 1.
Table 1. Properties of adhesive for CFRP sheets.
Tensile
Strength (MPa)
Flexural
Strength (MPa)
Compressive
Strength (MPa)
Shear Strength
(MPa)
Tensile
Modulus of
Elasticity (GPa)
Elongation (%)
Bond Strength
to Concrete
(MPa)
Non-Volatile
Matter
Content (%)
51.5 75.4 89.6 18.3 31.0 1.70 3.60 99.5
The concrete was designed at strength grade of C50 with a characteristic cubic com-
pressive strength of 50 MPa and a target cubic compressive strength of 58.2 MPa for the
mix proportion design [
34
]. It was made of ordinary silicate cement of 42.5 strength grade,
river sand with modulus of 2.82, continuous grading crushed limestone with maximum
particle size of 20 mm, tap water and high-performance water reducer. The properties of
these raw materials met related specifications in Chinese codes [
35
–
37
]. The weight method
was used to mix proportion of concrete with assumption that the unit weight per volume
was 2450 kg/m
3
[
34
,
38
]. The dosages of cement, water, sand, crushed limestone and water
reducer were 530 kg/m
3
, 185 kg/m
3
, 606 kg/m
3
, 1127 kg/m
3
and 2.65 kg/m
3
. The slump
of fresh concrete was controlled at (100
±
20) mm. The test beams were cast in four batches
of concrete; each batch was accompanied by six cubic specimens of 150 mm dimension.
The test beams and cubic specimens were cured under the same conditions for the same
period until testing. The cubic compressive strength (f
cu
) and splitting tensile strength (f
t
)
of concrete were tested on a electro-hydraulic servo testing machine according to Chinese
code GB50081 [
39
]; results are presented in Table 2. For the relevant analysis, the cylinder
compressive strength (fc) can be taken as 0.76fcu [40].

Buildings 2023,13, 2115 4 of 19
Table 2. Tested concrete strengths and strengthening parameters of test beams.
Beam No. fcu
(MPa)
ft
(MPa) Layer of CFRP Pre-Tension Force (kN) Loading Method
JZL-0a 60.1 3.71 0 0 Statistic to determine the cracking load
JZL-0b 60.1 3.71 0 0 Statistic to determine the cracking load
JZCL-1a 60.1 3.71 0 0 Repeated as a reference
JZCL-1b 56.7 3.64 0 0 Repeated as a reference
YJCL-2a 56.7 3.64 1 20 Repeated
YJCL-2b 56.7 3.64 1 20 Repeated
YJCL-3a 59.6 3.69 2 40 Repeated
YJCL-3b 59.6 3.69 2 40 Repeated
YJCL-4a 59.6 3.69 1 30 Repeated
YJCL-4b 61.5 3.78 1 30 Repeated
YJCL-5a 61.5 3.78 2 60 Repeated
YJCL-5b 61.5 3.78 2 60 Repeated
The hot-rolled ribbed steel bar with strength grade HRB335 and diameter of 12 mm
was used for the longitudinal tensile reinforcement and the construction in compression,
the measured yield strength f
y
= 455 MPa, with an ultimate strength of 530 MPa and a
total elongation of 18.4%. The hot-rolled ribbed steel bar with strength grade HRB235 and
diameter of 8 mm was used for the stirrups; the measured yield strength f
yv
= 347 MPa
with an ultimate strength of 496 MPa and a total elongation of 23.6%. The mechanical
properties were measured according to Chinese code GB/T 228.1 [41].
2.2. Preparation of Test Beams
Considering the prestress degrees of the CFRP sheets expressed by one-layer and
two-layer CFRP sheets with two levels of prestress, four groups of test beams were prepared
for testing. Additionally, two groups of reinforced concrete beams were required: one was
used for determining the actual cracking loads of pure bending concrete of test beams
under static loads, and the other was used for a reference of the flexural performance of
reinforced concrete beams without strengthening by CFRP sheets under repeated loads.
Therefore, 6 groups of test beams were needed in this experiment. With two beams under
the same conditions as a group, twelve test beams were designed, as presented in Table 2.
The width b= 150 mm and depth h= 300 mm for a rectangular section, and the length was
3.0 m for a span l
0
= 2.7 m. Meanwhile, the pre-tension forces 20 kN and 30 kN of a layer of
CFRP corresponded to 29.5% and 44.2% of tested ultimate tensile strength f
fu
= 4060 MPa.
That is, the control tensioning stresses of CFRP sheets were 0.295ffu and 0.442ffu.
To ensure the beams strengthened with prestressed CFRP sheets failed in flexure as
expected, rational longitudinal tensile reinforcement with enough stirrups was used to
avoid the shear failure [
3
,
4
]. The reinforced concrete beams were designed with reinforce-
ments of three longitudinal tensile steel bars with concrete cover of 25 mm and stirrups
with spacing of 100 mm. The effective sectional depth h
0
= 269 mm. The reinforcement ratio
of longitudinal tensile steel bars was 0.84%, and the stirrups ratio was 0.67%. The designed
bending capacity of reinforced concrete beams was 38.6 kN
·
m, while the designed shear
capacity was 142.1 kN. Details of the beams are exhibited in Figure 1.
Buildings 2023, 13, x FOR PEER REVIEW 5 of 20
Figure 1. Geometry and reinforcement details of reinforced concrete beam (unit: mm).
2.3. Pre-Crack Loading and Strengthening Method
The target pre-crack width was the limit of 0.20 mm for reinforced concrete beams in
normal environment [40]. The cracks were made by four-point bending test [42,43]. The
loads were multi-step with increments of 20% of the ultimate, except for the step that was
10% of the ultimate when the loads reached the cracking resistance. The crack width was
detected by reading microscope during the loading process.
Two beams marked as JZL-0a/b were loaded until failure to obtain the reference data
about the loads corresponding to the cracking resistance, the limit crack width of 0.20 mm,
the yield of longitudinal tensile steel bars, the ultimate and the failure paern.
The strengthening process of pre-cracked beams can be summarized as follows: (1)
We polished the concrete surface that was bonded with CFRP sheets, made smooth arcs
at the places with U-jacket CFRP sheets with grinding machine and cleaned them with
acetone solution; (2) we smeared the adhesive on concrete surface with a roller; (3) we
bonded one end of CFRP sheet on the boom surface and fixed it with a U-jacket CFRP
sheet; (4) we created tension with the CFRP sheet with a special prestress device that slips
along the test beam and fixes the CFRP sheet at other end of test beam; (5) after the tensile
strain of CFRP sheet remained constant, we smeared adhesive on CFRP sheet and bonded
them on the boom concrete surface; (6) we bonded the U-jackets of CFRP sheets at shear-
span to fix the longitudinal CFRP sheets.
The tensile force was measured by the strain gauges pasted on the CFRP sheet and
verified by the elongation of the CFRP sheet. After the strengthening work, the beams
were cured indoors for 3 days to ensure the adhesive solidification. Figure 2 shows the
CFRP sheets bonded on test beams.
Figure 2. Test beams strengthened by CFRP sheets.
2.4. Repeated Loading Method and Instruments
As exhibited in Figure 3, test beam was simply supported on supports with steel roll-
ers, and two hydraulic jacks on top surface of test beam at the one-third points were used
to exert the repeated load. The hydraulic jacks were hung by steel frames, which were
fixed on the steel foundation. The repeated load of each step was measured by the load
meters linked to the hydraulic jacks, while it was verified by the meter of a hydraulic oil
pump, which provided oil pressure to the parallelly connected jacks. The load meters were
automatically recorded by a DH3818-type data acquisition system produced by Shanghai
Donghua Test Machine Co., Ltd. (Shanghai, China).
Figure 1. Geometry and reinforcement details of reinforced concrete beam (unit: mm).

Buildings 2023,13, 2115 5 of 19
2.3. Pre-Crack Loading and Strengthening Method
The target pre-crack width was the limit of 0.20 mm for reinforced concrete beams in
normal environment [
40
]. The cracks were made by four-point bending test [
42
,
43
]. The
loads were multi-step with increments of 20% of the ultimate, except for the step that was
10% of the ultimate when the loads reached the cracking resistance. The crack width was
detected by reading microscope during the loading process.
Two beams marked as JZL-0a/b were loaded until failure to obtain the reference data
about the loads corresponding to the cracking resistance, the limit crack width of 0.20 mm,
the yield of longitudinal tensile steel bars, the ultimate and the failure pattern.
The strengthening process of pre-cracked beams can be summarized as follows:
(1) We polished the concrete surface that was bonded with CFRP sheets, made smooth
arcs at the places with U-jacket CFRP sheets with grinding machine and cleaned them
with acetone solution; (2) we smeared the adhesive on concrete surface with a roller;
(3) we bonded one end of CFRP sheet on the bottom surface and fixed it with a U-jacket
CFRP sheet; (4) we created tension with the CFRP sheet with a special prestress device that
slips along the test beam and fixes the CFRP sheet at other end of test beam; (5) after the
tensile strain of CFRP sheet remained constant, we smeared adhesive on CFRP sheet and
bonded them on the bottom concrete surface; (6) we bonded the U-jackets of CFRP sheets
at shear-span to fix the longitudinal CFRP sheets.
The tensile force was measured by the strain gauges pasted on the CFRP sheet and
verified by the elongation of the CFRP sheet. After the strengthening work, the beams were
cured indoors for 3 days to ensure the adhesive solidification. Figure 2shows the CFRP
sheets bonded on test beams.
Buildings 2023, 13, x FOR PEER REVIEW 5 of 20
Figure 1. Geometry and reinforcement details of reinforced concrete beam (unit: mm).
2.3. Pre-Crack Loading and Strengthening Method
The target pre-crack width was the limit of 0.20 mm for reinforced concrete beams in
normal environment [40]. The cracks were made by four-point bending test [42,43]. The
loads were multi-step with increments of 20% of the ultimate, except for the step that was
10% of the ultimate when the loads reached the cracking resistance. The crack width was
detected by reading microscope during the loading process.
Two beams marked as JZL-0a/b were loaded until failure to obtain the reference data
about the loads corresponding to the cracking resistance, the limit crack width of 0.20 mm,
the yield of longitudinal tensile steel bars, the ultimate and the failure paern.
The strengthening process of pre-cracked beams can be summarized as follows: (1)
We polished the concrete surface that was bonded with CFRP sheets, made smooth arcs
at the places with U-jacket CFRP sheets with grinding machine and cleaned them with
acetone solution; (2) we smeared the adhesive on concrete surface with a roller; (3) we
bonded one end of CFRP sheet on the boom surface and fixed it with a U-jacket CFRP
sheet; (4) we created tension with the CFRP sheet with a special prestress device that slips
along the test beam and fixes the CFRP sheet at other end of test beam; (5) after the tensile
strain of CFRP sheet remained constant, we smeared adhesive on CFRP sheet and bonded
them on the boom concrete surface; (6) we bonded the U-jackets of CFRP sheets at shear-
span to fix the longitudinal CFRP sheets.
The tensile force was measured by the strain gauges pasted on the CFRP sheet and
verified by the elongation of the CFRP sheet. After the strengthening work, the beams
were cured indoors for 3 days to ensure the adhesive solidification. Figure 2 shows the
CFRP sheets bonded on test beams.
Figure 2. Test beams strengthened by CFRP sheets.
2.4. Repeated Loading Method and Instruments
As exhibited in Figure 3, test beam was simply supported on supports with steel roll-
ers, and two hydraulic jacks on top surface of test beam at the one-third points were used
to exert the repeated load. The hydraulic jacks were hung by steel frames, which were
fixed on the steel foundation. The repeated load of each step was measured by the load
meters linked to the hydraulic jacks, while it was verified by the meter of a hydraulic oil
pump, which provided oil pressure to the parallelly connected jacks. The load meters were
automatically recorded by a DH3818-type data acquisition system produced by Shanghai
Donghua Test Machine Co., Ltd. (Shanghai, China).
Figure 2. Test beams strengthened by CFRP sheets.
2.4. Repeated Loading Method and Instruments
As exhibited in Figure 3, test beam was simply supported on supports with steel
rollers, and two hydraulic jacks on top surface of test beam at the one-third points were
used to exert the repeated load. The hydraulic jacks were hung by steel frames, which
were fixed on the steel foundation. The repeated load of each step was measured by the
load meters linked to the hydraulic jacks, while it was verified by the meter of a hydraulic
oil pump, which provided oil pressure to the parallelly connected jacks. The load meters
were automatically recorded by a DH3818-type data acquisition system produced by
Shanghai Donghua Test Machine Co., Ltd. (Shanghai, China).
Buildings 2023, 13, x FOR PEER REVIEW 6 of 20
Figure 3. Repeated loading devices for test beams.
Figure 4 shows the repeated load history of this study; each step is in a static manner
[44]. The yield load (P
y
) and the yield deflection (a
f,y
) at mid-span are a load and a mid-
span deflection that correspond to the yield of the longitudinal tensile steel bars. Before
the yield of longitudinal steel bars, the load was exerted in steps at 50%, 75% and 100% of
the yield load P
y
; each step was repeated two times. During the process of each step, the
load was divided several times to simulate a continuously increased = decreased load sub-
jected by the test beam. After the yield of longitudinal steel bars, the load was controlled
in steps by the deflection, which increased at intervals by multiples of the yield deflection
a
f,y
; each step was also repeated two times.
Figure 4. Steps of repeated loads.
Before casting concrete of test beams, the strain gauges with size of 1 mm × 1 mm
were bonded on the longitudinal tensile steel bars at mid-span section and covered by
epoxy resin with a bed of gauze. The wires of strain gauges were guided outside of con-
crete with enough length to be linked to the strain meter. In addition, the strain of longi-
tudinal CFRP sheet was measured by three gauges that were pasted on surface of CFRP
along the length direction, and the concrete strain changed at mid-span section was meas-
ured by three strain gauges pasted on top surface and five strain gauges pasted along
depth direction on side surface of mid-span section, as presented in Figure 5. The deflec-
tion of test beam was measured by five displacement meters placed on supports, mid-
span and loading sections. All test data were recorded by the data acquisition system. The
size of strain gauges pasted on concrete was 100 mm × 3 mm, and that which was pasted
on CFRP was 5 mm × 3 mm. The crack width was detected by a reading microscope.
Figure 5. Arrangement of the displacement meters and the strain gauges pasted on concrete and
CFRP sheets.
Figure 3. Repeated loading devices for test beams.

Buildings 2023,13, 2115 6 of 19
Figure 4shows the repeated load history of this study; each step is in a static manner [
44
].
The yield load (P
y
) and the yield deflection (a
f,y
) at mid-span are a load and a mid-span
deflection that correspond to the yield of the longitudinal tensile steel bars. Before the yield
of longitudinal steel bars, the load was exerted in steps at 50%, 75% and 100% of the yield
load P
y
; each step was repeated two times. During the process of each step, the load was
divided several times to simulate a continuously increased = decreased load subjected by
the test beam. After the yield of longitudinal steel bars, the load was controlled in steps by
the deflection, which increased at intervals by multiples of the yield deflection a
f,y
; each
step was also repeated two times.
Buildings 2023, 13, x FOR PEER REVIEW 6 of 20
Figure 3. Repeated loading devices for test beams.
Figure 4 shows the repeated load history of this study; each step is in a static manner
[44]. The yield load (P
y
) and the yield deflection (a
f,y
) at mid-span are a load and a mid-
span deflection that correspond to the yield of the longitudinal tensile steel bars. Before
the yield of longitudinal steel bars, the load was exerted in steps at 50%, 75% and 100% of
the yield load P
y
; each step was repeated two times. During the process of each step, the
load was divided several times to simulate a continuously increased = decreased load sub-
jected by the test beam. After the yield of longitudinal steel bars, the load was controlled
in steps by the deflection, which increased at intervals by multiples of the yield deflection
a
f,y
; each step was also repeated two times.
Figure 4. Steps of repeated loads.
Before casting concrete of test beams, the strain gauges with size of 1 mm × 1 mm
were bonded on the longitudinal tensile steel bars at mid-span section and covered by
epoxy resin with a bed of gauze. The wires of strain gauges were guided outside of con-
crete with enough length to be linked to the strain meter. In addition, the strain of longi-
tudinal CFRP sheet was measured by three gauges that were pasted on surface of CFRP
along the length direction, and the concrete strain changed at mid-span section was meas-
ured by three strain gauges pasted on top surface and five strain gauges pasted along
depth direction on side surface of mid-span section, as presented in Figure 5. The deflec-
tion of test beam was measured by five displacement meters placed on supports, mid-
span and loading sections. All test data were recorded by the data acquisition system. The
size of strain gauges pasted on concrete was 100 mm × 3 mm, and that which was pasted
on CFRP was 5 mm × 3 mm. The crack width was detected by a reading microscope.
Figure 5. Arrangement of the displacement meters and the strain gauges pasted on concrete and
CFRP sheets.
Figure 4. Steps of repeated loads.
Before casting concrete of test beams, the strain gauges with size of 1 mm
×
1 mm
were bonded on the longitudinal tensile steel bars at mid-span section and covered by
epoxy resin with a bed of gauze. The wires of strain gauges were guided outside of concrete
with enough length to be linked to the strain meter. In addition, the strain of longitudinal
CFRP sheet was measured by three gauges that were pasted on surface of CFRP along
the length direction, and the concrete strain changed at mid-span section was measured
by three strain gauges pasted on top surface and five strain gauges pasted along depth
direction on side surface of mid-span section, as presented in Figure 5. The deflection of
test beam was measured by five displacement meters placed on supports, mid-span and
loading sections. All test data were recorded by the data acquisition system. The size of
strain gauges pasted on concrete was 100 mm
×
3 mm, and that which was pasted on CFRP
was 5 mm ×3 mm. The crack width was detected by a reading microscope.
Buildings 2023, 13, x FOR PEER REVIEW 6 of 20
Figure 3. Repeated loading devices for test beams.
Figure 4 shows the repeated load history of this study; each step is in a static manner
[44]. The yield load (P
y
) and the yield deflection (a
f,y
) at mid-span are a load and a mid-
span deflection that correspond to the yield of the longitudinal tensile steel bars. Before
the yield of longitudinal steel bars, the load was exerted in steps at 50%, 75% and 100% of
the yield load P
y
; each step was repeated two times. During the process of each step, the
load was divided several times to simulate a continuously increased = decreased load sub-
jected by the test beam. After the yield of longitudinal steel bars, the load was controlled
in steps by the deflection, which increased at intervals by multiples of the yield deflection
a
f,y
; each step was also repeated two times.
Figure 4. Steps of repeated loads.
Before casting concrete of test beams, the strain gauges with size of 1 mm × 1 mm
were bonded on the longitudinal tensile steel bars at mid-span section and covered by
epoxy resin with a bed of gauze. The wires of strain gauges were guided outside of con-
crete with enough length to be linked to the strain meter. In addition, the strain of longi-
tudinal CFRP sheet was measured by three gauges that were pasted on surface of CFRP
along the length direction, and the concrete strain changed at mid-span section was meas-
ured by three strain gauges pasted on top surface and five strain gauges pasted along
depth direction on side surface of mid-span section, as presented in Figure 5. The deflec-
tion of test beam was measured by five displacement meters placed on supports, mid-
span and loading sections. All test data were recorded by the data acquisition system. The
size of strain gauges pasted on concrete was 100 mm × 3 mm, and that which was pasted
on CFRP was 5 mm × 3 mm. The crack width was detected by a reading microscope.
Figure 5. Arrangement of the displacement meters and the strain gauges pasted on concrete and
CFRP sheets.
Figure 5.
Arrangement of the displacement meters and the strain gauges pasted on concrete and
CFRP sheets.
3. Test Results and Analyses
3.1. Concrete Strain along Depth of Mid-Span Section
Figure 6presents the concrete strains along the depth of the mid-span section on part
of the test beams under the peak repeated load at the second repetition of each loading step.
The concrete strain at the key loading steps, including the decompression of the bottom
surface, the yield of longitudinal tensile steel bars and the ultimate state, can be determined
from the figures. Similar to that of reinforced concrete beams under static and repeated
loads, the concrete strain along the depth of the mid-span section of the strengthened
reinforced concrete beams still maintained a linear variation close to a plane. Therefore, the
assumption of plane section is also adaptable to reinforced concrete beams strengthened
with prestressed CFRP sheets.

Buildings 2023,13, 2115 7 of 19
Buildings 2023, 13, x FOR PEER REVIEW 7 of 20
3. Test Results and Analyses
3.1. Concrete Strain along Depth of Mid-Span Section
Figure 6 presents the concrete strains along the depth of the mid-span section on part
of the test beams under the peak repeated load at the second repetition of each loading
step. The concrete strain at the key loading steps, including the decompression of the bot-
tom surface, the yield of longitudinal tensile steel bars and the ultimate state, can be de-
termined from the figures. Similar to that of reinforced concrete beams under static and
repeated loads, the concrete strain along the depth of the mid-span section of the strength-
ened reinforced concrete beams still maintained a linear variation close to a plane. There-
fore, the assumption of plane section is also adaptable to reinforced concrete beams
strengthened with prestressed CFRP sheets.
-1000 0 1000 2000 3000 4000 5000
50
100
150
200
250
300
h (mm)
ε (10
-6
)
YJCL-2a
50%P
y
75%P
y
100%P
y
Δ
2
Δ
-1000 0 1000 2000 3000 4000 5000
50
100
150
200
250
300
h (mm)
ε (10
-6
)
YJCL-3a
50%P
y
75%P
y
100%P
y
Δ
2
Δ
-1000 0 1000 2000 3000 4000 5000
50
100
150
200
250
300
h (mm)
ε (10
-6
)
YJCL-4a
50%P
y
75%P
y
100%P
y
Δ
2
Δ
-1000 0 1000 2000 3000 4000 5000
50
100
150
200
250
300
h (mm)
ε (10
-6
)
YJCL-5a
50%P
y
75%P
y
100%P
y
Δ
2
Δ
Figure 6. Concrete strain along depth of mid-span section on part of test beams.
3.2. Crack Distribution and Development
The cracks distributed on test beams are presented in Figure 7. Beams JZCL-1a/b in-
itially cracked at a load of 12.5 kN with a crack width of 0.03 mm and a height of about
130 mm. The number of cracks increased, while the widths increased on the beams during
the first two steps of repeated loads. Beams JZCL-1a and JZCL-1b, respectively, reached
the yield at loads of 37.0 kN and 41.7 kN. At this point, the beams underwent a large de-
formation with the extension and opening of cracks and formed main cracks with large
widths and heights. Under the repeated load controlled by deformation, the widths of the
cracks continuously increased. The compression concrete crushed at 44.8 kN and 49.6 kN,
respectively, for beams JZCL-1a and JZCL-1b underwent the 3af,y repetition.
Figure 6. Concrete strain along depth of mid-span section on part of test beams.
3.2. Crack Distribution and Development
The cracks distributed on test beams are presented in Figure 7. Beams JZCL-1a/b
initially cracked at a load of 12.5 kN with a crack width of 0.03 mm and a height of about
130 mm. The number of cracks increased, while the widths increased on the beams during
the first two steps of repeated loads. Beams JZCL-1a and JZCL-1b, respectively, reached
the yield at loads of 37.0 kN and 41.7 kN. At this point, the beams underwent a large
deformation with the extension and opening of cracks and formed main cracks with large
widths and heights. Under the repeated load controlled by deformation, the widths of the
cracks continuously increased. The compression concrete crushed at 44.8 kN and 49.6 kN,
respectively, for beams JZCL-1a and JZCL-1b underwent the 3af,y repetition.
Beams YJCL-2a/b initially cracked with a width of 0.02–0.03 mm and a height of
about 90 mm at loads of 9 kN and 11 kN, respectively. When the load reached 39.6 kN,
several microcracks along prestressed CFRP sheets appeared, along with slight crackles of
transversal wires. When the load reached 55.5 kN, the beams reached the yield with many
more cracks even on the shear-span. After the repetition was controlled by deformation,
the deflection increased with the extension of cracks.
Beams YJCL-3a/b initially cracked with a width of 0.05 mm and a height of about
100 mm at loads of 12.5 kN and 13.5 kN, respectively. Under the repeated loads, the number
of cracks increased with slight extension and opening. The CFRP sheets presented a slight
crackle at the load of 63.4 kN, and some wires of the outside layer broke at the loads of
67.4 kN and 75.5 kN, respectively, for beams YJCL-3a and YJCL-3b. After the repetition
was controlled by deformation, the crackling of the CFRP sheets continuously sounded,
while the cracks slightly extended.
Beams YJCL-4a/b initially cracked with a width of 0.02 mm and height of about
70–140 mm at a load of 9 kN. The cracks extended obviously with a crackle of the bond ma-
trix under repeated loads of 47.5 kN. The longitudinal tensile steel bars yielded respectively

Buildings 2023,13, 2115 8 of 19
at the loads of 63.4 kN and 59.5 kN. After that, new cracks appeared on the shear-span and
extended rapidly.
Buildings 2023, 13, x FOR PEER REVIEW 8 of 20
Figure 7. The crack distributed on test beams.
Beams YJCL-2a/b initially cracked with a width of 0.02–0.03 mm and a height of about
90 mm at loads of 9 kN and 11 kN, respectively. When the load reached 39.6 kN, several
microcracks along prestressed CFRP sheets appeared, along with slight crackles of trans-
versal wires. When the load reached 55.5 kN, the beams reached the yield with many more
cracks even on the shear-span. After the repetition was controlled by deformation, the
deflection increased with the extension of cracks.
Beams YJCL-3a/b initially cracked with a width of 0.05 mm and a height of about 100
mm at loads of 12.5 kN and 13.5 kN, respectively. Under the repeated loads, the number
of cracks increased with slight extension and opening. The CFRP sheets presented a slight
crackle at the load of 63.4 kN, and some wires of the outside layer broke at the loads of
67.4 kN and 75.5 kN, respectively, for beams YJCL-3a and YJCL-3b. After the repetition
was controlled by deformation, the crackling of the CFRP sheets continuously sounded,
while the cracks slightly extended.
Beams YJCL-4a/b initially cracked with a width of 0.02 mm and height of about 70–
140 mm at a load of 9 kN. The cracks extended obviously with a crackle of the bond matrix
Figure 7. The crack distributed on test beams.
Beams YJCL-5a/b initially cracked with a width of 0.02 mm and a height of about
70–120 mm at load of 13.5 kN. Beam YJCL-5a showed a crackle of the CFRP sheets at a
load of 39.6 kN, and the cracks extended rapidly afterward; the yield load reached 77.3 kN.
Beam YJCL-5b had a little change in the cracks under the repeated load before the yield
at a load of 71.4 kN. Continuously, the cracks extended rapidly in both the pure bending
segment and the shear-span.
Generally, the cracks in the pure bending segment of test beams basically elongated
vertically along the depth direction and forked at the end part. The inclined cracks in
the shear-span segment extended at a slant while confined by the U-jacket CFRP sheet.
According to the statistical results of the cracks summarized in Table 3, the number of
cracks increased while the spacing of cracks decreased with the increase of the prestressing
of CFRP sheets. This minimized the maximum crack width of the strengthened beams

Buildings 2023,13, 2115 9 of 19
at the serviceability limit state due to the confinement of the prestressed CFRP sheets to
the extension and opening of cracks. Meanwhile, the crack width at the ultimate state
presented a downward tendency with the increased prestressing of the CFRP sheets.
Table 3. Cracks at the serviceability limit, the yield and the ultimate state of test beams.
Test Beam
Serviceability Limit State Yield State Ultimate State
wmax (mm) Number of Cracks lm(mm) wmax (mm) wmax (mm)
JZCL-1a 0.45 19 105 0.55 0.95
JZCL-1b 0.47 20 100 0.65 1.03
YJCL-2a 0.40 19 105 0.49 0.85
YJCL-2b 0.40 17 111 0.51 0.91
YJCL-3a 0.27 21 100 0.45 0.79
YJCL-3b 0.28 24 88 0.52 0.82
YJCL-4a 0.24 22 91 0.49 0.78
YJCL-4b 0.28 24 88 0.51 0.75
YJCL-5a 0.19 28 75 0.47 0.68
YJCL-5b 0.17 26 81 0.50 0.65
3.3. Mid-Span Deflection
The skeleton load vs. mid-span deflection curves of test beams are presented in
Figure 8, while those corresponding to the complete load vs. mid-span deflection curves of
test beams are drawn in Figure 9. Obviously, beams JZCL-1a/b, which were not strength-
ened by CFRP sheets, presented a larger deformability after the yield of longitudinal tensile
steel bars, which could undergo the 2a
f,y
repetition. Comparatively, beams YJCL-2a/b,
which were strengthened by one layer of prestressed CFRP sheets with a tensioning force
of 20 kN, could withstand the 2a
f,y
repetition, while other beams all failed during the
2af,y repetition.
Buildings 2023, 13, x FOR PEER REVIEW 10 of 20
0 5 10 15 20 25 30 35 40 45 50 55 60 65
10
20
30
40
50
60
70
80
90 JZCL-1a/b
YJCL-2a/b
YJCL-3a/b
YJCL-4a/b
YJCL-5a/b
P (kN)
a
f
(mm)
Figure 8. Skeleton load mid-span deflection curves of test beams.
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-2a
YJCL-2b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-3a
YJCL-3b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-4a
YJCL-4b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-5a
YJCL-5b
P (kN)
a
f
(mm)
Figure 9. The complete load mid-span deflection curves of test beams.
The skeleton curves contain the ascending portion without a tendency of descending.
This is similar to that of the reinforced concrete beams strengthened with prestressed
CFRP sheets under static loads [45,46]. With the increase in the prestress degree of CFRP
sheets, the flexural stiffness of test beams in the sequence of YJCL-2a/b, YJCL-4a/b, YJCL-
3a/b and YJCL-5a/b increased in portions not only before but after the yield of the longi-
tudinal tensile steel bars. This indicates that the entirety of the flexural segment of the test
beams was ensured due to the confinement effect of prestressed CFRP sheets on the de-
velopment of cracks under repeated loads. As a result, the normal serviceability was im-
proved due to a small deflection that happened on the beams under the same loads. Mean-
while, the bearing capacity of test beams was improved with relatively small deflection
due to the strengthening of prestressed CFRP sheets. Compared to the greater deflection
that happened on the beams without strengthening, the CFRP-strengthened beams had
Figure 8. Skeleton load mid-span deflection curves of test beams.
The skeleton curves contain the ascending portion without a tendency of descending.
This is similar to that of the reinforced concrete beams strengthened with prestressed CFRP
sheets under static loads [
45
,
46
]. With the increase in the prestress degree of CFRP sheets,
the flexural stiffness of test beams in the sequence of YJCL-2a/b, YJCL-4a/b, YJCL-3a/b
and YJCL-5a/b increased in portions not only before but after the yield of the longitudinal
tensile steel bars. This indicates that the entirety of the flexural segment of the test beams
was ensured due to the confinement effect of prestressed CFRP sheets on the development
of cracks under repeated loads. As a result, the normal serviceability was improved due
to a small deflection that happened on the beams under the same loads. Meanwhile,

Buildings 2023,13, 2115 10 of 19
the bearing capacity of test beams was improved with relatively small deflection due to
the strengthening of prestressed CFRP sheets. Compared to the greater deflection that
happened on the beams without strengthening, the CFRP-strengthened beams had almost
similar deflection at the ultimate state due to the failure that came from the fracturing of
the CFRP sheets.
Buildings 2023, 13, x FOR PEER REVIEW 10 of 20
0 5 10 15 20 25 30 35 40 45 50 55 60 65
10
20
30
40
50
60
70
80
90 JZCL-1a/b
YJCL-2a/b
YJCL-3a/b
YJCL-4a/b
YJCL-5a/b
P (kN)
a
f
(mm)
Figure 8. Skeleton load mid-span deflection curves of test beams.
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-2a
YJCL-2b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-3a
YJCL-3b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-4a
YJCL-4b
P (kN)
a
f
(mm)
0 5 10 15 20 25 30 35 40
10
20
30
40
50
60
70
80
90
YJCL-5a
YJCL-5b
P (kN)
a
f
(mm)
Figure 9. The complete load mid-span deflection curves of test beams.
The skeleton curves contain the ascending portion without a tendency of descending.
This is similar to that of the reinforced concrete beams strengthened with prestressed
CFRP sheets under static loads [45,46]. With the increase in the prestress degree of CFRP
sheets, the flexural stiffness of test beams in the sequence of YJCL-2a/b, YJCL-4a/b, YJCL-
3a/b and YJCL-5a/b increased in portions not only before but after the yield of the longi-
tudinal tensile steel bars. This indicates that the entirety of the flexural segment of the test
beams was ensured due to the confinement effect of prestressed CFRP sheets on the de-
velopment of cracks under repeated loads. As a result, the normal serviceability was im-
proved due to a small deflection that happened on the beams under the same loads. Mean-
while, the bearing capacity of test beams was improved with relatively small deflection
due to the strengthening of prestressed CFRP sheets. Compared to the greater deflection
that happened on the beams without strengthening, the CFRP-strengthened beams had
Figure 9. The complete load mid-span deflection curves of test beams.
3.4. Failure Patterns
The failure patterns of test beams are recorded in photos in Figure 10. Beams
JZCL-1a/b without strengthening failed with crushed compressive concrete after the
yield of longitudinal tensile steel bars. This is a normal failure pattern with sufficient
ductility [
47
]. Comparatively, beams YJCL-2a/b and YJCL-4a/b, which were strengthened
by using one layer of prestressed CFRP sheets with tensioning forces of 20 kN and 30 kN,
failed, with the CFRP sheets breaking after the yield of longitudinal tensile steel bars. Beams
YJCL-3a/b and YJCL-5a/b, which were strengthened by using two layers of prestressed
CFRP sheets with tensioning forces of 40 kN and 60 kN, failed, with the CFRP sheets
breaking into filaments and peeling off the concrete, while part of the tensile concrete
bonded to the CFRP sheets in the pure bending segment and longitudinally cracked due
to the drawing of fractured CFRP sheets, and the U-jacket CFRP sheets near the loading
sections peeled off or were partially broken. This shows that a reliable bond between the
CFRP sheets and concrete is a premise of synergistic action [
48
]. Strengthened by a layer
of CFRP sheets with control tensioning stresses of 0.295f
fu
and 0.442f
fu
, the load capacity
of test beams at the yield of longitudinal tensile steel bars increased by 41.0% and 56.2%,
while the ultimate load capacity increased by 41.9% and 43.8%. Strengthened by two layers
of CFRP sheets with control tensioning stresses of 0.295f
fu
and 0.442f
fu
, the load capacity
of test beams at the yield of longitudinal tensile steel bars increased by 81.4% and 88.8%,
while the ultimate load capacity increased by 56.5% and 74.8%.

Buildings 2023,13, 2115 11 of 19
Buildings 2023, 13, x FOR PEER REVIEW 12 of 20
Figure 10. Failure paerns on bending segment of test beams.
Table 4. Test strains of longitudinal tensile steel bars, CFRP sheets and compressive concrete under
repeated load.
Beam No. Longitudinal Tensile Steel Bars (×10−6) Strain of CFRP Sheets (×10−6) Compressive Concrete (×10−6)
εs,0 εs,y εs,u εcf,0 εcf,y εcf,u εc,0 εc,y εc,u
YJCL-2a −24 2033 4205 4732 7272 10,153 47 335 862
YJCL-2b −24 2045 4255 4726 7258 10,041 49 346 878
YJCL-3a −46 2195 4463 4745 7550 9547 98 467 904
YJCL-3b −48 2198 4465 4750 7577 9547 100 456 945
YJCL-4a −36 2115 4310 7162 9857 12,542 73 394 868
YJCL-4b −38 2122 4332 7162 9837 12,441 75 366 915
YJCL-5a −72 2235 4660 7182 9962 12,044 147 445 1065
YJCL-5b −744 2265 4678 7180 9935 12,015 147 505 1105
4. Prediction of Bending Performances
Based on the stress process of reinforced concrete beams strengthened by prestressed
CFRP sheets, prediction formulas are proposed according to the design principles of pre-
stressed concrete beams [3,40]. To put it simply, the stresses of CFRP sheets and
Figure 10. Failure patterns on bending segment of test beams.
Table 4lists the maximum strains of longitudinal tensile steel bars, CFRP sheets
and the compressive concrete of test beams at the states of the decompression of bottom
surface concrete, the yield of longitudinal tensile steel bars and the ultimate loads. The
strains of CFRP sheets are synergetic with those of bottom surface concrete in the repeated
loading process until the CFRP sheet peeled off the concrete. With the increase in the
prestress of the CFRP sheets, the ultimate strains of longitudinal tensile steel bars became
larger. This means that a higher prestress of CFRP sheets led to a greater pre-compressed
strain of longitudinal tensile steel bars. Meanwhile, a lower tensile strain of CFRP sheets
accompanied by a higher prestress at the broken state, due to the larger strain of CFRP
sheets, occurred under the higher pretension force. If the prestress loss of CFRP sheets
was about 10% tension stress, the effective pretension strains at 20 kN and 30 kN were
4355
µε
and 6532
µε
, respectively. However, the broken strain of CFRP sheets only reached
9547–12,542
µε
, which was about 57–75% of the ultimate strain 16,776
µε
. This indicates
that the CFRP sheets bonded on test beams could not reach the ultimate strength due to the
breaking of some wires with uneven tensioning.
Due to the precompression of longitudinal steel bars under the tensioning of pre-
stressed CFRP sheets, the longitudinal steel tensile bars were still in compression when
the concrete was decompressed at bottom surface of the bending section. Due to the

Buildings 2023,13, 2115 12 of 19
compression zone of test beams under repeated load being the tensile zone under the
prestressed CFRP sheets, the compression of concrete was postponed by the pre-tensioning
of prestressed CFRP sheets. This led to the compressive strain of concrete at the failure state
not being able to reach the ultimate, and no crushing of compressive concrete took place.
Table 4.
Test strains of longitudinal tensile steel bars, CFRP sheets and compressive concrete under
repeated load.
Beam No. Longitudinal Tensile Steel Bars (×10−6) Strain of CFRP Sheets (×10−6) Compressive Concrete (×10−6)
εs,0 εs,y εs,u εcf,0 εcf,y εcf,u εc,0 εc,y εc,u
YJCL-2a −24 2033 4205 4732 7272 10,153 47 335 862
YJCL-2b −24 2045 4255 4726 7258 10,041 49 346 878
YJCL-3a −46 2195 4463 4745 7550 9547 98 467 904
YJCL-3b −48 2198 4465 4750 7577 9547 100 456 945
YJCL-4a −36 2115 4310 7162 9857 12,542 73 394 868
YJCL-4b −38 2122 4332 7162 9837 12,441 75 366 915
YJCL-5a −72 2235 4660 7182 9962 12,044 147 445 1065
YJCL-5b −744 2265 4678 7180 9935 12,015 147 505 1105
4. Prediction of Bending Performances
Based on the stress process of reinforced concrete beams strengthened by prestressed
CFRP sheets, prediction formulas are proposed according to the design principles of pre-
stressed concrete beams [
3
,
40
]. To put it simply, the stresses of CFRP sheets and longitudinal
tensile steel bars are positive in tension, and that of concrete is positive in compression.
4.1. Assumptions for Predicting Flexural Performance
The assumptions for predicting flexural performance are summarized as follows:
(1)
The strain along normal-section depth fits for the plane section;
(2)
The constitutive model of concrete can be expressed as:
σc=2"εc
ε0−εc
ε02#fc(εc<ε0)(1)
σc=fc(ε0<εc≤εcu)(2)
where ε0= 0.002, εcu = 0.0033.
(3)
No bond slip exists between longitudinal tensile steel bars and concrete. The constitu-
tive model of steel bars can be expressed as:
σs=Esεs(εs≤εy)(3)
σs=fy(εs>εy)(4)
(4) No bond slip exists along the interface between prestressed CFRP sheets and concrete.
The linear constitutive model of CFRP sheets can be used as:
σf=Efεf(εf≤εfu)(5)
(5) The flexural stiffness is the same along the span of beam, and the depths of prestressed
CFRP sheets can be neglected.
(6) The residual strains of concrete and steel bars are neglected due to the pre-crack width
being controlled within the limit.

Buildings 2023,13, 2115 13 of 19
4.2. Prestress Loss of CFRP Sheet
The control stress for the pre-tensioning of CFRP sheets can be calculated as:
σf,con =Nf/EfAf(6)
The prestress loss of CFRP sheets due to compression of anchorage locked on the ends
of reinforced concrete beams can be calculated as:
σl1=∆
Lf
σf,con (7)
The prestress loss due to the relaxation of CFRP sheets and the long-term deformation
of RC beams can be calculated as [29]:
σl2=2%σf,con (8)
The prestress loss due to the shrinkage and creep of concrete can be calculated refer-
encing the prestressed concrete with steel reinforcement [40]:
σl3=35 +280σpc /fcu
1+15ρ(9)
σpc =(σf,con −σl1−σl2)Af
A0
+(σf,con −σl1−σl2)Af(h−y0)2
I0
(10)
I0=(0.0833 +0.1αEsρ)bh3(11)
y0=(0.5 +0.42αEsρ)h(12)
Therefore, the effective prestress of CFRP sheets can be calculated as:
σpf =σf,con −σl1−σl2−σl3(13)
4.3. Initial Prestress of Concrete and Steel Bars
The prestress of concrete on the top and bottom surfaces of reinforced concrete beams
can be calculated as:
σpc,T =σpf Af
A0
−σpfAf(h−y0)y0
I0
(14)
σpc,B =σpf Af
A0
+σpfAf(h−y0)2
I0
(15)
The prestress of the longitudinal tensile steel bars of reinforced concrete beams can be
calculated as:
σps =−αEsσpfAf
A0
−αEsσpf Af(h−y0)(h0−y0)
I0
(16)
4.4. Stress When Prestress Disappeared at the Bottom Concrete Surface
With the increase in repeated loads, the pre-compression of concrete at the bottom
surface of RC beams will be reduced. The prestress disappears when the tensile stress
produced by the load reaches the pre-compressive stress, and the corresponding bending
moment on the cross-section of RC beam can be obtained with the following formula:
M0=σpc,BI0/(h−y0)(17)

Buildings 2023,13, 2115 14 of 19
With the assumption that no bond slip exists along the interface between CFRP sheets
and concrete, the changes in strain are the same between CFRP sheets and concrete at the
bottom surface. Therefore, the stress of CFRP sheets is:
σ0f =σpf +αEfσpc,B (18)
The stress of longitudinal tensile steel bars of RC beams can be calculated as:
σ0s =σps +αEs M0(h0−y0)
I0
(19)
In this study, due to the compression of the anchorage only taking place at the ten-
sioning end, the compression of the anchorage locked on the reinforced concrete beams
∆
= 2 mm. The length of the CFRP sheets L
f
= 2.3 m, which was determined by the total
length 2.7 m minus the fixed ends of 200 mm. Table 5summarizes the control prestress,
prestress losses and effective prestress of the CFRP sheets, the initial prestresses of concrete
and steel bars and the stresses of CFRP sheets and steel bars at the prestress disappearance
at the bottom concrete surface.
Table 5. Control prestress, prestress losses and effective prestress of CFRP sheets.
Tension Force (kN) Layer of CFRP Sheet σf,con
(MPa)
σl1
(MPa)
σl1
(MPa)
σl1
(MPa)
σpf
(MPa)
σpc,T
(MPa)
σpc,B
(MPa)
σps
(MPa)
σ0f
(MPa)
σ0s
(MPa)
20 1 1197.6 1.0 24.0 38.0 1134.6 −0.77 1.52 −7.20 1145.0 −5.09
40 2 1197.6 1.0 24.0 44.2 1128.4 −1.53 3.02 −
14.18
1148.8 −
10.03
30 1 1796.4 1.6 36.0 40.7 1718.1 −1.17 2.30 −
10.76
1733.6 −7.61
60 2 1796.4 1.6 36.0 50.1 1708.7 −2.32 4.58 −
21.36
1739.4 −
15.10
4.5. Flexural Stiffness
Due to the reinforced concrete beams being loaded with cracks before strengthening, if
the prestress of concrete disappeared at the bottom surface of the strengthened beams, the
flexural behavior is similar to that of conventional reinforced concrete beams after concrete
cracking. Therefore, the cracking moment corresponds to the prestress disappearance at the
bottom surface of strengthened beams. Similar to the prestressed concrete beams [
40
,
48
],
the flexural stiffness can be calculated at two stages.
When M≤M0,
B0=0.85EcI0(20)
When M>M0,
B=0.85EcI0
M0
M+1−M0
M1+0.21
αEsρ−0.7(21)
Based on the principle of equivalent flexural stiffness for reinforced concrete beams,
the mid-span deflection can be calculated as:
af=0.1132(M/B−M0/B0)l2
0(22)
With the above formulas for calculating the mid-span deflection at normal serviceabil-
ity state, the predicted results agree well with the tested ones.
4.6. Stress at the Yield of Longitudinal Steel Bars
With the assumption of the plane section, the strains of concrete at the top surface and
the longitudinal tensile steel bars are calculated as:
εc,y =0.002 −σpc,T
Ec(23)

Buildings 2023,13, 2115 15 of 19
εs,y =fy−σps
Es(24)
4.7. Flexural Bearing Capacity
Considering that the prestressed CFRP sheets broke without reaching the ultimate
tensile strain, the flexural bearing capacity of the strengthened beams under repeated
loads can be conservatively predicted at the yield of longitudinal tensile steel bars. The
compression zone of concrete can be simplified as a linear stress distribution along the
sectional depth, and the length of internal force arm is taken respectively as 0.92h
0
for
calculating the bearing capacity at the yield of longitudinal tensile steel bars and that at the
ultimate state [
40
,
48
]. Based on this experimental study, the tensile stress of CFRP sheets
at the ultimate state of strengthened beams can be taken as 0.65f
fu
. Therefore, based on
the fore and moment equilibrium conditions on the bending section, the flexural bearing
capacity of the strengthened beams at the yield of longitudinal tensile steel bars and that at
the ultimate state can be predicted as follows:
My=0.92 fyAsh0+Afσ0f +fy−σ0s(1+2as/h0)(0.92h0+as)(25)
Mu=0.92 fyAsh0+0.65 ffu Af(0.92h0+as)(26)
The prediction results of the flexural capacity of the test strengthened beams are
presented in Table 6. Good prediction can be performed with the fitness of test results.
Table 6. Comparison of test to predicted flexural capacity of test beams.
Beam No.
The Yield Load (kN) The Ultimate Load (kN)
Test Predicted Ratio Test Predicted Ratio
YJZL-2a 55.5 51.2 1.084 67.6 56.1 1.206
YJZL-2b 55.5 51.2 1.084 66.4 56.1 1.184
YJZL-3a 67.4 60.1 1.122 68.5 69.7 0.982
YJZL-3b 75.4 60.1 1.255 79.2 69.7 1.136
YJZL-4a 63.4 54.3 1.168 67.4 56.1 1.202
YJZL-4b 59.5 54.3 1.096 68.4 56.1 1.220
YJZL-5a 77.3 66.2 1.168 82.5 69.7 1.183
YJZL-5b 71.3 66.2 1.077 82.5 69.7 1.183
4.8. Ductility
Due to the yield of test beams being able to be determined by experiments in this
study, the ductility of test beams is represented by the deflection ductility facto (
µ
). This is
a ratio of the mid-span deflection at the ultimate state to that at the yield state of test beams.
The results are presented in Table 7. The ductility of test beams without strengthening
by CFRP sheets was ideal, while that of test beams strengthened with prestressed CFRP
sheets obviously reduced by 54.9% to 186%. This is due to the greater brittleness of CFRP
sheets impregnated with epoxy resin. The breaking of CFRP sheets took place after the
beams reached the yield state, with most of them accompanied by peeling off from the
concrete and rapidly reaching the ultimate. The ductility was reduced with the increase
in the layers of CFRP sheets under the conditions of the same prestress and were reduced
with the prestress increase in the case of the same layers of CFRP sheets. Therefore, the
sharp reduction in the ductility of beams strengthened with prestressed CFRP sheets needs
to be addressed.

Buildings 2023,13, 2115 16 of 19
Table 7. Deflection ductility of test beams.
Beam No. af,y (mm) af,u (mm) µAverage µ
JZCL-1a 11.4 44.4 3.89 3.78
JZCL-1b 16.3 59.6 3.66
YJCL-2a 15.0 34.9 2.33 2.44
YJCL-2b 13.3 34.0 2.56
YJCL-3a 18.1 37.8 2.09 1.78
YJCL-3b 22.7 33.4 1.47
YJCL-4a 31.2 36.4 1.17 1.32
YJCL-4b 18.5 27.2 1.47
YJCL-5a 22.1 35.2 1.59 1.52
YJCL-5b 23.2 33.4 1.44
5. Conclusions
Based on this experimental study of cracked reinforced concrete beams strengthened
with prestressed CFRP sheets under repeated loads, conclusions can be drawn as follows:
(1)
The assumption of the plane section is adaptable for reinforced concrete beams with
prestressed CFRP sheets under repeated load. This provides a foundation for building
the calculation methods for the bending behaviors of strengthened beams.
(2) Attributed to the precompression of concrete in the tensile zone of reinforced concrete
beams under repeated loads, the spacing and widths of cracks decreased with the
increased prestress of CFRP sheets. The flexural stiffness increased correspondingly
to reduce the mid-span deflection. The strengthened beams with the highest prestress
degree of CFRP sheets could limit the maximum crack width to within 0.2 mm, while
others could limit the maximum crack width to within 0.3 mm. Therefore, the normal
serviceability of the strengthened beams can be improved with small crack width and
deflection.
(3)
The strengthened beams could reach a higher bearing capacity with the increase in
the prestress of CFRP sheets. With the control tensioning stresses of CFRP sheets at
0.295f
fu
and 0.442f
fu
, the strengthened beams with a layer of CFRP sheets presented
an increased load capacity at the yield of longitudinal tensile steel bars by 41.0% and
56.2%, while the ultimate load capacity increased by 41.9% and 43.8%; the strength-
ened beams with two layers of CFRP sheets had an increased load capacity at the yield
of longitudinal tensile steel bars by 81.4% and 88.8%, while the ultimate load capacity
increased by 56.5% and 74.8%. However, the reduction in ductility represented by the
ratio of mid-span deflection at the ultimate to that at the yield of longitudinal tensile
steel bars needs to be addressed.
(4)
The ideal strengthening effect depends on the reliable bonding of CFRP sheets to
concrete. Measurement should be further studied to prevent the CFRP sheets from
peeling off of bonded concrete, and the breaking of CFRP sheets resulted from the
damage of cracked concrete along longitudinal tensile steel bars.
Author Contributions:
Conceptualization and methodology, F.L. and Y.W.; validation, H.W., C.L.
and Q.M.; formal analysis and investigation, writing—original draft preparation, H.W., S.S. and C.L.;
writing—review and editing, Y.W. and Q.M.; supervision and funding acquisition, F.L. All authors
have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the CSCEC Technology Research and Development Project,
China (CSCEC-2021-Z-24), and Special Joint Research Project of Zhengzhou City and NCWU,
China (2021013).
Data Availability Statement:
Data are available with the first author and can be shared with anyone
upon reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.

Buildings 2023,13, 2115 17 of 19
Glossary
Nfthe pre-tensioning force of CFRP sheets;
Mthe moment on the pure bending segment of RC beams under repeated loads;
M0the bending moment produced by repeated load when the prestress of concrete disappeared
at the bottom surface;
Mythe yield moment of the beams at the yield of longitudinal tensile steel bars;
Muthe ultimate moment of the beams at the ultimate state;
bthe sectional width of RC beams;
bfthe width of CFRP sheets;
hthe sectional depth of RC beams;
h0the effective sectional depth of RC beams;
l0the span of RC beams;
lmthe average spacing of cracks on the side surface at central height of the longitudinal tensile
steel bars of RC beams;
tfthe depth of CFRP sheets;
y0the distance from central axis to top section-edge of the transferred section of RC beams;
Asthe sectional area of longitudinal tensile steel bars;
Afthe sectional area of CFRP sheets;
Bthe flexural stiffness of RC beams strengthened by prestressed CFRP sheets;
I0the inertia moment of the transferred section of RC beam;
Lfthe length of CFRP sheets;
fcthe axial compressive strength of concrete;
fythe yield strength of steel bars;
ffu the ultimate tensile strength of CFRP sheets;
σcthe stress of concrete;
σsthe stress of longitudinal tensile steel bars;
σfthe stress of CFRP sheets;
σf,con the control stress of pre-tensioning for CFRP sheets;
σl1the prestress loss of CFRP sheets due to the compression of the anchorage locked on RC beams;
σl2the prestress loss due to the relaxation of CFRP sheets;
σl3the prestress loss due to the shrinkage and creep of concrete;
σpc the prestress of concrete after the prestress loss at the first stage;
σpf the effective prestress of CFRP sheets after the whole prestress loss;
σpc,T the prestress at the top edge of cross-section of RC beam;
σpc,B the prestress at the bottom edge of cross-section of RC beam;
σps the prestress of longitudinal tensile steel bars;
σ0f the stress of CFRP sheets when the prestress of concrete disappeared at the bottom edge;
σ0s the stress of longitudinal steel bars when prestress of concrete disappeared at the bottom edge;
εcthe strain of concrete;
εsthe strain of longitudinal steel bars;
εfthe strain of CFRP sheets;
εf,y the strain of CFRP sheets corresponding to the yield of longitudinal tensile steel bars;
εf,max the maximum strain of CFRP sheets at fracture;
Ecthe modulus of elasticity of concrete;
Esthe modulus of elasticity of steel bars;
Efthe modulus of elasticity of CFRP sheet;
αEs the elastic modulus ratio between steel bars and concrete;
αEf the elastic modulus ratio between CFRP sheet and concrete;
afthe mid-span deflection;
afy the mid-span deflection at the yield state of RC beam;
afu the mid-span deflection at the ultimate state of RC beam;
ρthe ratio of longitudinal tensile steel bars;
∆the compression value of anchorage locked on RC beam;
µthe ductility factor.
wmax the maximum crack width on the side surface at central height of the longitudinal tensile
steel bars of RC beam.

Buildings 2023,13, 2115 18 of 19
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