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Design method for calculating load-carrying capacity of reinforced concrete beams strengthened with external FRP
Construction and Building Materials 50 (2014) 577–583
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Design method for calculating load-carrying capacity of reinforced concrete beams strengthened with external FRP Tomas Skuturna ⇑, Juozas Valivonis Department of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University, Saule˙tekio al. 11, LT–10223 Vilnius, Lithuania
h i g h l i g h t s The analytical design method for calculating the load-carrying capacity of strengthened beams. Evaluation of the influence of the stiffness of the bond between concrete and external FRP. Comparison of the experimental and calculated results. Statistical assessment of proposed design method.
a r t i c l e
i n f o
Article history: Received 16 April 2013 Received in revised form 1 October 2013 Accepted 4 October 2013 Available online 26 October 2013 Keywords: Reinforced concrete FRP Strengthening Analytical modeling
a b s t r a c t This article analyses the load-carrying capacity of reinforced concrete beams strengthened with external FRP reinforcement. It provides a method for calculating the load-carrying capacity of strengthened beams. The calculation method enables to assess the influence of the bond slip on the load-carrying capacity of beams. A database has been created for experimental results. Calculations of the load-carrying capacity of strengthened beams have been made. The received theoretical results have been compared to the experimental ones. Statistical evaluation of the calculation method for load-carrying capacity has been carried out. The comparative and statistical analysis of the received results shows that the calculated load-carrying capacity of strengthened beams is similar to the one received in the course of experiments. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Strengthening of reinforced structures with FRP reinforcement is a method which is quite widely applied and is supposed to be efficient and convenient for the purpose of strengthening [1,2]. The efficiency of using FRP can only be ensured by its common work with the strengthened structure. There are a few factors that have an impact on the bond between concrete and FRP. These are the dimensions of the concrete element and FRP, the properties of concrete and the fibre, additional methods of anchorage, environmental conditions [3]. Very often the bond between concrete and FRP reinforcement in strengthened structures is not rigid [4,5]. Under load, due to shear strains in the bond, external FRP reinforcement may move in respect to concrete. This causes decrease in the efficiency of strengthening, increase in deflection and reduction of the load-carrying capacity. Generally, the methodology for making theoretical calculations of the load-carrying capacity of strengthened flexural ⇑ Corresponding author. Tel.: +370 52745225. E-mail addresses: [email protected] (T. Skuturna), [email protected] (J. Valivonis). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.10.015
members is based on the assumption that the hypothesis of plane sections is valid. Two equations of equilibrium, the total of internal axial forces and the total of moments, are used to calculate the strains of concrete in compression and those of the external FRP reinforcement under tension. This calculation method is also provided in the design guidances for calculating strengthened structures [6–8]. These recommendations are based on the assumption that the bond between concrete and FRP is rigid. Displacement of FRP in respect to concrete is not taken into account here. The experimental research, however, shows that the bond slip has an impact on the load-carrying capacity of the elements under flexure and the type of failure. It is not possible to evaluate the influence of the bond slip on the load-carrying capacity of strengthened structures with the existing methodology of design guidances. In order to get more accurate calculation results, the rigidity of the bond should be evaluated. 2. Development of the theoretical model The suggested method for calculating reinforced concrete flexural structures strengthened with external FRP reinforcement is based on the theory of built-up bars [9,10]. The following
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εc1
assumptions are made: the beam is consisted of two bars: concrete and FRP; bars are with partial shear connection. According to the theory of built-up bars, generally the displacement in the bond between the layers of two bars is described with the following equation:
u ¼ u1 u2
x1
ð1Þ
du e¼ dx
3 εs3
du ¼ e1 e2 dx
εf3
1 εs1
εf2
εf1
Fig. 2. Distribution of longitudinal deformations in experimental beams when M/ MR = 0.3 (1), when M/MR = 0.6 (2) and when M/MR = 0.9 (3).
ð3Þ
The stiffness of the composite section of two bars is calculated as follows:
2 εs2
ð2Þ
In this case, according to Eqs. (1) and (2), the relationship of the bond displacement and the strains can be written as follows:
2
e1 ¼ e01 þ
d y 2
dx
a1
ð9Þ
a2
ð10Þ
2
2 X E1 I1 þ E2 I2 ¼ Ei I i
ð4Þ
i¼1
where E1, I1, E2, I2 is the moduli of elasticity of the first and the second bars as well as moments of inertia. The moment acting in the composite section of two bars: 2 X M i T 2 a2 T 1 a1
ð5Þ
e2 ¼ e02
where Mi is the moment caused by external loading on one separate bar in a composite element, T1, T2 the shear forces acting in the bond; a1 and a2 is the distances from the centre of the bar to the centre of the bond between two bars (Fig. 1). Since shear forces acting in the bond between the bars are equal, and a = a1 + a2 the expression of the bending moment in a composite element: 2 X Mi Ta
ð6Þ
d y 2
dx
where e01 and e02 is the strains caused by forces acting along the axes of the bars. Since in this case there are no axial forces, these strains are caused only by shear forces T1 and T2. Therefore, the strains can be calculated according to the following equations::
e01 ¼
T1 E 1 A1
ð11Þ
e02 ¼
T 2 E 2 A2
ð12Þ
i¼1
M¼
εc3
x3
x2
here u1 and u2 is the displacements on the surfaces of the upper and the lower layers of the bars. The relationship between displacement and strains is obtained as follows:
M¼
εc2
where A1 and A2 is the cross-sectional areas of bars. When the expressions of axial strains in bars are known, the equations of the strains in the bond between bars can be rewritten as follows:
e1 ¼
T1 M P2 a1 E 1 A1 i¼1 Ei I i
ð13Þ
e2 ¼
T 2 M þ P2 a2 E 2 A2 i¼1 Ei I i
ð14Þ
i¼1
Shear force T can be expressed through tangential stresses s:
T¼
Z
x
sdx
ð7Þ
0
The curvature of the axis in an element under a bending moment can be described with the following expression: 2
d y
M ¼ P2 dx i¼1 Ei Ii
ð8Þ
2
Strains on the surface of the bond of the bars are defined with the help of the equations:
A rearranged Eq. (3) can be written as follows:
du T1 M T2 M ¼ P2 a1 þ P2 a2 dx E1 A1 E A 2 2 E I i¼1 i i i¼1 Ei Ii
ð15Þ
If Eq. (6) is inserted into Eq. (15), a rearranged Eq. (15) can be rewritten in the form (16):
du ¼ dx
! P2 1 1 a2 Mi T P2i¼1 a þ þ P2 E1 A1 E2 A2 i¼1 Ei Ii i¼1 Ei I i
ð16Þ
The expression of the displacement in the area of the bond of the bars can also be written as follows:
u¼ M1
T1
a1
a
a2
T2
M2
Fig. 1. Calculation scheme.
s 1 dT ¼ l l dx
ð17Þ
where l is the characteristics which estimates the rigidity of the bond of the bars. Displacement in the area of the bond of two bars can be described as follows:
du ¼ dx
2 1 d T
l dx2
ð18Þ
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
fc
b
εc d2
fs2As2
εs2
0.8x
x
0.8fcd bx
d
h
εs1
d1
εf
bf
fs1 As1 kff Af
kε f
Fig. 3. Strain and stress distribution in a reinforced concrete beam section.
F/2
F/2
b As2
d2
h As1 Af la
la
tf d1
bf
l Fig. 4. Test setup and reinforcement details of the beam.
bcf Geff a
ð19Þ
where bcf is the width of the joint of the bars; Geff is the characteristics of the shear in the bond between bars. When both expressions (16) and (18) of the displacement in the area of the bond between two bars are equated:
2 1 d T
l dx2
! P2 1 1 a2 Mi T P2i¼1 a þ þ P2 E 1 A1 E 2 A2 E I E i¼1 i i i¼1 i Ii
¼
ð20Þ
1.40 Mexp/Mcalc Mexp/Mcalc*
1.20
Mexp / Mcalc
l¼
1.00 0.80
The variables of the received equation can be marked:
c¼
1 1 a2 þ þ P2 E 1 A1 E 2 A2 i¼1 Ei I i
0.60
ð21Þ 0.40 0
Eq. (20) can be written:
P2 Mi ¼ T c P2i¼1 a E dx i¼1 i I i
2 1 d T
l
2
ð22Þ
2
2
dx
¼ p
ð23Þ
If Eq. (22) is differentiated two times, the following expression is obtained: 4
d T 4
dx
2
lc
d T dx
2
lap
P2
i¼1 Ei I i
¼0
k2 ¼ lc
ð24Þ
ð25Þ
When Eq. (24) is solved, the expression of the shear force is got if the simply supported flexural element is loaded by two concentrated forces:
Fla a chðkð0:5l la ÞÞ T ¼ P2 1 shðkla Þ kla chð0:5klÞ c i¼1 Ei Ii
200
300
400
500
600
Mexp
The external loading and the bending moment have a relationship:
d M
100
Fig. 5. The compare of theoretical results Mexp and Mexp .
The variables of the received equation can be marked as follows:
T ¼ Tsk Ts ¼
ð27Þ Fla a
k¼1
chðkð0:5l la ÞÞ shðkla Þ kla chð0:5klÞ
ð29Þ
where Ts is the shear force when the connection of the bars is rigid; k the coefficient which evaluates the rigidity of the bond of the bars; l the length of the beam span; la is the distance from support to the concentrated force. If the flexural element is loaded by uniformly distributed load: 2
ð26Þ
ð28Þ
cðE1 I1 þ E2 I2 Þ
T¼
pl a P2
8c
i¼1 Ei Ii
" 1
# 8ðchð0:5klÞ 1Þ 2
k2 l chð0:5klÞ
ð30Þ
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
Table 1 Database of experimental and theoretical results of reinforced concrete beams strengthened with external CFRP reinforcement. Reference
Beam number
Beam notation
Mexp (kN m)
Mcalc (kN m)
Mexp/Mcalc
M calc (kN m)
Mexp =Mcalc
[12] [13]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
G1 A1.1 A3.1 CFC30 A10 A20 B10 B20 B4 B5 B6 B3 B4 B5 B6 C3 C4 C5 C6 A0 A3 A4 B3 B4 B5 B6 C3 C4 C5 5 DF2 A2 A3 B3 B2 B3 VR3 VR4 VR5 VR6 A950 A1100 A1150 C5 C10 C20 B08S B08M B083m B2 B3 B4 B6 B7 B8 B10 B11 B12 BR1-2 BR1-3 BR2-2 BR3-2
35.0 78.1 67.3 53.7 23.5 28.4 30.9 31.9 137.8 103.0 108.0 20.7 19.7 26.1 26.1 28.1 29.0 38.7 38.0 20.2 19.4 18.9 131.8 130.2 147.4 142.4 522.3 535.4 520.1 19.8 30.0 16.5 17.6 18.8 42.4 45.8 25.5 24.3 40.0 39.4 12.4 12.6 13.0 15.6 15.0 13.9 67.2 98.0 64.4 18.4 21.3 22.3 25.5 28.5 31.8 31.9 32.7 34.0 15.8 15.6 15.9 13.6
37.8 83.6 81.1 52.4 24.7 26.3 31.4 32.5 116.9 101.4 101.4 17.5 19.0 24.0 24.0 30.6 30.6 38.9 38.9 18.5 17.0 17.0 133.4 133.4 179.0 179.0 550.3 550.3 733.9 16.0 26.7 13.6 14.6 16.6 42.4 50.0 28.5 28.5 35.4 35.4 14.2 14.2 14.2 15.1 14.5 13.4 76.0 93.8 64.7 24.9 25.8 25.7 28.9 29.6 30.2 33.7 34.2 34.6 15.2 15.2 15.4 13.4 Mean Standard deviation Coefficient of variation (%)
0.93 0.93 0.83 1.03 0.95 1.08 0.98 0.98 1.18 1.02 1.07 1.18 1.04 1.09 1.09 0.92 0.94 0.99 0.98 1.09 1.14 1.11 0.99 0.98 0.82 0.80 0.95 0.97 0.72 1.23 1.12 1.21 1.20 1.13 1.00 0.92 0.90 0.85 1.13 1.11 0.87 0.89 0.92 1.03 1.03 1.03 0.88 1.04 1.00 0.74 0.82 0.87 0.88 0.96 1.05 0.95 0.95 0.98 1.04 1.03 1.03 1.01 0.99 0.11 11.36
37.8 92.1 81.1 52.4 27.0 31.8 33.0 36.8 201.9 152.9 152.9 19.0 19.0 34.5 34.5 34.8 34.8 45.5 45.5 18.5 19.3 19.3 151.7 151.7 179.0 179.0 623.8 623.8 733.9 21.2 26.7 14.9 18.1 19.8 52.6 54.4 50.6 50.6 75.6 75.6 21.7 21.7 21.7 21.3 20.7 19.6 82.1 122.9 64.7 29.2 35.2 39.2 30.6 32.8 37.4 35.0 36.6 40.3 19.1 19.1 19.3 17.1
0.93 0.85 0.83 1.03 0.87 0.89 0.94 0.87 0.68 0.67 0.71 1.09 1.04 0.76 0.76 0.81 0.83 0.85 0.84 1.09 1.01 0.98 0.87 0.86 0.82 0.80 0.84 0.86 0.72 0.93 1.12 1.10 0.97 0.95 0.81 0.84 0.50 0.48 0.53 0.52 0.57 0.58 0.60 0.73 0.72 0.71 0.82 0.80 1.00 0.63 0.61 0.57 0.83 0.87 0.85 0.91 0.89 0.85 0.83 0.82 0.82 0.80 0.82 0.15 18.69
[14] [15]
[16]
[17]
[18] [19]
[20] [21] [22]
[23] [24]
[25]
[26]
[27]
[10]
The variables of the received equation can be marked as follows: 2
Ts ¼
pl a 8cðE1 I1 þ E2 I2 Þ
ð31Þ
k¼1
8ðchð0:5klÞ 1Þ 2
k2 l chð0:5klÞ
ð32Þ
When calculating the load-carrying capacity of the bent elements made of two bars, the rigidity of the bond between bars of the element can be estimated by coefficient k. This estimation of the
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583 Table 2 Sample statistics and results of Shapiro-Wilk test. Sample number
n
Beam number
As1/db
d (mm)
Variable
Mean
v
Min
Max
W
Wa
Distribution
1
10
12, 13, 14, 15, 42, 43, 44, 45, 46, 41
0.65–1.64
120–140
7
16, 17, 18, 19, 21, 22, 30
1.68–1.71
120
3
10
20, 32, 33, 50, 51, 52, 59, 60, 61, 62
0.56–1.05
150–175
4
7
34, 53, 54, 55, 56, 57, 58
1.51–2.58
150–164
5
7
1, 31, 36, 37, 38, 39, 40
0.58–0.83
200–225
6
6
4, 5, 6, 7, 8, 35
0.86–1.52
206–220
7
5
2, 3, 47, 48, 49
0.82–1.06
270–275
8
4
23, 24, 25, 26
1.70
240
9
3
9, 10, 11
0.61–0.73
309
10
3
27, 28, 29
1.71
384
1.02 0.76 1.05 0.89 1.00 0.82 0.99 0.88 0.99 0.70 1.00 0.90 0.94 0.86 0.90 0.84 1.09 0.69 0.88 0.81
9.62 23.63 11.24 8.96 15.30 23.01 8.22 4.75 12.26 36.57 4.44 8.26 9.11 9.20 11.23 4.03 7.66 2.49 15.86 9.26
0.87 0.57 0.92 0.81 0.74 0.57 0.88 0.83 0.85 0.48 0.95 0.81 0.83 0.80 0.80 0.80 1.02 0.67 0.72 0.72
1.18 1.09 1.23 1.01 1.21 1.10 1.13 0.95 1.13 1.12 1.08 1.03 1.04 1.00 0.99 0.87 1.18 0.71 0.97 0.86
0.921 0.842 0.917 0.872 0.932 0.918 0.925 0.938 0.810 0.827 0.938 0.947 0.983 0.771 0.824 0.938 0.953 0.940 0.820 0.861
0.842
2
Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc
NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL
Table 3 Results of t test. Sample number 1 2 3 4 5 6 7 8 9 10
Variable
t
n1
ta/2,(n1)
H0
Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc
0.56 4.34 1.03 3.58 0.09 2.92 0.41 7.72 0.14 3.05 0.21 3.29 1.63 4.03 2.08 9.72 1.80 31.65 1.48 4.52
9.00 9.00 6.00 6.00 9.00 9.00 6.00 6.00 6.00 6.00 5.00 5.00 4.00 4.00 3.00 3.00 2.00 2.00 2.00 2.00
2.262 2.262 2.447 2.447 2.262 2.262 2.447 2.447 2.447 2.447 2.571 2.571 2.776 2.776 3.182 3.182 4.303 4.303 4.303 4.303
ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED
rigidity of the bond can be used in calculating flexural reinforced structures strengthened with external FRP reinforcement. Experimental research performed [11] shows that the bond between the external FRP reinforcement and the strengthened reinforced concrete element is not rigid (Fig. 2). When the bending moment increases, the strains in the bond between concrete and FRP also increase and as a result FRP reinforcement is more and more displaced in respect to concrete. This is why it is suggested to estimate the rigidity of the bond with coefficient k which can be calculated according to the expressions (29) or (32). The load-carrying capacity of reinforced concrete beams strengthened with external FRP reinforcement is calculated (Fig. 3) by summing the moments of internal forces about the centroid of the concrete compressive stress block:
M R ¼ As1 fyd ðd 0:4xÞ þ Af f fd kðh 0:4xÞ þ As2 fscd ð0:4x d2 Þ
ð33Þ
As1 is the cross-sectional area of the steel bar reinforcement in tension; As2 the cross-sectional area of steel bar reinforcement in
0.803 0.842 0.803 0.803 0.788 0.762 0.748 0.767 0.767
compression; Af the cross-sectional area of FRP reinforcement; fyd the design strength of steel bar reinforcement in tension; fscd the design strength of the steel bar reinforcement in compression; ffd the design strength of FRP reinforcement; h the overall depth of the cross-section; d the effective depth of the cross-section; d1 the distance from the bottom of the element to the centre of reinforcement in tension; d2 is the distance from the top of the element to the centre of reinforcement in compression. The depth of the neutral axis is calculated by summing internal forces and taking into account the stresses and strains in the steel bar reinforcement as well as carbon fibre plastic:
x¼
As1 fyd þ Af f fd k As2 fscd 0:8f cd b
ð34Þ
Here b is the width of the element; fcd is the design compressive strength of concrete. The load-carrying capacity of reinforced beams strengthened with CFRP reinforcement is calculated according to the assumptions that the ultimate deformation of concrete ecu = 0.0035; the ultimate FRP strain [10], at the moment of separation efs = 0.008; the hypothesis of planes sections is valid for concrete element, however, the shear strains acting between the strengthened element and the external reinforcement are assessed by coefficient k, since the bond of concrete and fibre plastic is not rigid. On the ground of the analysis of the results of the experimental research it was determined that the coefficient k which assesses the rigidity between concrete and external FRP reinforcement can be calculated according to Eqs. (29) or (32) if:
As1 fyk <1 Af f fk
ð35Þ
or
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm < 6 103
ðwhere f cm ; fctm in kPaÞ
ð36Þ
where fyk is the characteristic strength of steel bar reinforcement in tension; ffk the characteristic strength of FRP reinforcement; fctm the mean strength of concrete in tension; fcm is the mean compressive strength of concrete. Coefficient k = 1, when:
As1 fyk P1 Af f fk
ð37Þ
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and
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm > 6 103
ðwherefcm ; fctm in kPaÞ
ð38Þ
Coefficient k = 0.5, when conditions (39) and (40) are valid.
As1 fyk P2 Af f fk
ð39Þ
and
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm > 8:5 103
ðwherefcm ; fctm in kPaÞ
ð40Þ
According to the results of experimental research of reinforced concrete beams strengthened with external FRP reinforcement, the characteristics of the shear in Eq. (19) can be calculated as follows:
Geff
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1 fctm fcm ¼ qffiffiffiffiffiffiffiffiffiffiffiffi
ð41Þ
3. Results of calculations and statistical verification of the theoretical model In order to carry out theoretical calculations a database of experimental results was created. The database comprises the results of research performed by different authors on 62 different beams. The data was collected according to the following criteria: the cross-section of beams is rectangular; beams were reinforced with steel bar reinforcement and strengthened with an external CFRP reinforcement; the external reinforcement was glued in the tension zone of the beams (Fig. 4); carbon fibre was not prestressed and not anchored additionally; the ultimate strength of concrete up to 50 MPa; the samples were not additionally preloaded prior to strengthening. The loading scheme for reinforced concrete beams strengthened with an external CFRP reinforcement is show in Fig. 4. During the research all beams were tested in four-point bending. Table 1 provides the experimental Mexp and the calculated Mcalc load carrying capacities of strengthened beams. It also provides the ratio of the experimental and the theoretical results Mexp/Mcalc. The theoretical load carrying capacities Mexp were calculated when the coefficient k = 1. This allows to show the effect of slip for results of calculated load carrying capacities Fig. 5. In order to determine the feasibility of the suggested method a statistical analysis of calculation results has been performed. The calculated load-carrying capacity of strengthened beams will always be different from the one received in the course of the experiment. However, this difference may be statistically insignificant. The ideal case is when the means of the Mexp/Mcalc values equals to 1:
ð42Þ Mexp =Mcalc , Mexp =M calc
where M exp =Mcalc are the means of the ratios of the experimental and theoretical results. In order to compare the mean of theoretical and experimental results [28], a statistical hypothesis is formulated:
: Mexp =Mcalc ¼ 1
:
: Mexp =Mcalc –1
H1
ð44Þ
Here W is the calculated value in Shapiro-Wilk test, Wa is the ultimate value received from tables. In order to check the hypothesis the t test is applied since the data has normal distribution. When the variances are unknown and not equal, t statistics is calculated as follows:
M exp =M calc 1 qffiffiffiffiffiffiffiffiffiffiffi 2 sexp=calc
ð45Þ
n
The hypothesis H0 is rejected when:
where Ecm is the secant modulus of elasticity of concrete; the width, thickness of external FRP reinforcement as well as the number of layers respectively bf, tf, nf.
8 < H0
W P Wa
t¼
Ef bf t f nf Ecm b
M exp =M calc ¼ 1
Table 2 provides: the size of samples n, the means, the minimal and maximal values, the coefficient of variation v. With the help of Shapiro-Wilk test [29] for samples of small size (Eq. (44)) it was determined that it can be stated that when the level of significance is a = 0,05, the data has normal distribution (Table 2).
ð43Þ
On the ground of the gathered data, statistical samples of results have been made. The samples have been made according to the ratio of steel reinforcement of the beams and the effective depth of the cross-section.
jtj > ta=2;n1
ð46Þ
where ta/2,n1 is the critical value of the Stjudent‘s distribution with n 1 degrees of freedom, when a = 0.05; sexp/calc the standard deviations sexp/calc and sexp=calc of the ratios of the experimental and the theoretical results; n is the size of the samples. The received statistical results are provided in Table 3. Statistical analysis allows to state that the difference between the theoretical and the experimental load-carrying capacity in samples 1–10 is statistically insignificant if the coefficient k is applied. This shows that with the help of the suggested methodology it is possible to make quite accurate calculations of the load carrying capacity of beams strengthened with external CFRP reinforcement. However the means of the ratios of the experimental and theoretical results statistically significantly differ from 1 when the coefficient k = 1. 4. Conclusions A method for calculating the load-carrying capacity of reinforced concrete beams strengthened with external CFRP reinforcement provided in this work allows to perform the evaluation of the rigidity of the bond between the external CFRP reinforcement and concrete. This ensures performance of more accurate calculations of the load-carrying capacity of strengthened beams. The accuracy of the method has been assessed having performed calculations of 62 experimental beams strengthened with external CFRP reinforcement. A statistical evaluation of the received theoretical and the experimental results has also been performed. The performed analysis showed that, if we assess the rigidity of the bond between external reinforcement and concrete, with the help of the suggested calculation method it is possible to make quite precise calculations of the load-carrying capacity. Statistical evaluation of the results shows that the difference between the load-carrying capacity received during the course of the experiment and the calculated ones is statistically insignificant. References [1] Trapko W, Trapko T. Load-bearing capacity of compressed concrete elements subjected to repeated load strengthened with CFRP materials. J Civ Eng Manage 2012;18(2):590–9. [2] Hajsadeghi M, Alaee FJ, Shahmohammadi A. Investigation on behaviour of square/rectangular reinforced concrete columns retrofitted with FRP jacket. J Civ Eng Manage 2011;17(3):400–8. [3] Silva MAG, Biscaia HC. Effects of exposure to saline humidity on bond between GFRP and concrete. Compos Struct 2010;93(1):216–24. [4] Biscaia HC, Chastre C, Silva MAG. Double shear tests to evaluate the bond strength between GFRP/concrete elements. Compos Struct 2012;94(2): 681–94.
T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583 [5] Biscaia HC, Chastre C, Silva MAG. Linear and nonlinear analysis of bond–slip models for interfaces between FRP composites and concrete. Composites Part B 2013;45(1):1554–68. [6] ACI 440.2R-08. Guide for the design and construction of externally bonded FRP. Systems for strengthening concrete structures. American Concrete Institute; 2008. [7] FIB Bulletin 14. Externally bonded FRP reinforcement for RC structures. International federation for structural concrete; 2001. [8] Technical Report 55. Design guidance for strengthening concrete structures using fibre composite materials. Concrete Society; 2012. [9] Rzhanitsyn AR. Built-up bars and plates. Strojizdat; 1986. [10] Skuturna T. Research of flexural reinforced concrete members strengthened with external carbon fibre. PhD thesis, Vilnius Gediminas technical University; 2009 [in Lithuanian]. nas P, Marcˇiukaitis G, Daugevicˇius M. Analysis of [11] Skuturna T, Valivonis J, Vainiu deflections of bridge girders strengthened by carbon fibre reinforcement. Balt J Road Bridge Eng 2008;3(3):145–51. [12] Brosens K. Anchorage of Externally Bonded Steel Plates and CFRP Laminates for the Strengthening of Concrete Elements. PhD thesis, K.U.Leuven; 2001. [13] Spadea G, Bencardino F, Swamy RN. Optimizing the performance characteristics of beams strengthened with bonded CFRP laminates. Mater Struct 2000;33(2):119–26. [14] Wenwei W, Guo L. Experimental study and analysis of RC beams strengthened with CFRP laminates under sustaining load. Int J Solids Struct 2006;43(6): 1372–87. [15] Zhang A, Jin W, Li G. Behavior of preloaded RC beams strengthened with CFRP laminates. J Zhejiang Univ Sci A 2006;7(3):436–44. [16] Kim D, Sebastian WM. Parametric study of bond failure in concrete beams externally strengthened with fibre reinforced polymer plates. Mag Concr Res 2002;54(1):47–59. [17] Rahimi H, Hutchinson A. Concrete beams strengthened with externally bonded FRP plates. J Compos Constr 2001;5(1):44–56.
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[18] Gao B, Kim JK, Leung CKY. Experimental study on RC beams with FRP strips bonded with rubber modified resins. Compos Sci Technol 2004;64(16): 2557–64. [19] Maalej M, Leong KS. Effect of beam size and FRP thickness on interfacial shear stress concentration and failure mode of FPP-strengthened beams. Compos Sci Technol 2005;65(7–8):1148–58. [20] Maalej M, Bian Y. Interfacial shear stress concentration in FRP strengthened beams. Compos Struct 2001;54(4):417–26. [21] Ahmed O, VanGemert D, Vandewalle L. Improved model for plate-end shear of CFRP strengthened RC beams. Cem Concr Compos 2001;23(1):3–19. [22] Ceroni F. Experimental performances of RC beams strengthened with FRP materials. Constr Build Mater 2010;24(9):1547–59. [23] Garden HN, Quantrill RJ, Hollaway LC, Thorne AM, Parke GAR. An experimental study of the anchorage length of carbon fibre composite plates used to strengthen reinforced concrete beams. Constr Build Mater 1998;12(4):203–19. [24] Beber AJ, Filho AC, Campagnolo JL. Flexural strengthening of r/c beams with CFRP sheets. In: Structural faults and Repair 99, proceedings of 8th international conference, London, July, 1999. [25] Nguyen DM, Chan TK, Cheong HK. Brittle failure and bond development length of CFRP concrete beams. J Compos Constr 2001;5(1):12–7. [26] Kotynia R, Baky HA, Neale KW, Ebead UA. Flexural strengthening of RC beams with externally bonded CFRP systems: test results and 3D nonlinear FE analysis. J Compos Constr 2008;12(2):190–201. [27] Esfahani MR, Kianoush MR, Tajari AR. Flexural behaviour of reinforced concrete beams strengthened by CFRP sheets. Eng Struct 2007;29(10): 2428–44. [28] Montgomery DC, Runger GC. Applied statistics and probability for engineers. John Wiley & Sons; 2002. [29] Shapiro SS, Wilk MB. An Analysis of variance test for normality. Biometrika 1965;52(3–4):591–611.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Design method for calculating load-carrying capacity of reinforced concrete beams strengthened with external FRP Tomas Skuturna ⇑, Juozas Valivonis Department of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University, Saule˙tekio al. 11, LT–10223 Vilnius, Lithuania
h i g h l i g h t s The analytical design method for calculating the load-carrying capacity of strengthened beams. Evaluation of the influence of the stiffness of the bond between concrete and external FRP. Comparison of the experimental and calculated results. Statistical assessment of proposed design method.
a r t i c l e
i n f o
Article history: Received 16 April 2013 Received in revised form 1 October 2013 Accepted 4 October 2013 Available online 26 October 2013 Keywords: Reinforced concrete FRP Strengthening Analytical modeling
a b s t r a c t This article analyses the load-carrying capacity of reinforced concrete beams strengthened with external FRP reinforcement. It provides a method for calculating the load-carrying capacity of strengthened beams. The calculation method enables to assess the influence of the bond slip on the load-carrying capacity of beams. A database has been created for experimental results. Calculations of the load-carrying capacity of strengthened beams have been made. The received theoretical results have been compared to the experimental ones. Statistical evaluation of the calculation method for load-carrying capacity has been carried out. The comparative and statistical analysis of the received results shows that the calculated load-carrying capacity of strengthened beams is similar to the one received in the course of experiments. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Strengthening of reinforced structures with FRP reinforcement is a method which is quite widely applied and is supposed to be efficient and convenient for the purpose of strengthening [1,2]. The efficiency of using FRP can only be ensured by its common work with the strengthened structure. There are a few factors that have an impact on the bond between concrete and FRP. These are the dimensions of the concrete element and FRP, the properties of concrete and the fibre, additional methods of anchorage, environmental conditions [3]. Very often the bond between concrete and FRP reinforcement in strengthened structures is not rigid [4,5]. Under load, due to shear strains in the bond, external FRP reinforcement may move in respect to concrete. This causes decrease in the efficiency of strengthening, increase in deflection and reduction of the load-carrying capacity. Generally, the methodology for making theoretical calculations of the load-carrying capacity of strengthened flexural ⇑ Corresponding author. Tel.: +370 52745225. E-mail addresses: [email protected] (T. Skuturna), [email protected] (J. Valivonis). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.10.015
members is based on the assumption that the hypothesis of plane sections is valid. Two equations of equilibrium, the total of internal axial forces and the total of moments, are used to calculate the strains of concrete in compression and those of the external FRP reinforcement under tension. This calculation method is also provided in the design guidances for calculating strengthened structures [6–8]. These recommendations are based on the assumption that the bond between concrete and FRP is rigid. Displacement of FRP in respect to concrete is not taken into account here. The experimental research, however, shows that the bond slip has an impact on the load-carrying capacity of the elements under flexure and the type of failure. It is not possible to evaluate the influence of the bond slip on the load-carrying capacity of strengthened structures with the existing methodology of design guidances. In order to get more accurate calculation results, the rigidity of the bond should be evaluated. 2. Development of the theoretical model The suggested method for calculating reinforced concrete flexural structures strengthened with external FRP reinforcement is based on the theory of built-up bars [9,10]. The following
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
εc1
assumptions are made: the beam is consisted of two bars: concrete and FRP; bars are with partial shear connection. According to the theory of built-up bars, generally the displacement in the bond between the layers of two bars is described with the following equation:
u ¼ u1 u2
x1
ð1Þ
du e¼ dx
3 εs3
du ¼ e1 e2 dx
εf3
1 εs1
εf2
εf1
Fig. 2. Distribution of longitudinal deformations in experimental beams when M/ MR = 0.3 (1), when M/MR = 0.6 (2) and when M/MR = 0.9 (3).
ð3Þ
The stiffness of the composite section of two bars is calculated as follows:
2 εs2
ð2Þ
In this case, according to Eqs. (1) and (2), the relationship of the bond displacement and the strains can be written as follows:
2
e1 ¼ e01 þ
d y 2
dx
a1
ð9Þ
a2
ð10Þ
2
2 X E1 I1 þ E2 I2 ¼ Ei I i
ð4Þ
i¼1
where E1, I1, E2, I2 is the moduli of elasticity of the first and the second bars as well as moments of inertia. The moment acting in the composite section of two bars: 2 X M i T 2 a2 T 1 a1
ð5Þ
e2 ¼ e02
where Mi is the moment caused by external loading on one separate bar in a composite element, T1, T2 the shear forces acting in the bond; a1 and a2 is the distances from the centre of the bar to the centre of the bond between two bars (Fig. 1). Since shear forces acting in the bond between the bars are equal, and a = a1 + a2 the expression of the bending moment in a composite element: 2 X Mi Ta
ð6Þ
d y 2
dx
where e01 and e02 is the strains caused by forces acting along the axes of the bars. Since in this case there are no axial forces, these strains are caused only by shear forces T1 and T2. Therefore, the strains can be calculated according to the following equations::
e01 ¼
T1 E 1 A1
ð11Þ
e02 ¼
T 2 E 2 A2
ð12Þ
i¼1
M¼
εc3
x3
x2
here u1 and u2 is the displacements on the surfaces of the upper and the lower layers of the bars. The relationship between displacement and strains is obtained as follows:
M¼
εc2
where A1 and A2 is the cross-sectional areas of bars. When the expressions of axial strains in bars are known, the equations of the strains in the bond between bars can be rewritten as follows:
e1 ¼
T1 M P2 a1 E 1 A1 i¼1 Ei I i
ð13Þ
e2 ¼
T 2 M þ P2 a2 E 2 A2 i¼1 Ei I i
ð14Þ
i¼1
Shear force T can be expressed through tangential stresses s:
T¼
Z
x
sdx
ð7Þ
0
The curvature of the axis in an element under a bending moment can be described with the following expression: 2
d y
M ¼ P2 dx i¼1 Ei Ii
ð8Þ
2
Strains on the surface of the bond of the bars are defined with the help of the equations:
A rearranged Eq. (3) can be written as follows:
du T1 M T2 M ¼ P2 a1 þ P2 a2 dx E1 A1 E A 2 2 E I i¼1 i i i¼1 Ei Ii
ð15Þ
If Eq. (6) is inserted into Eq. (15), a rearranged Eq. (15) can be rewritten in the form (16):
du ¼ dx
! P2 1 1 a2 Mi T P2i¼1 a þ þ P2 E1 A1 E2 A2 i¼1 Ei Ii i¼1 Ei I i
ð16Þ
The expression of the displacement in the area of the bond of the bars can also be written as follows:
u¼ M1
T1
a1
a
a2
T2
M2
Fig. 1. Calculation scheme.
s 1 dT ¼ l l dx
ð17Þ
where l is the characteristics which estimates the rigidity of the bond of the bars. Displacement in the area of the bond of two bars can be described as follows:
du ¼ dx
2 1 d T
l dx2
ð18Þ
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
fc
b
εc d2
fs2As2
εs2
0.8x
x
0.8fcd bx
d
h
εs1
d1
εf
bf
fs1 As1 kff Af
kε f
Fig. 3. Strain and stress distribution in a reinforced concrete beam section.
F/2
F/2
b As2
d2
h As1 Af la
la
tf d1
bf
l Fig. 4. Test setup and reinforcement details of the beam.
bcf Geff a
ð19Þ
where bcf is the width of the joint of the bars; Geff is the characteristics of the shear in the bond between bars. When both expressions (16) and (18) of the displacement in the area of the bond between two bars are equated:
2 1 d T
l dx2
! P2 1 1 a2 Mi T P2i¼1 a þ þ P2 E 1 A1 E 2 A2 E I E i¼1 i i i¼1 i Ii
¼
ð20Þ
1.40 Mexp/Mcalc Mexp/Mcalc*
1.20
Mexp / Mcalc
l¼
1.00 0.80
The variables of the received equation can be marked:
c¼
1 1 a2 þ þ P2 E 1 A1 E 2 A2 i¼1 Ei I i
0.60
ð21Þ 0.40 0
Eq. (20) can be written:
P2 Mi ¼ T c P2i¼1 a E dx i¼1 i I i
2 1 d T
l
2
ð22Þ
2
2
dx
¼ p
ð23Þ
If Eq. (22) is differentiated two times, the following expression is obtained: 4
d T 4
dx
2
lc
d T dx
2
lap
P2
i¼1 Ei I i
¼0
k2 ¼ lc
ð24Þ
ð25Þ
When Eq. (24) is solved, the expression of the shear force is got if the simply supported flexural element is loaded by two concentrated forces:
Fla a chðkð0:5l la ÞÞ T ¼ P2 1 shðkla Þ kla chð0:5klÞ c i¼1 Ei Ii
200
300
400
500
600
Mexp
The external loading and the bending moment have a relationship:
d M
100
Fig. 5. The compare of theoretical results Mexp and Mexp .
The variables of the received equation can be marked as follows:
T ¼ Tsk Ts ¼
ð27Þ Fla a
k¼1
chðkð0:5l la ÞÞ shðkla Þ kla chð0:5klÞ
ð29Þ
where Ts is the shear force when the connection of the bars is rigid; k the coefficient which evaluates the rigidity of the bond of the bars; l the length of the beam span; la is the distance from support to the concentrated force. If the flexural element is loaded by uniformly distributed load: 2
ð26Þ
ð28Þ
cðE1 I1 þ E2 I2 Þ
T¼
pl a P2
8c
i¼1 Ei Ii
" 1
# 8ðchð0:5klÞ 1Þ 2
k2 l chð0:5klÞ
ð30Þ
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
Table 1 Database of experimental and theoretical results of reinforced concrete beams strengthened with external CFRP reinforcement. Reference
Beam number
Beam notation
Mexp (kN m)
Mcalc (kN m)
Mexp/Mcalc
M calc (kN m)
Mexp =Mcalc
[12] [13]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
G1 A1.1 A3.1 CFC30 A10 A20 B10 B20 B4 B5 B6 B3 B4 B5 B6 C3 C4 C5 C6 A0 A3 A4 B3 B4 B5 B6 C3 C4 C5 5 DF2 A2 A3 B3 B2 B3 VR3 VR4 VR5 VR6 A950 A1100 A1150 C5 C10 C20 B08S B08M B083m B2 B3 B4 B6 B7 B8 B10 B11 B12 BR1-2 BR1-3 BR2-2 BR3-2
35.0 78.1 67.3 53.7 23.5 28.4 30.9 31.9 137.8 103.0 108.0 20.7 19.7 26.1 26.1 28.1 29.0 38.7 38.0 20.2 19.4 18.9 131.8 130.2 147.4 142.4 522.3 535.4 520.1 19.8 30.0 16.5 17.6 18.8 42.4 45.8 25.5 24.3 40.0 39.4 12.4 12.6 13.0 15.6 15.0 13.9 67.2 98.0 64.4 18.4 21.3 22.3 25.5 28.5 31.8 31.9 32.7 34.0 15.8 15.6 15.9 13.6
37.8 83.6 81.1 52.4 24.7 26.3 31.4 32.5 116.9 101.4 101.4 17.5 19.0 24.0 24.0 30.6 30.6 38.9 38.9 18.5 17.0 17.0 133.4 133.4 179.0 179.0 550.3 550.3 733.9 16.0 26.7 13.6 14.6 16.6 42.4 50.0 28.5 28.5 35.4 35.4 14.2 14.2 14.2 15.1 14.5 13.4 76.0 93.8 64.7 24.9 25.8 25.7 28.9 29.6 30.2 33.7 34.2 34.6 15.2 15.2 15.4 13.4 Mean Standard deviation Coefficient of variation (%)
0.93 0.93 0.83 1.03 0.95 1.08 0.98 0.98 1.18 1.02 1.07 1.18 1.04 1.09 1.09 0.92 0.94 0.99 0.98 1.09 1.14 1.11 0.99 0.98 0.82 0.80 0.95 0.97 0.72 1.23 1.12 1.21 1.20 1.13 1.00 0.92 0.90 0.85 1.13 1.11 0.87 0.89 0.92 1.03 1.03 1.03 0.88 1.04 1.00 0.74 0.82 0.87 0.88 0.96 1.05 0.95 0.95 0.98 1.04 1.03 1.03 1.01 0.99 0.11 11.36
37.8 92.1 81.1 52.4 27.0 31.8 33.0 36.8 201.9 152.9 152.9 19.0 19.0 34.5 34.5 34.8 34.8 45.5 45.5 18.5 19.3 19.3 151.7 151.7 179.0 179.0 623.8 623.8 733.9 21.2 26.7 14.9 18.1 19.8 52.6 54.4 50.6 50.6 75.6 75.6 21.7 21.7 21.7 21.3 20.7 19.6 82.1 122.9 64.7 29.2 35.2 39.2 30.6 32.8 37.4 35.0 36.6 40.3 19.1 19.1 19.3 17.1
0.93 0.85 0.83 1.03 0.87 0.89 0.94 0.87 0.68 0.67 0.71 1.09 1.04 0.76 0.76 0.81 0.83 0.85 0.84 1.09 1.01 0.98 0.87 0.86 0.82 0.80 0.84 0.86 0.72 0.93 1.12 1.10 0.97 0.95 0.81 0.84 0.50 0.48 0.53 0.52 0.57 0.58 0.60 0.73 0.72 0.71 0.82 0.80 1.00 0.63 0.61 0.57 0.83 0.87 0.85 0.91 0.89 0.85 0.83 0.82 0.82 0.80 0.82 0.15 18.69
[14] [15]
[16]
[17]
[18] [19]
[20] [21] [22]
[23] [24]
[25]
[26]
[27]
[10]
The variables of the received equation can be marked as follows: 2
Ts ¼
pl a 8cðE1 I1 þ E2 I2 Þ
ð31Þ
k¼1
8ðchð0:5klÞ 1Þ 2
k2 l chð0:5klÞ
ð32Þ
When calculating the load-carrying capacity of the bent elements made of two bars, the rigidity of the bond between bars of the element can be estimated by coefficient k. This estimation of the
581
T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583 Table 2 Sample statistics and results of Shapiro-Wilk test. Sample number
n
Beam number
As1/db
d (mm)
Variable
Mean
v
Min
Max
W
Wa
Distribution
1
10
12, 13, 14, 15, 42, 43, 44, 45, 46, 41
0.65–1.64
120–140
7
16, 17, 18, 19, 21, 22, 30
1.68–1.71
120
3
10
20, 32, 33, 50, 51, 52, 59, 60, 61, 62
0.56–1.05
150–175
4
7
34, 53, 54, 55, 56, 57, 58
1.51–2.58
150–164
5
7
1, 31, 36, 37, 38, 39, 40
0.58–0.83
200–225
6
6
4, 5, 6, 7, 8, 35
0.86–1.52
206–220
7
5
2, 3, 47, 48, 49
0.82–1.06
270–275
8
4
23, 24, 25, 26
1.70
240
9
3
9, 10, 11
0.61–0.73
309
10
3
27, 28, 29
1.71
384
1.02 0.76 1.05 0.89 1.00 0.82 0.99 0.88 0.99 0.70 1.00 0.90 0.94 0.86 0.90 0.84 1.09 0.69 0.88 0.81
9.62 23.63 11.24 8.96 15.30 23.01 8.22 4.75 12.26 36.57 4.44 8.26 9.11 9.20 11.23 4.03 7.66 2.49 15.86 9.26
0.87 0.57 0.92 0.81 0.74 0.57 0.88 0.83 0.85 0.48 0.95 0.81 0.83 0.80 0.80 0.80 1.02 0.67 0.72 0.72
1.18 1.09 1.23 1.01 1.21 1.10 1.13 0.95 1.13 1.12 1.08 1.03 1.04 1.00 0.99 0.87 1.18 0.71 0.97 0.86
0.921 0.842 0.917 0.872 0.932 0.918 0.925 0.938 0.810 0.827 0.938 0.947 0.983 0.771 0.824 0.938 0.953 0.940 0.820 0.861
0.842
2
Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc Mexp/Mcalc Mexp =Mcalc
NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL NORMAL
Table 3 Results of t test. Sample number 1 2 3 4 5 6 7 8 9 10
Variable
t
n1
ta/2,(n1)
H0
Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc Mexp/Mcalc M exp =M calc
0.56 4.34 1.03 3.58 0.09 2.92 0.41 7.72 0.14 3.05 0.21 3.29 1.63 4.03 2.08 9.72 1.80 31.65 1.48 4.52
9.00 9.00 6.00 6.00 9.00 9.00 6.00 6.00 6.00 6.00 5.00 5.00 4.00 4.00 3.00 3.00 2.00 2.00 2.00 2.00
2.262 2.262 2.447 2.447 2.262 2.262 2.447 2.447 2.447 2.447 2.571 2.571 2.776 2.776 3.182 3.182 4.303 4.303 4.303 4.303
ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED ACCEPTED REJECTED
rigidity of the bond can be used in calculating flexural reinforced structures strengthened with external FRP reinforcement. Experimental research performed [11] shows that the bond between the external FRP reinforcement and the strengthened reinforced concrete element is not rigid (Fig. 2). When the bending moment increases, the strains in the bond between concrete and FRP also increase and as a result FRP reinforcement is more and more displaced in respect to concrete. This is why it is suggested to estimate the rigidity of the bond with coefficient k which can be calculated according to the expressions (29) or (32). The load-carrying capacity of reinforced concrete beams strengthened with external FRP reinforcement is calculated (Fig. 3) by summing the moments of internal forces about the centroid of the concrete compressive stress block:
M R ¼ As1 fyd ðd 0:4xÞ þ Af f fd kðh 0:4xÞ þ As2 fscd ð0:4x d2 Þ
ð33Þ
As1 is the cross-sectional area of the steel bar reinforcement in tension; As2 the cross-sectional area of steel bar reinforcement in
0.803 0.842 0.803 0.803 0.788 0.762 0.748 0.767 0.767
compression; Af the cross-sectional area of FRP reinforcement; fyd the design strength of steel bar reinforcement in tension; fscd the design strength of the steel bar reinforcement in compression; ffd the design strength of FRP reinforcement; h the overall depth of the cross-section; d the effective depth of the cross-section; d1 the distance from the bottom of the element to the centre of reinforcement in tension; d2 is the distance from the top of the element to the centre of reinforcement in compression. The depth of the neutral axis is calculated by summing internal forces and taking into account the stresses and strains in the steel bar reinforcement as well as carbon fibre plastic:
x¼
As1 fyd þ Af f fd k As2 fscd 0:8f cd b
ð34Þ
Here b is the width of the element; fcd is the design compressive strength of concrete. The load-carrying capacity of reinforced beams strengthened with CFRP reinforcement is calculated according to the assumptions that the ultimate deformation of concrete ecu = 0.0035; the ultimate FRP strain [10], at the moment of separation efs = 0.008; the hypothesis of planes sections is valid for concrete element, however, the shear strains acting between the strengthened element and the external reinforcement are assessed by coefficient k, since the bond of concrete and fibre plastic is not rigid. On the ground of the analysis of the results of the experimental research it was determined that the coefficient k which assesses the rigidity between concrete and external FRP reinforcement can be calculated according to Eqs. (29) or (32) if:
As1 fyk <1 Af f fk
ð35Þ
or
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm < 6 103
ðwhere f cm ; fctm in kPaÞ
ð36Þ
where fyk is the characteristic strength of steel bar reinforcement in tension; ffk the characteristic strength of FRP reinforcement; fctm the mean strength of concrete in tension; fcm is the mean compressive strength of concrete. Coefficient k = 1, when:
As1 fyk P1 Af f fk
ð37Þ
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T. Skuturna, J. Valivonis / Construction and Building Materials 50 (2014) 577–583
and
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm > 6 103
ðwherefcm ; fctm in kPaÞ
ð38Þ
Coefficient k = 0.5, when conditions (39) and (40) are valid.
As1 fyk P2 Af f fk
ð39Þ
and
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:7 fctm fcm > 8:5 103
ðwherefcm ; fctm in kPaÞ
ð40Þ
According to the results of experimental research of reinforced concrete beams strengthened with external FRP reinforcement, the characteristics of the shear in Eq. (19) can be calculated as follows:
Geff
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1 fctm fcm ¼ qffiffiffiffiffiffiffiffiffiffiffiffi
ð41Þ
3. Results of calculations and statistical verification of the theoretical model In order to carry out theoretical calculations a database of experimental results was created. The database comprises the results of research performed by different authors on 62 different beams. The data was collected according to the following criteria: the cross-section of beams is rectangular; beams were reinforced with steel bar reinforcement and strengthened with an external CFRP reinforcement; the external reinforcement was glued in the tension zone of the beams (Fig. 4); carbon fibre was not prestressed and not anchored additionally; the ultimate strength of concrete up to 50 MPa; the samples were not additionally preloaded prior to strengthening. The loading scheme for reinforced concrete beams strengthened with an external CFRP reinforcement is show in Fig. 4. During the research all beams were tested in four-point bending. Table 1 provides the experimental Mexp and the calculated Mcalc load carrying capacities of strengthened beams. It also provides the ratio of the experimental and the theoretical results Mexp/Mcalc. The theoretical load carrying capacities Mexp were calculated when the coefficient k = 1. This allows to show the effect of slip for results of calculated load carrying capacities Fig. 5. In order to determine the feasibility of the suggested method a statistical analysis of calculation results has been performed. The calculated load-carrying capacity of strengthened beams will always be different from the one received in the course of the experiment. However, this difference may be statistically insignificant. The ideal case is when the means of the Mexp/Mcalc values equals to 1:
ð42Þ Mexp =Mcalc , Mexp =M calc
where M exp =Mcalc are the means of the ratios of the experimental and theoretical results. In order to compare the mean of theoretical and experimental results [28], a statistical hypothesis is formulated:
: Mexp =Mcalc ¼ 1
:
: Mexp =Mcalc –1
H1
ð44Þ
Here W is the calculated value in Shapiro-Wilk test, Wa is the ultimate value received from tables. In order to check the hypothesis the t test is applied since the data has normal distribution. When the variances are unknown and not equal, t statistics is calculated as follows:
M exp =M calc 1 qffiffiffiffiffiffiffiffiffiffiffi 2 sexp=calc
ð45Þ
n
The hypothesis H0 is rejected when:
where Ecm is the secant modulus of elasticity of concrete; the width, thickness of external FRP reinforcement as well as the number of layers respectively bf, tf, nf.
8 < H0
W P Wa
t¼
Ef bf t f nf Ecm b
M exp =M calc ¼ 1
Table 2 provides: the size of samples n, the means, the minimal and maximal values, the coefficient of variation v. With the help of Shapiro-Wilk test [29] for samples of small size (Eq. (44)) it was determined that it can be stated that when the level of significance is a = 0,05, the data has normal distribution (Table 2).
ð43Þ
On the ground of the gathered data, statistical samples of results have been made. The samples have been made according to the ratio of steel reinforcement of the beams and the effective depth of the cross-section.
jtj > ta=2;n1
ð46Þ
where ta/2,n1 is the critical value of the Stjudent‘s distribution with n 1 degrees of freedom, when a = 0.05; sexp/calc the standard deviations sexp/calc and sexp=calc of the ratios of the experimental and the theoretical results; n is the size of the samples. The received statistical results are provided in Table 3. Statistical analysis allows to state that the difference between the theoretical and the experimental load-carrying capacity in samples 1–10 is statistically insignificant if the coefficient k is applied. This shows that with the help of the suggested methodology it is possible to make quite accurate calculations of the load carrying capacity of beams strengthened with external CFRP reinforcement. However the means of the ratios of the experimental and theoretical results statistically significantly differ from 1 when the coefficient k = 1. 4. Conclusions A method for calculating the load-carrying capacity of reinforced concrete beams strengthened with external CFRP reinforcement provided in this work allows to perform the evaluation of the rigidity of the bond between the external CFRP reinforcement and concrete. This ensures performance of more accurate calculations of the load-carrying capacity of strengthened beams. The accuracy of the method has been assessed having performed calculations of 62 experimental beams strengthened with external CFRP reinforcement. A statistical evaluation of the received theoretical and the experimental results has also been performed. The performed analysis showed that, if we assess the rigidity of the bond between external reinforcement and concrete, with the help of the suggested calculation method it is possible to make quite precise calculations of the load-carrying capacity. Statistical evaluation of the results shows that the difference between the load-carrying capacity received during the course of the experiment and the calculated ones is statistically insignificant. References [1] Trapko W, Trapko T. Load-bearing capacity of compressed concrete elements subjected to repeated load strengthened with CFRP materials. J Civ Eng Manage 2012;18(2):590–9. [2] Hajsadeghi M, Alaee FJ, Shahmohammadi A. Investigation on behaviour of square/rectangular reinforced concrete columns retrofitted with FRP jacket. J Civ Eng Manage 2011;17(3):400–8. [3] Silva MAG, Biscaia HC. Effects of exposure to saline humidity on bond between GFRP and concrete. Compos Struct 2010;93(1):216–24. [4] Biscaia HC, Chastre C, Silva MAG. Double shear tests to evaluate the bond strength between GFRP/concrete elements. Compos Struct 2012;94(2): 681–94.
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