Prediction of Deflection of Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymer
Abstract
:1. Introduction
2. Analyzed Beams
3. Calculation of Deflection
4. Results
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Notation
A, B, C | the designation of the equation for the depth of the neutral axis; |
Ac | the cross section of the compressed concrete layer; |
Af | the cross-section area of carbon fiber; |
Ared | the transformed cross section of the beam; |
As1 and As2 | the cross-section area of the tensioned and compressed reinforcement; |
Ecm | the modulus of elasticity of concrete; |
Ef | the modulus of elasticity of carbon fiber; |
Es1, Es2 | the modulus of elasticity of steel bars; |
III, IIII | the effective moment of inertia at stages 2 and 3; |
II.red, III.red, IIII.red | the moment of the inertia of the transformed cross-section at stages 1, 2, and 3; |
MI and MI.S | the cracking moment of the unstrengthened beam and strengthened beam respectively; |
MII.S | the bending moment of the strengthened beam when the yielding of reinforcement is reached; |
MR, MII | the maximum carrying bending moment of the unstrengthened beam; |
MR.S, MIII | the maximum carrying bending moment of the strengthened beam; |
Sred | the static moment of the transformed cross-section; |
a | distance from the support to loading; |
b | the width of the beam; |
d1 and d2 | distance from the beam edge to the center of the tensioned and compressed reinforcement; |
fc | the compressive strength of concrete cylinders; |
fct | the tensile strength of concrete; |
h | the height of the beam; |
k1 | the coefficient evaluating the shape of stress distribution; |
l | the span length of the beam; |
tf | the thickness of the carbon fiber layer; |
xI, xII and xIII | the depth of the neutral axis at stages 1, 2 and 3; |
yc.I, yc.II, yc.III, | the centre of the gravity of the beam cross-section at stages 1, 2 and 3; |
αf, αs1, αs2 | relative coefficients; |
γ1.c, γ1.t | relative coefficients evaluating a change in the depth of the neutral axis at stage 2; |
γ2.c, γ2.t | relative coefficients evaluating a change in the depth of the neutral axis at stage 3; |
εc | the strain of the compressed concrete; |
εc1 | strain when the maximum strength of concrete material is reached; |
εf | the strain of the carbon fiber layer; |
εs | the strain of the tensioned reinforcement; |
εs2 | the strain of the compressed reinforcement; |
σc | stresses in the layer of the compressed concrete; |
σf | stresses in the layer of carbon fiber; |
σs | stresses in the tensioned reinforcement; |
σs2 | stresses in the compressed reinforcement; |
ωI, ωII | the deflection of the control beam up to the end of stages I and II. |
ωI.S, ωII.S, ωIII.S | the deflection of the strengthened beam up to the end of stages I, II, and III. |
Appendix A
References
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Author | Beam Name | l, m | Load Positions, m | b, m | h, m | As1 | As2 | d1, m | d2, m | Af |
---|---|---|---|---|---|---|---|---|---|---|
Barros et al., 2005 [24] | V1 | 1.5 | 0.5 + 0.5 + 0.5 | 0.1 | 0.178 | 2Ø6 | 2Ø8 | 0.024 | 0.025 | – |
V1R1 | 0.17 | 2Ø6 | 1 × 1.45 × 9.59 | |||||||
V2 | 0.173 | 3Ø6 | – | |||||||
V2R2 | 0.177 | 3Ø6 | 2 × 1.45 × 9.59 | |||||||
V3 | 0.175 | 2Ø6 | – | |||||||
V3R2 | 0.175 | 2Ø6 + Ø8 | 2 × 1.45 × 9.59 | |||||||
V4 | 0.175 | 3Ø8 | 0.025 | – | ||||||
V4R3 | 0.18 | 3Ø8 | 3 × 1.45 × 9.59 | |||||||
Bilotta et al., 2015 [25] | Ref_c_no_1 | 2.1 | 0.925 + 0.25 + 0.925 | 0.12 | 0.16 | 2Ø10 | 2Ø10 | 0.05 | 0.035 | – |
Ref_d_no_1 | Distributed load | – | ||||||||
EBR_c_1.4 × 40_1 | 0.925 + 0.25 + 0.925 | 56 mm2 | ||||||||
EBR_c_1.4 × 40_2 | 56 mm2 | |||||||||
EBR_d_1.4 × 40_1 | Distributed load | 56 mm2 | ||||||||
EBR_d_1.4 × 40_2 | 56 mm2 | |||||||||
NSM_c_2_1.4 × 10_1 | 0.925 + 0.25 + 0.925 | 28 mm2 | ||||||||
NSM_d_2_1.4 × 10_1 | Distributed load | 28 mm2 | ||||||||
NSM_c_3_1.4 × 10_1 | 0.925 + 0.25 + 0.925 | 42 mm2 | ||||||||
NSM_d_3_1.4 × 10_1 | Distributed load | 42 mm2 | ||||||||
David et al., 2003 [26] | P1 | 2.8 | 0.9 + 1.0 + 0.9 | 0.15 | 0.3 | 2Ø14 | 2Ø8 | 0.027 | 0.024 | – |
P2 | 1.2 (cm2) | |||||||||
P5 | 2.4 (cm2) | |||||||||
EL-Gamal et al., 2016 [27] | REF | 2.36 | 0.93 + 0.5 + 0.93 | 0.2 | 0.3 | 2Ø12 | 2Ø8 | 0.04 | 0.032 | – |
CN1 | 71.26 (mm2) | |||||||||
CN2 | 2 × 71.26 (mm2) | |||||||||
GN1 | 71.3 (mm2) | |||||||||
GN2 | 2 × 71.3 (mm2) | |||||||||
CHYB | 71.26 + 25.8 (mm2) | |||||||||
GHYB | 71.3 + 25.8 (mm2) | |||||||||
REF-II | 4Ø12 | – | ||||||||
CN1-II | 71.26 (mm2) | |||||||||
CN2-II | 2 × 71.26 (mm2) | |||||||||
Ferrier et al., 2003 [28] | A1 | 2.0 | 0.7 + 0.6 + 0.7 | 0.15 | 0.25 | 2Ø14 | 2Ø8 | 0.025 | 0.025 | – |
A2 | 120 (mm2) | |||||||||
Gao et al., 2004 [29] | CON1 | 1.5 | 0.5 | 0.15 | 0.2 | 2Ø10 | 2Ø8 | 0.038 | 0.027 | – |
A0 | 0.22 × 75 | |||||||||
A10 | 0.22 × 75 | |||||||||
A20 | 0.22 × 75 | |||||||||
B0 | 0.44 × 75 | |||||||||
B10 | 0.44 × 75 | |||||||||
B20 | 0.44 × 75 | |||||||||
Gao et al., 2006 [30] | 2O | 1.5 | 0.5 | 0.15 | 0.2 | 2Ø10 | 2Ø8 | 0.038 | 0.027 | – |
2N6 | 6 × 0.11 × 150 | |||||||||
2T625-1 | ||||||||||
2T650-1 | ||||||||||
2T675-1 | ||||||||||
2N4 | 4 × 0.11 × 150 | |||||||||
2T450-1 | ||||||||||
2T4100-1 | ||||||||||
Heffernan 1997 [31] | Conventional | 4.8 | 1.6 + 1.6 + 1.6 | 0.3 | 0.5739 | 2Ø25 + Ø20 | 2Ø10 | 0.074 | 0.067 | – |
CFRP strengthened | 65.5 (mm2) | |||||||||
Heffernan and Erki 2004 [32] | Conventional | 2.85 | 1.1 + 0.65 + 1.1 | 0.15 | 0.3 | 2Ø20 + Ø10 | 2Ø10 | 0.041 | 0.037 | – |
CFRP strengthened | 89.4 (mm2) | |||||||||
Hosseini et al., 2014 [33] | SREF | 2.4 | 0.9 + 0.6 + 0.9 | 0.6 | 0.12 | 4Ø8 | 3Ø6 | 0.024 | 0.023 | – |
S2L-0 | 2 × 1.4 × 20 | |||||||||
S2L-20 | ||||||||||
S2L-40 | ||||||||||
Khalifa et al., 2016 [34] | B-C | 2.2 | 0.95 + 0.3 + 0.95 | 0.15 | 0.26 | 2Ø12 | 2Ø12 | 0.041 | 0.031 | – |
B-S-2 | 60 (mm2) | |||||||||
B-S-4 | 120 (mm2) | |||||||||
B-N-1-2 | 60 (mm2) | |||||||||
B-N-2-2 | 60 (mm2) | |||||||||
B-N-2-4 | 120 (mm2) | |||||||||
Kotynia et al., 2008 [35] | B-08S | 4.2 | 1.4 + 1.4 + 1.4 | 0.15 | 0.3 | 3Ø12 | 2Ø10 | 0.03 * | 0.03 ** | 60 (mm2) |
B-083m | 58.5 (mm2) | |||||||||
Kotynia et al., 2011 [36] | G1 | 6.0 | 1.2 + 1.2 + 1.2 + 1.2 + 1.2 | 1.0 | 0.22 | 7Ø12 | 7Ø8 | 0.03143 * | 0.024 ** | – |
G2 | 120 (mm2) | |||||||||
G3 | 120 (mm2) | |||||||||
G4 | 120 (mm2) | |||||||||
Kotynia et al., 2014 [37] | B12-a | 6.0 | 1.2 + 1.2 + 1.2 + 1.2 + 1.2 | 0.5 | 0.22 | 4Ø12 | 4Ø8 | 0.031 | 0.029 | 1.2 × 100 |
B12-asp | 1.2 × 100 | |||||||||
B16-asp | 1.2 × 100 | |||||||||
Omran et al., 2012 [38] | B0 | 5.0 | 2 + 1 + 2 | 0.2 | 0.4 | 3Ø15 | 2Ø10 | 0.057 | 0.036 | – |
B1-NP | 2 × 2 × 16 | |||||||||
B1-P1 | ||||||||||
B1-P2 | ||||||||||
B1-P3 | ||||||||||
Rezazadeh et al., 2014 [39] | Control | 2.2 | 0.9 + 0.4 + 0.9 | 0.15 | 0.3 | 2Ø10 | 2Ø10 | 0.035 | 0.025 | – |
Non prestressed | 1.4 × 20 | |||||||||
20% prestressed | ||||||||||
30% prestressed | ||||||||||
40% prestressed | ||||||||||
Sharaky et al., 2014 [40] | CB | 2.4 | 0.8 + 0.8 + 0.8 | 0.16 | 0.28 | 2Ø12 | 2Ø8 | 0.036 | 0.034 | – |
LB1C1 | 1Ø8 | |||||||||
LB1G1 | 1Ø8 | |||||||||
LB2C1 | 2Ø8 | |||||||||
LB2G1 | 2Ø8 | |||||||||
LA2C1 | 2Ø8 | |||||||||
LA2G1 | 2Ø8 | |||||||||
LB1G2 | 1Ø12 | |||||||||
Soudki et al., 2007 [41] | C-0 | 2.25 | 0.75 | 0.15 | 0.25 | 2Ø10 | 2Ø6 | 0.025 | 0.023 | – |
T-0 | 4 × 0.11 | |||||||||
S-0 | 50 × 1.2 | |||||||||
Teng et al., 2006 [42] | B0 | 3.0 | 1.2 + 0.6 + 1.2 | 0.15 | 0.3 | 2Ø12 | 2Ø8 | 0.036 | 0.034 | – |
B500 | 2 × 16 | |||||||||
B1200 | ||||||||||
B1800 | ||||||||||
B2900 | ||||||||||
Valivonis et al., 2010 [14] | B6.1C | 1.2 | 0.4 + 0.4 + 0.4 | 100 | 200 | 2Ø6 | 2Ø6 | 0.025 | 0.025 | 0.167 (cm2) |
B6.2C | ||||||||||
B6.5 | – | |||||||||
B8.1C | 2Ø8 | 0.167 (cm2) | ||||||||
B8.2C | ||||||||||
B8.3 | – | |||||||||
B12.1C | 203 | 2Ø12 | 2Ø8 | 0.167 (cm2) | ||||||
B12.2C | 200 | |||||||||
B12.5 | 104 | 198 | – | |||||||
B12.6 | 105 | 201 | ||||||||
Wu et al., 2014 [43] | Control | 1.8 | 0.6 + 0.6 + 0.6 | 0.15 | 0.3 | 3Ø14 | 2Ø6 | 0.037 | 0.033 | – |
B11 | Ø7.9 | |||||||||
B21 | 2Ø7.9 | |||||||||
B22 | ||||||||||
BP11 | Ø7.9 | |||||||||
BP12 | ||||||||||
BP13 | ||||||||||
BP14 | ||||||||||
Xiong et al., 2007 [44] | Pa | 2.1 | 0.7 | 0.125 | 0.2 | 2 × 10 | 2×8 | 0.03 | 0.024 | – |
2C | 0.22 × 100 | |||||||||
Pb | 2 × 12 | 0.031 | – |
Author | Beam Name | fc, MPa | fct, MPa | Ec, GPa | fy1, MPa | fy2, MPa | Es1, GPa | Es2, GPa | ff,fe, MPa | Ef,fe, GPa |
---|---|---|---|---|---|---|---|---|---|---|
Barros et al., 2005 [24] | V1 | 46.1 | 3.37 | 33.35 | 730 | 554.32 | 200 | 200 | – | – |
V1R1 | 2740 | 158.8 | ||||||||
V2 | 46.1 | 3.58 | 36.5 | 730 | – | – | ||||
V2R2 | 2740 | 158.8 | ||||||||
V3 | 46.1 | 3.21 | 34.89 | 730 | – | – | ||||
V3R2 | 730; 554.32 | 2740 | 158.8 | |||||||
V4 | 46.1 | 3.43 | 35.86 | 554.32 | – | – | ||||
V4R3 | 2740 | 158.8 | ||||||||
Bilotta et al., 2015 [25] | Ref_c_no_1 | 17.4 | 1.34 | 25.98 | 540 | 540 | 200 | 200 | – | – |
Ref_d_no_1 | – | – | ||||||||
EBR_c_1.4 × 40_1 | 2052 | 171 | ||||||||
EBR_c_1.4 × 40_2 | ||||||||||
EBR_d_1.4 × 40_1 | ||||||||||
EBR_d_1.4 × 40_2 | ||||||||||
NSM_c_2_1.4 × 10_1 | ||||||||||
NSM_d_2_1.4 × 10_1 | ||||||||||
NSM_c_3_1.4 × 10_1 | ||||||||||
NSM_d_3_1.4 × 10_1 | ||||||||||
David et al., 2003 [26] | P1 | 38.7 | 2.94 1 | 33.02 2 | 500 | 500 | 205 3 | 205 3 | – | – |
P2 | 39.2 | 2.97 1 | 33.14 2 | 2400 | 150 | |||||
P5 | 40.1 | 3.03 1 | 33.37 2 | |||||||
EL-Gamal et al., 2016 [27] | REF | 49.62 | 2.99 | 35.57 2 | 480 | 455 | 205 3 | 205 3 | – | – |
CN1 | 1588 | 119.4 | ||||||||
CN2 | ||||||||||
GN1 | 1185 | 52.34 | ||||||||
GN2 | ||||||||||
CHYB | 2096 * | 147.47 * | ||||||||
GHYB | 1800 * | 98.22 * | ||||||||
REF-II | – | – | ||||||||
CN1-II | 1588 | 119.4 | ||||||||
CN2-II | ||||||||||
Ferrier et al., 2003 [28] | A1 | 39 | 2.96 1 | 31 | 550 | 550 3 | 210 | 210 3 | – | – |
A2 | 650 | 80 | ||||||||
Gao et al., 2004 [29] | CON1 | 35.7 | 2.75 1 | 25 | 531 | 400 | 200 | 200 | – | – |
A0 | 4200 | 235 | ||||||||
A10 | ||||||||||
A20 | ||||||||||
B0 | ||||||||||
B10 | ||||||||||
B20 | ||||||||||
Gao et al., 2006 [30] | 2O | 62.1 | 4.29 1 | 37.1 | 460 | 460 | 200 | 205 | – | – |
2N6 | 4200 | 235 | ||||||||
2T625-1 | ||||||||||
2T650-1 | ||||||||||
2T675-1 | ||||||||||
2N4 | ||||||||||
2T450-1 | ||||||||||
2T4100-1 | ||||||||||
Heffernan 1997 [31] | Conventional | 32.9 | 2.56 1 | 31.45 2 | - | - | 200 | 200 | – | – |
CFRP strengthened | 325 | |||||||||
Heffernan and Erki 2004 [32] | Conventional | 37 | 2.83 1 | 32.57 2 | 511 & 411 | 411 | 210 | 210 | – | – |
CFRP strengthened | 233 | |||||||||
Hosseini et al., 2014 [33] | SREF | 46.7 | 3.43 1 | 29.7 | 486 | 464 | 200 | 200 | – | – |
S2L-0 | 2483.9 | 153.2 | ||||||||
S2L-20 | ||||||||||
S2L-40 | ||||||||||
Khalifa et al., 2016 [34] | B-C | 35 | 2.7 1 | 28 | 400 | 400 | 200 | 200 | 2800 | 165 |
B-S-2 | ||||||||||
B-S-4 | ||||||||||
B-N-1-2 | ||||||||||
B-N-2-2 | ||||||||||
B-N-2-4 | ||||||||||
Kotynia et al., 2008 [35] | B-08S | 32.3 | 2.52 1 | 31.27 2 | 490 | 524 | 195 | 209 | 2915 | 172 |
B-083m | 34.4 | 2.66 1 | 31.87 2 | 436 | 524 | 220 | 209 | 3500 | 230 | |
Kotynia et al., 2011 [36] | G1 | 45 | 3.33 1 | 34.55 2 | 554 | 561 | 200 | 200 | – | – |
G2 | 46.2 | 3.4 1 | 34.82 2 | 2800 | 165 | |||||
G3 | 45.9 | 3.39 1 | 34.75 2 | |||||||
G4 | 45.6 | 3.37 1 | 34.68 2 | 2235 | 149 | |||||
Kotynia et al., 2014 [37] | B12-a | 45.3 | 3.35 | 24.3 | 539.6 | 416.2 | 191.3 | 186.1 | 2800 | 173.3 |
B12-asp | 32.2 | 2.51 | 23.7 | 511.4 | 583.2 | 191.4 | 200.7 | |||
B16-asp | 49 | 3.57 | 25.4 | 595 | 555.8 | 198 | 196.4 | |||
Omran et al., 2012 [38] | B0 | 40 | 3.02 1 | 27.84 | 478 | 500 | 200 | 200 | – | – |
B1-NP | 2610 | 130.5 | ||||||||
B1-P1 | ||||||||||
B1-P2 | ||||||||||
B1-P3 | ||||||||||
Rezazadeh et al., 2014 [39] | Control | 32.2 | 2.51 1 | 27.4 | 585 | 585 | 208 | 208 | – | – |
Non prestressed | 1922 | 164 | ||||||||
20% prestressed | ||||||||||
30% prestressed | ||||||||||
40% prestressed | ||||||||||
Sharaky et al., 2014 [40] | CB | 32.4 | 2.8 | 31.7 | 545 | 545 | 205 | 205 | – | – |
LB1C1 | 2350 | 170 | ||||||||
LB1G1 | 1350 | 64 | ||||||||
LB2C1 | 2350 | 170 | ||||||||
LB2G1 | 1350 | 64 | ||||||||
LA2C1 | 2350 | 170 | ||||||||
LA2G1 | 1350 | 64 | ||||||||
LB1G2 | 1350 | 64 | ||||||||
Soudki et al., 2007 [41] | C-0 | 35 | 2.7 | 32.04 | 460 | 460 | 205 | 205 | – | – |
T-0 | 3480 | 230 | ||||||||
S-0 | 2800 | 165 | ||||||||
Teng et al., 2006 [42] | B0 | 44 | 3.27 1 | 34.31 2 | – | – | 210 | 210 | – | – |
B500 | 2068 | 131 | ||||||||
B1200 | ||||||||||
B1800 | ||||||||||
B2900 | ||||||||||
Valivonis et al., 2010 [14] | B6.1C | 34.4 | 2.93 | 32.45 | 358 | 358 | 205 | 205 | 4800 | 231 |
B6.2C | ||||||||||
B6.5 | – | – | ||||||||
B8.1C | 29.7 | 2.63 | 30.91 | 557 | 358 | 195 | 205 | 4800 | 231 | |
B8.2C | ||||||||||
B8.3 | – | – | ||||||||
B12.1C | 30.4 | 2.67 | 31.14 | 318 | 420 | 204.9 | 204.1 | 4800 | 231 | |
B12.2C | ||||||||||
B12.5 | 28.7 | 2.56 | 30.55 | – | – | |||||
B12.6 | – | – | ||||||||
Wu et al., 2014 [43] | Control | 34.4 | 2.66 1 | 31.87 2 | 340 | 240 | 200 | 200 | – | – |
B11 | 2629 | 170 | ||||||||
B21 | ||||||||||
B22 | ||||||||||
BP11 | ||||||||||
BP12 | ||||||||||
BP13 | ||||||||||
BP14 | ||||||||||
Xiong et al., 2007 [44] | Pa | 30.71 | 2.41 1 | 30.8 2 | 411 | 233 | 200 | 210 | – | – |
2C | 3652 | 252 | ||||||||
Pb | 606 | 210 | – | – |
© 2019 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 (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Daugevičius, M.; Valivonis, J.; Skuturna, T. Prediction of Deflection of Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymer. Materials 2019, 12, 1367. https://doi.org/10.3390/ma12091367
Daugevičius M, Valivonis J, Skuturna T. Prediction of Deflection of Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymer. Materials. 2019; 12(9):1367. https://doi.org/10.3390/ma12091367
Chicago/Turabian StyleDaugevičius, Mykolas, Juozas Valivonis, and Tomas Skuturna. 2019. "Prediction of Deflection of Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymer" Materials 12, no. 9: 1367. https://doi.org/10.3390/ma12091367