Abstract
Different methods for the estimation of fiber reinforced polymer (FRP) contribution to shear strength in reinforced concrete (RC) beams were evaluated and compared through reliability analysis using FRP rupture and debonding failure functions. The carbon fibers properties were obtained through laboratory tests allowing to define the probability distribution function of the fiber tensile strength and elastic modulus. Uncertainties associated with the load, material properties were taken into account in order to compare two standards (ACI 440.2R and CNR-DT200) and two more sophisticated methods proposed in the literature. The Monte Carlo method and an improved first order reliability method (FORM iHLRF) were used to determine the structural reliability of an RC beam with externally bonded FRP. The results show the effects of the different mechanical methods in the reliability indices of the strengthened RC beam. The performance function related to FRP rupture presented a much higher reliability index than the other failure function related to FRP debonding for all four methods. A reliability-based design optimization was also performed to measure the effects of the assumptions and uncertainties associated with each method regarding the design variable (width of the reinforcement) for the same target reliability. The most conservative methods were the standards (ACI 440.2R and CNR-DT200), leading to a wider FRP reinforcement than the ones estimated by the other methods for the same reliability safe level design.
Similar content being viewed by others
Abbreviations
- \(A_{fv}\) :
-
FRP area
- \(A_{sv}\) :
-
Cross-sectional area of steel stirrups intersected by the critical shear crack;
- \(b_w\) :
-
Beam thickness
- d :
-
Effective depth of reinforced concrete beam
- \(d_{fv}\) :
-
FRP effective depth
- \(D_{frp}\) :
-
Distribution factor of tensions in the fiber
- \(E_{f}\) :
-
Modulus of elasticity of FRP in the principal fibre-orientation
- \(f_{c}\) :
-
Concrete cylinder compressive strength
- \(f_{ctm}\) :
-
Average surface tensile strength of concrete
- \(f_{f}\) :
-
Tensile strength of FRP
- \(f_{fdd}\) :
-
FRP design strength
- \(f_{fe}\) :
-
Effective (average) stress in FRP strips
- \(f_{yw}\) :
-
Yield stress of steel stirrups
- \(f_{x}\) :
-
Probability density function
- \(G_{1}\) :
-
Failure mode of rupture
- \(G_{2}\) :
-
Failure mode of debonding
- h :
-
Depth of RC beam
- \(h_{fe}\) :
-
Effective depth of FRP
- \(h_{frp,e}\) :
-
Effective height of FRP bonded on beam sides
- K :
-
Shear interaction factor
- \(K_{v}\) :
-
Geometric coefficient
- \(K_{s}\) :
-
Mobilization factor for steel stirrups
- \(K_{f}\) :
-
Mobilization factor for the fiber
- \(k_{1}\) :
-
Geometric factor
- \(k_{2}\) :
-
Geometric factor
- \(L_e\) :
-
Effective bond length of FRP strips
- N :
-
Lifetime of the structure
- \(P_{f}\) :
-
Probability of failure
- \(P_{g}\) :
-
Dead load concentrated force
- \(P_{q}\) :
-
Live load concentrated force
- \(s_{f}\) :
-
FRP spacing
- \(t_{f}\) :
-
FRP thickness
- \(\mathbf{u }\) :
-
Random variables in the standard normal space
- \(V_{c}\) :
-
Shear resistance component contributed by concrete
- \(V_{E}\) :
-
Shear force on the RC beam
- \(V_{f}\) :
-
Shear resistance component contributed by FRP
- \(V_{f,p}\) :
-
Peak value of shear contribution of FRP strips
- \(V_{f_{R}}\) :
-
Shear contribution of FRP under rupture criterion
- \(V_{f_{D}}\) :
-
Shear contribution of FRP under debonding criterion
- \(V_{g}\) :
-
Dead load distributed force
- \(V_{q}\) :
-
Live load distributed force
- \(V_{s,p}\) :
-
Peak value of shear contribution of steel stirrups
- \(V_{s}\) :
-
Shear resistance component contributed by steel shear reinforcement
- \(\textbf{x}\) :
-
Random variables vector in the original space
- \(w_{e,p}\) :
-
Crack end width when the FRP shear contribution reaches its peak value
- \(w_{f}\) :
-
Width of FRP strip
- \(\alpha\) :
-
Inclination angle of fibers
- \(\alpha _{L}\) :
-
Bond length coefficient
- \(\alpha _{w}\) :
-
Strip width coefficient
- \(\boldsymbol{\hat{\alpha}}\) :
-
Normalized gradient vector in MPP
- \(\beta\) :
-
Reliability index
- \(\beta _{s}\) :
-
System reliability index
- \(\beta ^{t}\) :
-
Target reliability index
- \(\theta\) :
-
Angle of critical shear crack to longitudinal axis of beam
- \(\varepsilon _{fe}\) :
-
Strain of FRP
- \(\varepsilon _{fu}\) :
-
FRP ultimate tensile strain
- \(\lambda\) :
-
Normalized maximum bond length
- \(\mu\) :
-
Ratio of shear contribution between steel stirrups and FRP strips
- \(\rho\) :
-
Correlation coefficient of population
- \({\hat{\rho }}\) :
-
Sample Pearson correlation coefficient
- \(\Phi\) :
-
Cumulative Distributive Function
- \(\sigma _{f_{f}}\) :
-
Standard deviations of \(f_{f}\)
- \(\sigma _{E_{f}}\) :
-
Standard deviations of \(E_{f}\)
- \(\sigma _{f,e}\) :
-
Average stress in FRP strips intersected by the critical shear crack
- \(\sigma _{f,\mathrm{max}}\) :
-
Maximum achievable debonding stress in FRP strips intersected by the critical shear crack
References
ACI (2017) ACI committee 440 (2017) Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures (440.2R-17)
FIB (2001) Externally bonded FRP reinforcement for RC structures
CIDAR (2006) Design guideline for RC structures retrofitted with FRP and metal plates: beams and slabs. The Univ. of Adelaide, Adelaide
CAN/CSA (2002) Design and construction of building components with fibre-reinforced polymers S806-02. Number 2. Canadian Standards Association, Toronto
National Research Council (CNR) (2013) Guide for the design and construction of externally bonded FRP systems for strengthening existing structures, Rome. CNR-DT 200 R1/2013
Comité Européen de Normalisation (CEN) (1998) Eurocode 8: design of structures for earthquake resistance—part 3: assessment and retrofitting of buildings. En, 3:2005
Darby A, Ibell T, Clarke J (2004) TR55 design guidance for strengthening concrete structures using fibre composite materials. The Concrete Society, Camberley
German Committee for Structural Concrete (2012) Strengthening of concrete members with adhesively bonded reinforcement, DAfStb. Betonwerk+ Fertigteil-Technik
Plevris N, Triantafillou TC, Veneziano D (1995) Reliability of RC members strengthened with CFRP laminates. J Struct Eng 121(7):1037–1044
Chaallal O, Nollet M-J, Perraton D (1998) Strengthening of reinforced concrete beams with externally bonded fiber-reinforced-plastic plates: design guidelines for shear and flexure. Can J Civ Eng 25(4):692–704
Triantafillou TC (1998) Shear strengthening of reinforced concrete beams using epoxy-bonded FRP composites. ACI Struct J 95:107–115
Khalifa A, Gold WJ, Nanni A, Abdel Aziz MI (1998) Contribution of externally bonded FRP to shear capacity of RC flexural members. J Compos Constr 2(4):195–202
Khalifa A, Nanni A (2000) Improving shear capacity of existing RC T-section beams using CFRP composites. Cem Concr Compos 22(3):165–174
Täljsten B (2003) Strengthening concrete beams for shear with CFRP sheets. Constr Build Mater 17(1):15–26
Pellegrino C, Modena C (2006) Fiber-reinforced polymer shear strengthening of reinforced concrete beams: experimental study and analytical modeling. ACI Struct J 103(5):720
Mofidi A, Chaallal O (2014) Tests and design provisions for reinforced-concrete beams strengthened in shear using FRP sheets and strips. Int J Concr Struct Mater 8(2):117–128
Chen GM, Li SW, Fernando D, Liu PC, Chen JF (2017) Full-range FRP failure behaviour in RC beams shear-strengthened with FRP wraps. Int J Solids Struct 125:1–21
Bilotta A, Faella C, Martinelli E, Nigro E (2013) Design by testing procedure for intermediate debonding in EBR FRP strengthened RC beams. Eng Struct 46:147–154
Colotti V (2016) Mechanical shear strength model for reinforced concrete beams strengthened with FRP materials. Constr Build Mater 124:855–865
Ji C, Li W, Hu C, Xing F (2017) Data analysis on fiber-reinforced polymer shear contribution of reinforced concrete beam shear strengthened with U-jacketing fiber-reinforced polymer composites. J Reinf Plast Compos 36(2):98–120
Mofidi A, Chaallal O (2010) Shear strengthening of RC beams with EB FRP: influencing factors and conceptual debonding model. J Compos Constr 15(1):62–74
Rojas Nicolás Roa, de Albuquerque Nívea Gabriela Benevides, de Azevedo Melo Guilherme Sales Soares, Narváez Nathaly Sarasty (2017) Evaluation of estimation methods for the shear strengthening of RC beams with FRP composites (in Portuguese). REEC-Revista Eletrônica de Engenharia Civil 13(2)
D’Antino T, Triantafillou TC (2016) Accuracy of design-oriented formulations for evaluating the flexural and shear capacities of FRP-strengthened RC beams. Struct Concr 17(3):425–442
Lu R, Luo Y, Conte JP (1994) Reliability evaluation of reinforced concrete beams. Struct Saf 14(4):277–298
Belarbi A, Acun B (2013) FRP systems in shear strengthening of reinforced concrete structures. Procedia Eng 57:2–8
Chen JF, Teng JG (2001) Anchorage strength models for FRP and steel plates bonded to concrete. J Struct Eng 127(7):784–791
Chen JF, Teng JG (2003a) Shear capacity of fiber-reinforced polymer-strengthened reinforced concrete beams: fiber reinforced polymer rupture. J Struct Eng 129(5):615–625
Chen JF, Teng JG (2003b) Shear capacity of FRP-strengthened RC beams: FRP debonding. Constr Build Mater 17(1):27–41
Chen GM, Teng JG, Chen JF, Rosenboom OA (2010) Interaction between steel stirrups and shear-strengthening FRP strips in RC beams. J Compos Constr 14(5):498–509
Chen GM, Teng JG, Chen JF (2012) Process of debonding in RC beams shear-strengthened with FRP U-strips or side strips. Int J Solids Struct 49(10):1266–1282
Chen GM, Teng JG, Chen JF (2013) Shear strength model for FRP-strengthened RC beams with adverse FRP-steel interaction. J Compos Constr 1:50–66
Karbhari VM, Niu H, Sikorsky C (2006) Review and comparison of fracture mechanics-based bond strength models for FRP-strengthened structures. J Reinf Plast Compos 25(17):1757–1794
Mofidi A, Chaallal O (2011) Shear strengthening of RC beams with EB FRP: influencing factors and conceptual debonding model. J Compos Constr 1:62–74
Iervolino I, Galasso C (2012) Comparative assessment of load-resistance factor design of FRP-reinforced cross sections. Constr Build Mater 34:151–161
Galasso C, Maddaloni G, Cosenza E (2014) Uncertainly analysis of flexural overstrength for capacity design of RC beams. J Struct Eng 140(7):04014037
Atadero RA, Karbhari VM (2008) Calibration of resistance factors for reliability based design of externally-bonded FRP composites. Compos B Eng 39(4):665–679
Barbero EJ (2010) Introduction to composite materials design. CRC Press, Boca Raton
Behnam B, Eamon C (2013) Reliability-based design optimization of concrete flexural members reinforced with ductile FRP bars. Constr Build Mater 47:942–950
Cervenka V (2013) Reliability-based non-linear analysis according to FIB model code 2010. Struct Concr 14(1):19–28
Evangelista Jr F, Afanador-García N (2016) A polynomial chaos expansion approach to the analysis of uncertainty in viscoelastic structural elements. Dyna 83(199):172–182. https://doi.org/10.15446/dyna.v83n199.53834
Ortega JJ, Ruiz G, Rena CY, Afanador-García N, Tarifa M, Poveda E, Zhang X, Evangelista Jr F (2018) Number of tests and corresponding error in concrete fatigue. Int J Fatigue 116:210–219. https://doi.org/10.1016/j.ijfatigue.2018.06.022
Der Kiureghian A, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Struct Saf 31(2):105–112
Most T (2011) Assessment of structural simulation models by estimating uncertainties due to model selection and model simplification. Comput Struct 89(17):1664–1672
Oberkampf WL, Helton JC, Joslyn CA, Wojtkiewicz SF, Ferson S (2004) Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Saf 85(1):11–19
Keitel H, Jung B, Motra HB, Stutz H (2014) Quality assessment of coupled partial models considering soil-structure coupling. Eng Struct 59:565–573
Nowak AS, Szerszen MM (2003) Calibration of design code for buildings (ACI 318): part 1-statistical models for resistance. ACI Struct J 100(3):377–382
Szerszen MM, Nowak AS (2003) Calibration of design code for buildings (ACI318): part 2-reliability analysis and resistance factors. ACI Struct J 100(3):383–391
Nowak AS, Collins KR (2012) Reliability of structures. CRC Press, Boca Raton
Wieghaus KT, Atadero RA (2010) Effect of existing structure and FRP uncertainties on the reliability of FRP-based repair. Journal of Composites for Construction 15(4):635–643
Ribeiro SEC, Diniz SMC (2013) Reliability-based design recommendations for FRP-reinforced concrete beams. Eng Struct 52:273–283
Zadeh HJ, Nanni A (2013) Reliability analysis of concrete beams internally reinforced with fiber-reinforced polymer bars. ACI Struct J 110(6):1023
Baji H, Ronagh HR (2015) A reliability-based investigation into ductility measures of RC beams designed according to FIB model code 2010. Structural Concrete 16(4):546–557
Shafei E (2016) Deformation characteristics of shear-deficient RC beams wrapped with CFRP. Mater Struct 49(8):3119–3134
Zhang SS, Teng JG (2016) End cover separation in RC beams strengthened in flexure with bonded FRP reinforcement: simplified finite element approach. Mater Struct 49(6):2223–2236
Mansouri I, Ozbakkaloglu T, Kisi O, Xie T (2016) Predicting behavior of FRP-confined concrete using neuro fuzzy, neural network, multivariate adaptive regression splines and M5 model tree techniques. Mater Struct 49(10):4319–4334
Noh Y, Choi KK, Du L (2009) Reliability-based design optimization of problems with correlated input variables using a gaussian copula. Struct Multidiscip Optim 38(1):1–16
Lu DG, Song PY, Liu YF, Yu XH (2014) An extended first order reliability method based on generalized Nataf transformation. In: Deodatis G, Ellingwood BR, Frangopol DM (eds) Safety, reliability, risk and life-cycle performance of structures and infrastructures. CRC Press, London, pp 1177–1184
Du J, Li H, He Y (2017) The method of solving structural reliability with multiparameter correlation problem. Math Probl Eng 2017:1–12
Hasofer AM (1974) Reliability index and failure probability. J Struct Mech 3(1):25–27
Rackwitz R, Flessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494
Liu P-L, Der Kiureghian A (1991) Optimization algorithms for structural reliability. Struct Saf 9(3):161–177
Monti G et al (2007) Tests and design equations for FRP-strengthening in shear. Constr Build Mater 21(4):799–809
ASTM (2008) D3039/d 3039m-08, standard test method for tensile properties of polymer matrix composite materials. ASTM International, West Conshohocken
Montgomery DC, Runger GC, Hubele NF (2009) Engineering statistics. Wiley, Hoboken
Olea RA, Pawlowsky-Glahn V (2009) Kolmogorov–Smirnov test for spatially correlated data. Stoch Environ Res Risk Assess 23(6):749–757
Adjakossa EH, Sadissou I, Hounkonnou MN, Nuel G (2016) Multivariate longitudinal analysis with bivariate correlation test. PLoS ONE 11(8):e0159649
Melchers RE (1987) Structural reliability: analysis and prediction. Ellis Horwood series in civil engineering. Ellis Horwood, Wiley, Hoboken ISBN 9780853129301
Zhang Y, Der Kiureghian A (1995) Two improved algorithms for reliability analysis. In: Rackwitz R, Augusti G, Borri A (eds) Reliability and optimization of structural systems. IFIP—The international federation for information processing. Springer, Boston
Khalifa A, Nanni A (2002) Rehabilitation of rectangular simply supported RC beams with shear deficiencies using CFRP composites. Constr Build Mater 16(3):135–146
ACI (2011) Building code requirements for structural concrete (ACI 318-11). American Concrete Institute, Farmington Hills
ACI (2005) Evaluation of strength test results of concrete (ACI 214R-02). American Concrete Institute, Farmington Hills, p 6
Vrouwenvelder T (1997a) The JCSS probabilistic model code. Struct Saf 19(3):245–251
Vrouwenvelder T, Holicky M, Markova J (2002) JCSS probabilistic model code-example applications.Technical report, ISBN 978-3-909386-79-6
Vrouwenvelder T (1997b) The JCSS probabilistic model code. Part 3: resistance models, steel properties. Struct Saf 19(3):245–251
Vrouwenvelder T (1997c) The JCSS probabilistic model code. Part 2: load models, self weight. Struct Saf 19(3):245–251
Vrouwenvelder T (1997d) The JCSS probabilistic model code. Part 2: load models, live load. Struct Saf 19(3):245–251
JCSS (2002) Probabilistic model part I–III. Joint Committee on Structural safety. Technical report, ISBN 978-3-909386-79-6
Bourinet JM, Mattrand C, Dubourg V (2009) A review of recent features and improvements added to ferum software. In: Proceedings of the 10th international conference on structural safety and reliability (ICOSSAR’09)
Kiureghian AD, Haukaas T, Fujimura K (2006) Structural reliability software at the university of california, berkeley. Struct Saf 28(1–2):44–67
Haukaas T, Der Kiureghian A (2005) Parameter sensitivity and importance measures in nonlinear finite element reliability analysis. J Eng Mech 131(10):1013–1026
Rózsás Á, Sỳkora M (2015) Neglect of parameter estimation uncertainty can significantly overestimate structural reliability. Trans VŠB Tech Univ Ostrava Civ Eng Ser 15(2):1–10
Acknowledgements
The authors gratefully acknowledge the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (Grant No. 001). The first author would also like to thank the Laboratory LEM-DEC of PUC-RIO.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Narváez, N.S., Rojas, N.R. & Evangelista Jr, F. Reliability analyses of shear strengthened RC beams with externally bonded fiber reinforced polymer. Mater Struct 53, 31 (2020). https://doi.org/10.1617/s11527-020-01465-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1617/s11527-020-01465-0