Reliability analyses of shear strengthened RC beams with externally bonded fiber reinforced polymer


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.

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\(A_{fv}\) :

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


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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.

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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).

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  • Reliability analysis
  • FRP
  • Shear reinforcement
  • RBDO