Multiobjective Optimization of an Unbiased Model for Externally Bonded CFRP System Contributions to Shear Resistance in RC Beams

Data Availability Statement

Acknowledgments

Notation

A_fpl

ratio of CFRP reinforcement cross-sectional area provided per unit length (mm);

A_fw

area of FRP for shear strengthening measured perpendicular to the direction of the fibers (mm²);

A_sl

cross-sectional area of longitudinal steel reinforcement (mm²);

a

shear span (mm);

a_j

regression coefficient;

${\mathbf{a}}_{\mathit{j}}^{\mathrm{\prime }}$

regression coefficient;

B

regression coefficient;

b_j

regression coefficient;

b_w

breadth of beam web (mm);

C_f

strengthening configuration factor;

CoV

coefficient of variation;

D_f

stress distribution factor;

d_fv

effective depth of FRP shear reinforcement (mm);

d_s

internal arm of steel reinforcement (mm);

E_f

modulus of elasticity of the CFRP reinforcement (MPa);

E_s

modulus of elasticity of longitudinal steel (MPa);

f_ck

characteristic compressive strength of concrete (MPa);

f_cm

compressive strength of concrete (MPa);

f_ctk

characteristic value of tensile strength of concrete (MPa);

f_ctm

mean value of tensile strength of concrete (MPa);

f_fbwm

mean value bond strength of FRP reinforcement (MPa);

f_fe

effective stress in the FRP reinforcement as introduced in CNR-DT 200 R1 (MPa);

f_fem

mean value for the FRP effective stress as introduced in CIDAR (MPa);

f_fm

mean value of ultimate strength of FRP (MPa);

f_fm,max

mean value of maximum stress in FRP as introduced in CIDAR (MPa);

f_fmm

FRP debonding stress according to CNR-DT 200 R1 (MPa);

f_fu

ultimate strength of FRP (MPa);

f_fwm

mean value of the average stress in the FRP intersected by the shear crack in the ultimate limit state (MPa);

f_fwm,c

mean value of FRP strength attributed to a failure caused by FRP rupture (MPa);

H₀

null hypothesis;

H₁

alternative hypothesis;

h

height of beam (mm);

h_f

height of CFRP reinforcement (mm);

h_fe

effective height of FRP reinforcement as introduced in fib Bulletin 90 (mm);

h_w

height of beam web (mm);

K

composite feature of the CFRP reinforcement that is obtained from ${\rho }_f{E}_f/f_{cm}^{2/3}$;

k₁, k₂, k_v

parameters of the ACI PRC-440.2-23 model representing concrete strength, type of strengthening scheme, and bond reduction coefficient, respectively;

k_b, k_g

geometrical corrective factor and additional corrective factor, respectively, as introduced in CNR-DT 200 R1;

k_R

reduction factor to consider stress concentration at corners;

L_e(ACI)

active bond length as introduced in ACI PRC-440.2-23 (mm);

L_e(CIDAR)

effective bond length of FRP as introduced in CIDAR (mm);

L_e(fib)

effective bond length of FRP as introduced in fib Bulletin 90 (mm);

l_em

effective bond length as introduced in CNR-DT 200 R1 (mm);

l_t,max

anchorage length required to develop full anchorage capacity (mm);

eMAPE

mean absolute percentile error;

M_E

bending moment caused by external loads (N·m);

N

number of specimens;

n_f

number of FRP layers;

n_ss

parameters of the TR55 model representing type of strengthening scheme;

R²

coefficient of determination;

R_c

corner radius of beam cross section (mm);

RMSE

root mean squared error (kN);

r

Pearson coefficient of correlation;

s_f

spacing of CFRP reinforcement (mm);

s_u

interface slip corresponding to full debonding (mm);

s_0m

mean value of ultimate slip (mm);

t_f

thickness of CFRP reinforcement (mm);

$t_f^{\mathrm{\prime }}$

effective total thickness of FRP reinforcement as introduced in fib Bulletin 90 (mm);

t₀

thickness of each FRP layer (mm);

V_E

shear load caused by external loads (N·m);

V_f

contribution of CFRP reinforcement to shear resistance of RC beams (kN);

$V_f^{\exp }$

value of V_f obtained from experiments (kN);

$V_{{}_{R,f}}^{{}^{\mathrm{model}}}$

predicted value of V_f obtained from each specific model (kN);

w_f

width of CFRP reinforcement (mm);

x′, y′, m′, n′

parameters of the fib Bulletin 90 model;

x_j

input variable;

y_i

known value for specimen i;

$\overline{y}$

mean of known values;

$\overline{y}$

mean of predicted values;

α_f

fiber orientation angle;

β_L

coefficient to compensate for insufficient FRP anchorage length;

β_w

FRP width-to-spacing ratio;

Γ_fm

bonded joint specific fracture energy;

γ_i

predicted value for specimen i;

ε_f

strain in the FRP reinforcement;

ε_fe

effective strain in CFRP reinforcement;

${\epsilon }_{{}_{fe}}^{{}^{\exp }}$

value of ε_fe obtained from experiments;

${\epsilon }_{fe}^{{}^{\mathrm{pred}}}$

predicted value of ε_fe;

ε_fse

effective strain in the FRP for shear strengthening according to TR55;

ε_fu

ultimate strain of FRP at rupture (MPa);

ε_swy

steel yielding strain;

ε_x

longitudinal strain at the middepth of the section;

θ

inclination angle of CDC;

θ_min

minimum inclination angle of CDC;

κ_m

global model modification factor;

λ

ratio of the maximum available bond length to the effective bond length according to CIDAR;

ρ_f

ratio of CFRP reinforcement;

ρ_l

longitudinal tensile steel reinforcement ratio;

ρ_sw

ratio of steel stirrups;

ρ_xj,χ

coefficient of correlation between jth parameter and model error;

σ

standard deviation;

τ_b1m

mean value of ultimate bond strength (MPa);

χ_v

model error obtained for V_f;

χ_ε

model error obtained for effective strain; and

Ω

objective function.

References