Clustering-Based Active-Learning Kriging Reliability Analysis of FRP-Strengthened RC Beams with Random Finite-Element to Model Spatial Variability

Data Availability Statement

Acknowledgments

Notation

a_i

ith greatest eigenvalue of the standard normal field (ith out of r);

a_x

correlation length in x;

a_y

correlation length in y;

a_z

correlation length in z;

b

unknown coefficient;

b_c

width of the concrete beam;

b_f

width of the CFRP sheet;

${\mathbf{b}}^{\ast }$

vector of regression parameter;

${\mathbf{C}}_{\mathbf{Y}\mathbf{Y}}$

covariance matrix;

${\mathbf{C}}_{Y,Y_i}$

ith vector of the covariance matrix;

$c_i^j$

cluster centroid i of iteration j;

E_frp

FRP modulus of elasticity;

$\mathbf{F}$

regression term;

$\mathcal{F}(b,X)$

regression function of the random inputs, X;

$f_c^{\mathrm{\prime }}$

concrete compressive strength;

f_frpu

FRP tensile strength;

$f_t^{\mathrm{\prime }}$

concrete tensile strength;

f_y

steel yield strength;

G

performance function;

$\hat{G}$

surrogate performance function—estimate of G (kriging predictor);

$\hat{H}(\mathbf{Z},\theta )$

realization of a Gaussian random field;

$\widehat{H_{ln}}(\mathbf{Z},\theta )$

realizations of a lognormal random field;

$I[\hat{G}{(\mathbf{X},\mathbf{Y})}_j]$

jth value of the indicator function denoting the sign of the kriging estimated performance function;

K

concrete bulk modulus;

K_mean

number of clusters;

k

dimension;

$\mathbf{L}$

vector of autocorrelation length;

MD

moment due to dead load;

ML

moment due to live load;

MT

transformation of live load to load effects;

M_n

nominal flexural strength;

N_cluster

number of points included in each cluster;

NFLS

concrete–FRP bond normal strength;

n

number of realizations of the random variables;

P

lowest ranked U from ${\mathbf{U}}_{\mathbf{M}\mathbf{i}\mathbf{n}}$;

P_f0

probability of failure;

$P_f^0$

probability of failure calculated using the kriging predictor to evaluate the limit state;

$P_{f+}^k$

probability of failure corresponding to the mean kriging predictor value, $\hat{G}(X)$, plus the MSE, ${\sigma }_{\hat{G}}(x)$, multiplied by a constant, k;

$P_{f-}^k$

probability of failure corresponding to the mean kriging predictor value, $\hat{G}(X)$, minus the MSE, ${\sigma }_{\hat{G}}(x)$, multiplied by a constant, k;

Q

load model;

Q_t

ratio of the eigenvalues divided by the trace of the covariance matrix;

R

resistance model;

R_SNFE

nominal flexural strength of SNFE;

$R(\mathbf{S})$

correlation between the design sites ${\mathbf{S}}_{\mathbf{i}}$ and ${\mathbf{S}}_{\mathbf{j}}$;

r

number of standard normal variables;

$r(\mathbf{X})$

correlation between the unknown site, $\mathbf{X}$, and all known design sites, ${\mathbf{S}}_{\mathbf{i}}$;

SFLS

concrete–FRP bond shear strength;

$S(\mathbf{X})$

set of design sites;

s

conversion factor converting the lognormal field;

t

number of random variables;

U

reliability index of the U-learning function prediction;

${\mathbf{U}}_{\mathbf{M}\mathbf{i}\mathbf{n}}$

vector of descending order of U;

U_k

U of the kth trial;

u

vector used in forming the kriging predictor;

v_i

coefficient of variation of the mesh point, Y⁽ⁱ⁾;

v_j

coefficient of variation of the mesh point, Y^(j);

$v_{P_{f0}}$

coefficient of variation of P_f0;

$\mathbf{X}$

vector of resistance random variables;

${\mathbf{X}}_{\mathbf{k}}$

vector of inputs for the kth trial;

$\mathbf{Y}$

vector of load random variables;

$Y(\mathbf{X})$

corresponding outputs of $S(\mathbf{X})$;

Y_x

x coordinate of the geometric centroids of the SNFE;

Y_y

y coordinate of the geometric centroids of the SNFE;

Y_z

z coordinate of the geometric centroids of the SNFE;

$\mathbf{Z}$

vector of spatial coordinates;

z(X)

stochastic process of the random inputs, X;

β

reliability index;

β_w

shape factor that depends on the ratio of b_f/b_c;

${ϵ}_{\mathrm{SNFE}}$

model error;

${\epsilon }_{P_f}$

estimator error of the predator;

${\epsilon }_{\mathrm{stop}}$

stopping criteria for ending enrichment of the kriging predictor using active learning;

θ

independent identifier corresponding to a set of standard normal variables;

${\mu }_{\hat{G}}$

mean of the kriging predictor;

μ_lny

mean of a lognormally distributed field;

μ_y

mean;

ξ_i(θ)

ith randomly generated standard normal variable (ith out of r);

ρ_ij

squared exponential correlation to determine the correlation between two points Y⁽ⁱ⁾ and Y^(j);

${\rho }_{ij}^{\mathrm{\prime }}$

correlation for the standard normal field between two points Y⁽ⁱ⁾ and Y^(j);

${\sigma }_{\hat{G}}^2$

standard deviation of the kriging predictor;

σ_lny

standard deviation of a lognormally distributed field;

σ_y

standard deviation;

${\psi }_i$

ith greatest eigenvector of the standard normal field (ith out of r);

Ω_cluster

error between the previous and current set of cluster centroids, $c_i^{(j)}$ and $c_i^{(j+1)}$; and

Ω_stop

stopping value of Ω_cluster.

References