Recommendations for Active-Learning Kriging Reliability Analysis of Bridge Structures

Data Availability Statement

Acknowledgments

Notation

A_Ideal

area of idealized cross-sectional geometry;

A_g

gross cross-sectional area;

$A_{s_i}$

area of steel in the ith layer of rebar;

A_st

total area of steel in the cross-section;

B

width of the section;

C_r

compressive force;

d_b

diameter of steel bar reinforcements;

d_i

depth to ith layer of rebar;

EFF(x)

effective feasibility learning function;

$\mathcal{F}$

regression function;

F

regression realizations;

f_y

yield strength of steel;

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

compressive strength of concrete;

f(x)

regression evaluation of x;

G(x)

responses generated by original model;

$\hat{G}(x)$

responses generated by Kriging surrogate model;

$\overline{G}(x)$

mean value of Kriging prediction;

H

height of the cross-section;

H_c(x)

corrected H learning function;

H(x)

H learning function;

I₁

transformed RMSE;

I₂

transformed AAE;

I₃

transformed TTPs;

I₄

transformed DoC;

KO(x)

KO learning function;

L

correlation length vector;

L_effect

live load effect;

$L^{*}$

optimum correlation length vector;

$M_{D_{SW}}$

dead load moment due to self-weight;

$M_{D_{WS}}$

dead load moment due to wearing surface;

M_L

live load moment;

M_c

compressive moment from concrete zone;

M_in

vector of initial inputs;

M_out

vector of initial outputs;

M_r

moment resistance;

$M_{s_i}$

tensile moment generated by the ith layer of reinforcing bars;

m

number of design sites within design of experiments;

N

minimum number of trials;

N_AK-MCS

number of trials in the AK-MCS analysis;

N_added

number of added training points;

N_bl

number of reinforcing bar layers within the cross-section;

N_calls

summation number of initial and added training points;

N_cs

number of concrete sections with different base widths;

N_f

total number of failed trials;

N_I

total number of identical analyses;

N_initial

number of initial training set;

N_MCS

total number of trials in a Monte Carlo simulation;

N_s

number of successful reliability analyses;

n

number of random variables;

P

number of initial design points required to generate Kriging surrogate model;

PF

professional factor;

P_c

axial force in concrete under compression;

P_f

probability of failure;

$P_{f_{\mathrm{AK}-\mathrm{MCS}}}$

probability of failure of AK-MCS;

$P_{f_{\mathrm{AK}-\mathrm{MC}{\mathrm{S}}_{\mathrm{i}}}}$

Probability of failure of the ith identical AK configuration;

$P_{f_{\mathrm{MCS}}}$

probability of failure of Monte Carlo simulation;

P_r

axial load resistance;

$P_{r_{max}}$

maximum allowable axial load;

P_ro

axial load resistance with an eccentricity value of zero;

$P_{s_i}$

axial force in the ith steel rebar under tension;

P_y

axial force from interaction diagram;

$\hat{p}$

expected probability of failure;

q

number of model outputs;

$\hat{q}$

expected probability of survival;

R

correlation matrix;

REIF(x)

reliability based expected improvement learning function;

R(x)

resistance model;

r(x)

correlation evaluation for x;

S_i

vector containing the inputs from all design sites;

T_i

tensile force in the ith layer of rebar;

U(x)

U learning function;

$\epsilon (x)$

error measurements that determine the border of the desired range;

V

coefficient of variation;

v₁

linear scale;

v₂

exponential scale;

v₃

logarithmic scale;

X

design site;

x

vector made up of randomly generated variables;

Y_i

vector containing the outputs from all design sites;

$\mathcal{Z}$

stochastic process;

$Z_{\alpha }$

Z value for selected confidence level;

β

reliability index;

β_MCS

reliability index for the MCS for a given geometry;

β_j

matrix of regression coefficients;

${\beta }^{*}$

Kriging shape predictor factor;

${\gamma }^{*}$

Kriging shape prediction factor;

λ

bias;

μ

mean value;

${\mu }_{\hat{G}}(x)$

mean value of the Kriging predictor;

ρ

reinforcement ratio;

${\sigma }_{\hat{G}}(x)$

standard deviation of the Kriging predictor; and

φ(x)

variance of the Kriging predictor.

References