Identification of Influence Lines for Highway Bridges Using Bayesian Parametric Estimation Based on Computer Vision Measurements

Data Availability Statement

Acknowledgments

Notation

AAP

average accuracy of all categories;

A_i

the ith axis;

AP

accuracy of the model in a certain category;

AvgPool

channel-based global average pooling;

a_i

axle load distribution coefficient;

C

model class;

[C_b]

vehicle damping matrix in numerical model;

C_i

sampling point difference between the first axle and the ith axle;

c_s1, c_s2

suspension damping of the front and the rear axles in numerical model;

c_t1, c_t2

tire damping of the front and rear axles in numerical model;

D

dynamic data;

D_i

axle spacing;

D_Q

distance from Axle-1 to Axle-Q;

E

number of bridge response conditions included in the calculation;

E[·]

expectation;

F

input feature map;

F′

output feature map of the channel attention module;

F″

feature map output by the CBAM;

f

sampling frequency;

f^7×7

convolution operation with the filter size of 7 × 7;

FN

number of positive samples incorrectly predicted;

FP

number of negative samples incorrectly predicted;

{F_b}

vector of the wheel–road contact forces in numerical model;

G

number of detected categories;

G_q(·)

power spectral density function;

H

sampling number of spatial frequency;

h

number of parameters;

$I_{(K-C_i)}$

influence coefficient of the bridge corresponding to the ith axle;

$\{I{\}}_{K-C_Q,1}$

vector consisting of the bridge influence line (IL) ordinate;

$I_{N_0}$

N₀ × N₀ identity matrix;

I_i

IL ordinate of point i;

$I_i^{\mathrm{MAP}}$

maximum a posteriori IL ordinate of point i;

$I_{\alpha }$

inertia moment in numerical model;

I_Err

percentage error between maximum a posteriori influence line (MAPIL) and realistic IL;

$I_i^{\mathrm{sta}}$

ith ordinate values of realistic IL;

I_mid

midspan position IL ordinate;

{I^I}

interval of the IL vector;

$\{\overline{I^{\mathrm{I}}}\}$, {I^I}

calculated values of the upper and lower bounds of the IL;

K

sampling number of measurements;

[K_b]

vehicle stiffness matrix in numerical model;

k₀

normalizing constant;

k′

number of samples cutoff;

k_s1, k_s2

suspension stiffness of the front and the rear axles in numerical model;

k_t1, k_t2

tire stiffness of the front and rear axles in numerical model;

L

length of the bridge;

L₀

the transfer matrix representing the input–output relationship of the system;

L_I

identity matrix;

la_i

label value;

M

vehicle weight;

MaxPool

channel-based global maximum pooling;

[M_b]

vehicle mass matrix in numerical model;

M_c(F)

channel attention map output by the channel attention module;

M_s(F′)

spatial attention map output by the spatial attention module;

m₁, m₂

weight of the front and the rear axles in numerical model;

N

number of observations;

N₀

number of observed degrees of freedom;

N_a

length of the bridge IL vector;

N_d

sum of the number of degrees of freedom;

PF

penalty parameter of the error term defined by the users;

Q

number of vehicles axles;

${\overrightarrow{q}}_i$

the vector with label;

r(e)

road roughness at the coordinate of e;

TP

number of positive samples correctly predicted;

Δt

sampling time step;

V

vehicle velocity;

W_i

axle load;

$W_i^r$

virtual axle load;

[W^r]

virtual axis load matrix;

$[W{]}_{K,K-C_Q}$

axle load matrix;

w_i, $\overline{w_i}$

lower and upper bounds of weights matrix of the ith axle of a Q-axle vehicle;

x(t)

dynamical system, model response output vector at time t;

x(0)

initial condition of the model;

x_i0

central value of the affine form;

x_i1

partial deviations;

$y_k^T$

k-point load effect;

{Y}_K,1

responses vector collected at each time step;

y_n

measured response;

{Y_b}

vehicle displacement vector in numerical model

Y^m

displacement response vector.

α_i, α_j

Lagrange multipliers;

β_h

random phase angle uniformly distributed between $0\mathrm{and}2\pi$;

${\delta }_{n{n}^{\mathrm{\prime }}}$

Kronecker delta function;

$\epsilon$

prediction error;

${\epsilon }^m$

error vector;

${\epsilon }_i$

noise symbols;

θ

unknown parameters;

θ_m

model parameters;

κ

SVM kernel function;

σ

sigmoid function;

ψ

observed data;

ψ_i

value of the ith observation;

$\sum {}_{\epsilon }$

N₀ × N₀ covariance matrix of the prediction error process;

ϕ(q)

nonlinear function that maps the input data to the feature space;

${\mathrm{\Delta }}_{\mathrm{\Omega }}$

discrete sampling interval of spatial frequency;

Ω_h

spatial frequency; and

⊗

element-wise multiplication.

References