Analytical Evaluation of Ballasted Track Substructure Response under Repeated Train Loads

Data Availability Statement

Notation

A_b(z), A_c(z), A_s(z)

equivalent area of ballast, capping, and subgrade layers at depth z, respectively (m²);

$A_b^s$, $A_c^s$, $A_s^s$

equivalent shear area of ballast, capping, and subgrade layers, respectively (m²);

a, m, b

empirical parameters that depend on the type of subgrade soil;

a_t

total number of wheels/axles under consideration;

b_sl

width of sleeper (m);

c_b, c_c, c_s

damping coefficients of ballast, capping, and subgrade, respectively (N s/m);

$c_b^s$, $c_c^s$, $c_s^s$

shear damping coefficients of ballast, capping, and subgrade, respectively (N s/m);

D_w

wheel diameter (m);

E_r, E_b, E_c, E_s

elastic modulus of rail, ballast, capping, and subgrade, respectively (N/m²);

f_s,n, f_c,n, f_b,n

external forces acting on the subgrade, capping, and ballast, respectively (N);

g_t

center-to-center distance between the rails (m);

h_b, h_c, h_s

thickness of ballast, capping, and subgrade, respectively (m);

h_bl, h_bt

overlap height in ballast along longitudinal and transverse directions, respectively (m);

h_cl, h_ct

overlap height in capping along longitudinal and transverse directions, respectively (m);

h_i

thickness of the ith subgrade layer (m);

h_sl, h_st

overlap height in subgrade along longitudinal and transverse directions, respectively (m);

I

moment of inertia of rail (m⁴);

i₁, i₂

empirical parameters that depend on the axle load and subgrade type;

k

track modulus (N/m²);

k_b, k_c, k_s

stiffness of ballast, capping, and subgrade, respectively (N/m);

$k_b^s,k_c^s,k_s^s$

shear stiffness of ballast, capping, and subgrade, respectively (N/m);

k_p

spring constant of the rail pad (including sleeper) (N/m);

k₀

coefficient of lateral earth pressure;

$k_1^b,k_2^b,k_3^b,k_4^b$

empirical parameters for ballast;

$k_1^c,k_2^c,k_3^c,k_4^c$

empirical parameters for capping;

L

characteristic length (m);

l_e

effective length of sleeper (m);

l_sl

length of sleeper (m);

m_b, m_c, m_s

vibrating mass of ballast, capping, and subgrade, respectively (kg);

N

number of load cycles;

P_atm

atmospheric pressure (N/m²);

Q

static wheel load (N);

Q_a

static axle load (N);

Q_r,n

rail seat load at the nth sleeper (N);

S

sleeper spacing (m);

s_s

total irrecoverable deformation in the subgrade (m);

${\mathit{T}}_{\mathit{n}}^{\mathit{s}}$

average shear stress vector at the nth sleeper point in the yz plane;

V

train speed (m/s);

x_nj

distance between the nth sleeper and the jth wheel/axle (m);

ÿ_b,n, ẏ_b,n, y_b,n

acceleration, velocity, and displacement for ballast below the nth sleeper, respectively;

ÿ_c,n, ẏ_c,n, y_c,n

acceleration, velocity, and displacement for capping below the nth sleeper, respectively;

ÿ_s,n, ẏ_s,n, y_s,n

acceleration, velocity, and displacement of the subgrade below the nth sleeper, respectively;

α, β, γ

stress distribution angles for ballast, capping, and subgrade, respectively (°);

α₀, β₀

reference stress distribution angles for ballast and capping, respectively (°);

δ(x)

vertical deflection of the track at distance x (m);

${\epsilon }_b^p$, ${\epsilon }_c^p$, ${\epsilon }_s^p$

irrecoverable strain in ballast, capping, and subgrade, respectively (%);

$({\epsilon }_s^p{)}_i$

cumulative plastic strain in the ith subgrade layer;

ν_b, ν_c, ν_s

Poisson’s ratio of ballast, capping, and subgrade, respectively;

ρ_b, ρ_c, ρ_s

density of ballast, capping, and subgrade, respectively (kg/m³);

${\sigma }_d^{\mathrm{\prime }}$

deviator stress (N/m²);

σ_oct, τ_oct

octahedral normal and shear stresses, respectively (N/m²);

σ_s

compressive strength of the soil (N/m²);

${\sigma }_x^{\mathrm{\prime }},{\sigma }_y^{\mathrm{\prime }}$

lateral stresses in longitudinal and transverse directions, respectively (N/m²);

${\sigma }_{z,b}^{\mathrm{\prime }},{\sigma }_{z,c}^{\mathrm{\prime }},{\sigma }_{z,s}^{\mathrm{\prime }}$

vertical stresses in the ballast, capping, and subgrade layers, respectively (N/m²); and

φ′

friction angle (°).

References