Theoretical Solution for Long-Term Settlement of a Large Step-Tapered Hollow Pile in Karst Topography

Data Availability Statement

Acknowledgments

Notation

A, B

constants in the disturbance factor D;

A_p1, A_p2, …, A_p(n+1), A_p(n+2)

cross-sectional area;

c₁, c₂

constants;

D

disturbance factor;

D_u

ultimate value of the disturbance factor D;

d₁, …, d_n

diameter of the pile section;

E_M

viscoelastic modulus in the Maxwell model;

E_k

viscoelastic modulus in the Kelvin model;

E_p

elasticity modulus of the pile;

e

2.718281828;

G_k

shear modulus in the Kelvin model;

H, N

constants in the Weibull function;

l₁, L₂, …, l_n+1, l_n+2

length of the pile section;

M

point at the pile side;

n

number of times of variable cross section;

P(z)

axial force at depth z;

P_b

axial force of the pile tip;

P₀

axial force of the pile top;

R_bj

uniform annular load;

s

pile's displacement at a given depth z;

s_b

displacement of the pile tip;

s₀

displacement of the pile top;

t

time;

U₁, U₂, …, U_n+1, U_n+2

perimeter of the pile section;

v

Poisson's ratio of the soil;

α

enhancement effect coefficient;

γ

apparent shear strain of soil;

γ_c

shear strain of soil in the FA state;

γ_i

shear strain of soil in the RI state;

γ_p

unit weight of the pile;

ɛ

normal strain;

η_M

viscosity in the Maxwell model;

η_k

viscosity in the Kelvin model;

η_n

viscosity coefficient in the perfectly viscoplastic model;

ξ^c

cumulative creep strain;

ξ^p

cumulative plastic strain;

σ

normal stress;

σ_xM

additional lateral pressure at point M;

${\sigma }_0^{\mathrm{\prime }}$

pressure under the variable cross section;

τ(z)

shear stress of the pile–soil interface at a given depth z;

τ_c

stress in the perfectly viscoplastic model;

τ_i

stress in the Kelvin model;

τ_n

frictional resistance of friction plate in the perfectly viscoplastic model;

τ₁

shaft resistance on the pile side in the lower part of the variable cross section of the pile; and

φ

internal friction angle of soil.

References