Euler–Bernoulli Pile Element for Nonlinear Buckling Analysis of Single Piles in Slope

Acknowledgments

Notation

A

cross-sectional area;

A_ji

coefficients in the secant relations;

B_ji

coefficients in the secant relations;

C_ji

coefficients in the secant relations;

E

Young's modulus;

F_E

element resisting force vector from the element strains;

F_S

element resisting force vector from the soil resistance and sliding force;

F_T

total resisting force vector;

F_x1

force on the x-axis at the upper side;

F_x2

force on the x-axis at the lower side;

F_y1

force on the y-axis at the upper side;

F_y2

force on the y-axis at the lower side;

H_i

weight of the ith Gauss point;

I

moment of inertia;

[k]

global stiffness matrix;

[k]_E

tangent stiffness matrix of the whole element;

[k]_G

geometric stiffness matrix;

[k]_L

linear stiffness matrix;

[k]_LS

impaired stiffness matrix cause by the sliding force;

[k]_S

soil stiffness matrix;

k(v)

tangential value of the p–y curve at a specified lateral deflection;

L

element length;

M₁

bending moment at the upper side;

M₂

bending moment at the lower side;

P

axial load;

p

lateral soil resistance per unit length;

q(x)

value of the sliding force at a specified location of the element;

R

global resisting force vector;

TOL

value for the acceptable accuracy in the Newton–Raphson numerical procedure;

U

global nodal displacement vector;

U_E

potential energy accumulating in the pile;

U_S

potential energy accumulating in the surrounding soil;

U_SP

energy consumed by lateral soil resistance;

U_Sτ

energy consumed by shaft resistance;

u

axial deflection along the element;

u₁

axial deflection at the upper side;

u₂

axial deflection at the lower side;

u_i

axial defection at the ith Gauss point;

V₁

shear force at the upper side;

V₂

shear force at the lower side;

v

lateral deflection along the element;

v₁

lateral deflection at the upper side;

v₂

lateral deflection at the lower side;

v_i

lateral defection at the ith Gauss point;

W

work done by total external forces;

W_L

work done by the sliding force;

W_P

work done by external point load;

x_i

axial defection at the ith Gauss point;

[γ]_i

initial transformation matrix for describing the initial position of the element at during the ith step;

ΔF

incremental force or the unbalanced force vector;

Δr

element resisting force vector;

ΔU

incremental nodal displacement vector;

ɛ_x

element normal strain;

ɛ_xy

element shear strain;

θ₁

rotation at the upper side;

θ₂

rotation at the lower side;

κ_ij

coefficients in the stiffness matrix;

λ(u)

tangential value of the τ–z curve at a specified axial deflection;

μ_ij

coefficients in the stiffness matrix;

Π

total potential energy;

σ_x

element normal stress;

τ

shaft friction per unit length;

τ_xy

element shear stress; and

ω_ji

coefficients in the stiffness matrix.

References