Response of Inclined Loaded Pile in Layered Foundation Based on Principle of Minimum Potential Energy

Acknowledgments

Notation

A₀

the stiffness matrix;

a₀

the unknown coefficient;

a₁

the unknown coefficient;

a₂

the unknown coefficient;

a₃

the unknown coefficient;

b₀

the unknown coefficient;

b₁

the unknown coefficient;

b₂

the unknown coefficient;

b₃

the unknown coefficient;

C_j,k

the coefficient;

c

cohesive force;

D

the pile diameter;

d_j,k

the coefficient;

E

the elastic modulus;

E₀₁

the elastic modulus of the first soil;

E₀₂

the elastic modulus of the second soil;

E₀₃

the elastic modulus of the third soil;

E₀₄

the elastic modulus of the fourth soil;

E₀₅

the elastic modulus of the fifth soil;

E_m

the elastic modulus of the lower soil;

E_p

the elastic modulus for the concrete pile;

E_PI_P

the bending stiffness of pile;

E_s

the elastic modulus of the upper soil;

E_s1

the elastic modulus of the upper surface for the upper soil;

E_s2

the elastic modulus of the lower surface for the upper soil;

E_s3

he elastic modulus of the upper surface for the lower soil;

E_s4

the elastic modulus of the lower surface for the lower soil;

f

the lower soil;

H

the horizontal load;

H₀

the horizontal force at the pile tip;

L

the length of the pile;

L₁

the thickness of the upper soil;

L_2v

the thickness of the lower soil;

M

the bending moment;

M₀

the bending moment at the pile tip;

m₁

the coefficient of foundation reaction of upper soil;

m₂

the coefficient of foundation reaction of lower soil;

n

the order time;

n₁

the micro section number;

P₀

the axial force at the pile tip;

p

the vertical load;

R

pile radius;

r

the radial direction of the column coordinate;

u

the displacement along the radial;

u

the horizontal displacement;

u₀

the displacement of pile top;

u₁

the horizontal displacement of the upper soil;

u₂

the horizontal displacement o;

u(z)

the displacement function of pile changing with depth z;

u₁(z)

the horizontal displacement function of the upper part pile with z;

$u_1^{\mathrm{\prime }}$

the first derivative of the displacement function;

$u_1^{\mathrm{″}}$

the second derivative of the displacement function;

${u^{\mathrm{\prime }}}_1^{\mathrm{″}}$

the third derivative of the displacement function;

$u_1^{(4)}$

the fourth derivative of the displacement function;

u₂(z)

the displacement function of the lower part with z;

$u_2^{\mathrm{\prime }}$

the first derivative of the displacement function;

$u_2^{\mathrm{″}}$

the second derivative of the displacement function;

${u^{\mathrm{\prime }}}_2^{\mathrm{″}}$

the third derivative of the displacement function;

$u_2^{(4)}$

the fourth derivative of the displacement function;

v

the displacement along angular direction;

v

the Poisson’s ratio;

v_s1

the upper soil Poisson’s ratio;

v_s2

the lower soil Poisson’s ratio;

w

the displacement along axial;

w_p

the displacement of the pile top;

z

the angular direction of the column coordinate;

z

the depth of the pile;

α_s

the change coefficient of elastic modulus of surrounding soil along depth;

α_s1

the change coefficient of elastic modulus of the upper surrounding soil along depth;

α_s2

the change coefficient of elastic modulus of the lower surrounding soil along depth;

γ_rθ

the strain of the rθ direction;

γ_θz

the strain of the θz direction;

γ_zr

the strain of the zr direction;

ɛ_rr

the strain of the r direction;

ɛ_θθ

the strain of the θ direction;

ɛ_zz

the strain of the z direction;

θ

the axial direction of the column coordinate;

σ_rr

the stress of the r direction;

σ_θθ

the stress of the θ direction;

σ_zz

the stress of the z direction;

τ₁

the shear stress of the upper soil;

τ₂

the shear stress of the lower soil;

τ_rθ

the stress of the rθ direction;

τ_θz

the stress of the θz direction;

τ_zr

the stress of the zr direction;

φ(r)

the dimensionless displacement function of soil in the r direction;

Π

the total energy for the pile–soil system;

Π_pile1

the strain energy of the upper pile;

Π_pile2

the strain energy of the lower pile;

Π_soil1

the strain energy of the upper soil;

Π_soil2

the strain energy of the lower soil;

Π_f

the friction potential energy of the soil around the pile;

Π_f1

the friction potential energy of the upper soil around the pile;

Π_f2

the friction potential energy of the lower soil around the pile;

Π_H

horizontal load potential energy;

Π_M

bending moment potential energy on pile tip; and

Π_p

the vertical load potential energy.

References