Analysis of Rotary Vibration of Rigid Friction Pipe Pile in Unsaturated Soil

Table 1

Symbols table.

Symbol	Meaning
b^l,	Unit liquid and gas penetration force, equations (18) and (19)
G	Shear modulus
H	The effective pile length participating in the rotary vibration
	The first-order Hankel function
Ip	Moment of inertia of pile, equation (59)
I1, I2	The first and second orders of the first Bessel function of imaginary argument
J1, J2	The first- and second-order Bessel function of real arguments
K1, K2	The first and second orders of the second Bessel function of imaginary argument
K	Intrinsic permeability of solids
kM = kM1 + ikM2	The rotary complex stiffness of the pile body, equations (60) and (61)
M	Peak of the couple
Mp	The simple harmonic circumferential couple which the pile top sustains, equation (56)
Ms	The circumferential couple of the pile body under the coaction of soil inside and outside the pile, equation (57)
M	m represents solid, liquid, or gas, respectively, when m = s, l, or
N	Porosity
p^l,	Pressure of liquid and gas
Qi^m	Volume discharge, equations (5)∼(7)
,	Relative specific discharge, equations (8) and (9)
q	Equation (29)
qi^m	Mass discharge, equation (4)
RI, RO	Internal and external radii of the pipe pile
S	Saturation
ui	Displacement of the solid component
uθ	Amplitude of rotational vibration
	Velocity
,	Circumferential displacement of the liquid and gas phase components relative to the solid phase component
β	Volume compressibility of soil
βs	Compressibility of the solid phase component material
γ	Equation (17)
δij	Kronecker symbol
εij	Strain of solid
η	Equation (27)
θp	The maximum rotation angle of the pile body
λ	Lamé constants of dry soil
μ	Lamé constants of dry soil
νl,	Dynamic viscosity (absolute viscosity) of liquids or gases
ρ	Density of unsaturated soil, equation (25)
ρm	Mass density
ρp	Density of pile
	Relative density, equations (1) and∼(3)
σij	Total stress, equation (10)
	Stress corresponding to component m, equations (11) and (13)
τ	Shear stress, equation (41)
τij	, equation (11)
	Effective stress, equations (15) and (16)
ω	Circular frequency