## List of Foundation Symbols

Roman
A = a coefficient for estimating shaft load transfer factor
A(t) = time-dependent part of the shaft creep model
Ab, As = area of pile base and shaft, respectively
Ac = a parameter for the creep function of J(t)
Ag = constant for soil shear modulus distribution
Ah = a coefficient for estimating “A,” accounting for the
effect of H/L
AL, ALi = coefficient for the LFP; AL for the ith layer [FL−1-ni]
An, Bn = coefficients for predicting excess pore pressure
Aoh = the value of Ah at a ratio of H/L = 4
Ap = cross-sectional area of an equivalent solid cylinder
pile
Ar = coefficient for the LFP [FL-3]
As = 2(Bi + Li)L, perimeter area of pile group
Av = a constant for shaft limit stress distribution
B = a coefficient for estimating shaft load transfer factor
B(z) = sub-functions reflecting base effect due to lateral
Bc = a parameter for the creep function of J(t)
Bc = width of block
c, c′ = cohesion [FL−2]
C, Cy = zru*/ug (constant k), used for post-tip yield state, and
C at the tip-yield state
C(z) = subfunction reflecting base effect due to moment
CA, Cp = factors for relative density Dr
CB, CR, Cs = borehole diameter correction, rod length correction,
and sample correction
CN = a modification factor for overburden stress
COCR = overconsolidation correction factor
cub = undrained cohesion at pile-base level [FL−2]
cv = coefficient of soil consolidation
Cv(z) = a function for assessing pile stiffness at a depth of z,
Cvb = limiting value of the function for the ratio of base
Cvo = limiting value of the function, Cv(z), as z approaches
zero
Cλ = a coefficient for estimating “A,” accounting for the
effect of λ
D = outside diameter of a cylindrical pile [L]
D50 = maximum size of the smallest 50% of the sample [L]
dmax = depth of maximum bending moment in the shaft [L]
do = outside diameter of a pipe pile [L]
dr = reference width, 0.3 m
Dr = relative density of sand
E = Young’s modulus of soil or rock mass [FL−2]
Ec = modulus of elasticity of concrete [FL−2]
EcIp = EI, initial flexural rigidity of the shaft [FL2]
Em = hammer efficiency
Em = deformation modulus of an isotropic rock mass
[FL2]
Ep = Young’s modulus of an equivalent solid cylinder pile
[FL−2]
Er = deformation modulus of intact rock mass [FL2]
to the mudline; or e = Mo/Ht; or distance from point
O to incorporate dragging effect [L]
em, ep, ew = eccentricity of the location above the ground level
for measuring the Mmax, applying the P(Pg), and
measuring the wt
F(t) = the creep compliance derived from the visco-elastic
model
FixH = fixed-head, allowing translation but not rotation at
level
Fm = pile-pile interaction factor (passive pile tests)
f ′ c = characteristic value of compressive strength of the
concrete [FL−2]
f ′′ c = design value of concrete compressive strength
[FL−2]
fr = Mcryr/Ig, modulus of rupture [FL−2]
fr = friction ratio computed using the point and sleeve
friction (side friction), in percent
fy = yield strength of reinforcement [FL−2]
G, G = (initial) elastic shear modulus, average of the G
[FL−2]
G* = soil shear modulus, G* = (1 + 0.75νs)G [FL−2]
Gb, Gbj = shear modulus at just beneath pile base level; or initial
Gb for spring j (= 1, 2) [FL−2]
Gb(t) = time-dependent initial shear modulus at just beneath
pile base level [FL−2]
Gi, Gi
* = G, Gi
* for ith layer [FL−2]
Gj = the instantaneous and delayed initial shear modulus
for spring j (j = 1, 3) [FL−2]
GL = (initial) shaft soil shear modulus at just above the
pile base level [FL−2]
Gm = shear modulus of an isotropic rock mass [FL−2]
Gγj, G1% = shear modulus at a strain level of γj for spring j
(j = 1, 2) within the visco-elastic model, or shear
modulus at a shear strain of 1% [FL−2]
gs = pu/(qu)1/n, a factor featuring the impact of rock
roughness on the resistance pu
GSI = geology strength index
h = distance above pile-tip level
hd = the pile penetration into dense sand
H = the depth to the underlying rigid layer (Chapters 4
(Chapter 7) [F]
H, Hmax = lateral load exerted on a single pile, and maximum
imposed H [F]
H(z) = sub-function, due to lateral load (Chapter 7); lateral
force induced in a pile at a depth of z [F]
Hav, Hg = average load per pile in a group, and total load
imposed on a group [F]
He = lateral load applied when the slip depth is just initiated
at mudline [F]
Ho = H at a defined (tip-yield or YRP) state [F]
H2 = lateral load applied at a distance of “eo2” above point
O, and H1 = −H2 [L]
H z i( ) = normalized shear force induced in a pile at a normalized
depth of z for i = 1 (0 < z ≤ zo), 2 (zo < z ≤ z1), and
3 (z1 < z ≤ l), respectively
i = subscripts 1 and 2 denoting the upper sliding and
lower stable layer, respectively
I = settlement influence factor for single piles subjected
Icr = moment of inertia of cracked section [L4]
Ie = effective moment of inertia of the shaft after cracking
[L4]
Ig = moment of inertia of a gross section about centroidal
axis neglecting reinforcement [L4]
Ig = wgdEL/Pg, settlement influence factor for a pile
group
Im, Im-1 = modified Bessel functions of the first kind of noninteger
order, m and m − 1 respectively
Ip = moment of inertia of an equivalent solid cylinder pile
[L4]
Ip = the plasticity index
I(z) = sub-function due to moment
J = empirical factor lying between 0.5 and 3 for estimating
Ng
Ji = Bessel functions of the first kinds and of order i
(i = 0, 1)
J(t) = a creep function defined as ζc/G1
k = permeability of soil (Chapter 5)
k, ko = modulus of subgrade reaction [FL-3], k = kozm
,
m = 0
and 1 for constant and Gibson k, respectively; and
ko, a parameter [FL-3-m]
ki, kj = parameters for estimating load transfer factor, i = 1,
ki = modulus of subgrade reaction for ith layer (passive
piles)
kr = a constant for concrete rupture
ks = a factor representing pile–soil relative stiffness
ksg = ks for a pile in a two-pile group
kϕ = pile rotational (constraining) stiffness about the
K = average coefficient of earth pressure on pile shaft
with minimum and maximum values of Kmin and
Kmax
Ka = tan2 (45° − ϕ′/2), the coefficient of active earth
pressure
Kg = 0.6~1.5, group interaction factor, with higher values
for dense cohesionless or stiff cohesive soils, otherwise
for loose or soft soils
Ki(γ) = modified Bessel function of second kind of i-th order
Km, Km-1 = modified Bessel functions of the second kind of noninteger
order m and order m − 1, respectively
Kp = tan2 (45° + ϕ′/2), the coefficient of passive earth
pressure
L(l) = embedded pile length
Lc = length of block
Lc = critical embedded pile length beyond which the pile
is classified as infinitely long
LFP = net limiting force profile per unit length [FL−1]
LR, MR, TR = leading row, middle row, and trailing row
L1 = the depth of transition from elastic to plastic phase,
L2 = length of the elastic part of a pile under a given load
Lm = sliding depth during a passive soil test [L]
Ln = depth of neutral plane
m = 1/(2 + n), or number of rows of piles
m2 = ratio of shear moduli, Gγ1/Gγ2
M, M(x) = moment induced on a pile element, or M at a depth
of “x” [FL]
MA(x), MB(z) = moment induced in a pile element, at depth x and z,
respectively [FL]
MB = moment induced at pile-base level [FL]
Mcr = cracking moment [FL]
Mmax, Mm = maximum bending moment within a pile [FL]
Mn = nominal or calculated ultimate moment [FL]
Mo = moment applied on pile-head or at the mudline level
[FL]
Moi = Hieoi, bending moment about point O [FL]
Mt = moment applied at the shaft at the groundline [FL]
M(x) = M(x)λn+2/AL, normalized bending moment at
depth x
M z i( ) = bending moment induced in a pile at a normalized
depth of z for i = 1 (0 <z ≤ zo), 2(zo <z ≤ z1), and 3 (z1
<z ≤ l), respectively
Mmax = Mmaxλ2+n/AL, normalized Mmax
n = number of piles enclosed by the square (NSF)
n = number of piles in each row
n = power of the shear modulus distribution, nonhomogeneity
factor, or power for the LFP
nc = the safe cyclic load amplitude
nc
FreH, ns
respectively. The subscripts s and c refer to sand and
clay, respectively
ne = equivalent nonhomogeneity factor
ng = number of piles in the pile group
N = visco-elastic time factor, or blow count of the
Standard Penetration Test
N′ = corrected blow counts of SPT
N60 = blow counts of SPT at a standard rod energy ratio of
60%
Nc, Nq = bearing capacity factors
Nco = lateral capacity factor–correlated soil, undrained
strength, with the limiting pile–soil pressure at
mudline
Ng = gradient-correlated soil, undrained strength, with
the limiting pile–soil pressure
Ng
FreH, Ng
Nk = cone factor (a constant for each soil) ranging from
5 to 75
Np = a factor for limiting force per unit length (plastic
zone)
Np = fictitious tension for a strecthed membrane tied
together the springs around the pile shaft (elastic
zone)
N
c = equivalent lateral capacity factor correlated average
soil undrained strength with average net limiting
force per unit length by p s N d u u c =
p, pu = force per unit length, and limiting value of the p
[FL−1]
p′ = mean effective stress [FL−2]
P = vertical load on a pile head under passive tests [F]
pa = atmospheric pressure, ≈ 100 kPa
Pb(Pfb) = load of pile base (ultimate Pb) [F]
PBL = a total failure load of the group [F]
Pcap = the bearing capacity of the pile cap on the bearing
stratum
Pe = axial load at the depth of transition (L1) from elastic
to plastic phase [F]
Pex P
cap
in
cap , = components of capacity deduced from areas Aex
c
(lying outside the block) and Ain
c (inside block)
Pf(Pu) = ultimate pile bearing load [F]
Pfs = ultimate shaft load of a pile [F]
Pij = pile number ij in a group (i = 1–3, j = 1–2)
pm = p-multipliers used to reduce stiffness, and limiting
force for individual piles in a group
Pns = total downdrag load [F]
Pug = ultimate capacity of the pile group
pu
= average limiting force per unit length over the pile
embedment [FL−1]
p(x), Q(x) = net force per unit length, and shear force at the normalized
depth x
P(z) = axial force of pile body at a depth of z [F]
q′ = stress on the top of the weaker layer [FL−2]
qb, q′b = unit base resistance, net qb [FL−2]
qbmax = maximum end-bearing capacity [FL−2]
qc = cone resistance qc of a CPT test
qu = uniaxial compressive strength of the weaker material
(rock or concrete) [FL−2]
qui = uniaxial compressive strength of intact rock [FL−2]
Q, QB = shear force induced on a pile cross-section, and the
Q at pile-base level [F]
QA(x), QB(x) = shear force induced on a pile cross-section [F]
Qg = the perimeter of the pile group (equivalent large pile)
r = radial distance from pile axis [L]
r* = the radius at which the excess pore pressure, by the
time it reaches here, is small and can be ignored [L]
rm = radius of zone of shaft shear influence [L]
rmg = a radius of influence of pile group [L]
ro = radius of a cylindrical pile [L]
R = the radius beyond which the excess pore pressure is
initially zero [L]
R(R) = ratio of Ng of a single pile in a group over that of the
single pile (average of R)
RA, RB, RC,
RD, RE
= subratings for qui, RQD, spacing of discontinuities,
conditions of discontinuities, and groundwater,
respectively
Rb = ratio of settlement between that for pile and soil
caused by Pb, base settlement ratio
Rfb, Rfj = a hyperbolic curve-fitting constant for pile base load
settlement curve, or for the elastic element j within
the creep models
RMR = rock mass rating
RQD = rock quality designation
Rmax = roughness of the concrete
Rs = group settlement ratio
s = argument of the Laplace transform (Chapter 5), or
pile center to center spacing (Chapter 7)
sg = an integral factor to cater for all sorts of influence
su, su
= undrained shear strength, an average su over the pile
embedment [FL−2]
suL = undrained shear strength at pile tip or footing base
level [FL−2]
t (t*) = time elapsed
t = wall thickness of a pipe pile or thickness of concrete
cover [L]
t90 = time for (uo − u)/uo = 0.9
T = relaxation time, η/G2, or rate of consolidation
T50, T90 = time factor, T, for 50% and 90% degree of “consolidation,”
respectively
T2(T3) = relaxation time, ηγ2/Gγ2 (ηγ3/Gγ3)
Tmax = sliding force on a rigid pile
Tn(t) = the time for the solution of the reconsolidation theory,
also written as T(t)
TR = total resistance over the maximum slip depth xp
obtained under Pmax [F]
Tult = ultimate Tmax
u = excess pore water pressure [FL−2] or radial displacement
[L]
u, ug = lateral displacement, and u at mudline level [L]
u* = local threshold u* above which pile–soil relative slip
is initiated [L]
u(z) = axial pile displacement at a depth of z [L]
Uk = energy parameter for “y = 1” per unit pile length
Um = energy parameter for “dϕ/dr = 1” per unit radial
length
UN = energy parameter for “dy/dz = 1” per unit pile length
Un = energy parameter for “ϕ = 1” per unit radial length
uo = initial pore water pressure [FL−2]
uo(r) = initial excess pore water pressure at radius r [FL−2]
uv = vertical displacement along depth [L]
v = circumferential displacement [L]
Vi = cylinder function of i-th order
w, wc = local shaft deformation, and creep part of w [L]
w(or y), w(x),
w(z)
= lateral deflection of a pile, w in the plastic, and w in
elastic zone, respectively [L]
w(x), w′(x) = deflection [L] and rotation at the normalized depth x
w(z) = deformation of pile body at a depth of z for a given
time [L]
w(z) = pile body displacement at depth z, or simply written
as w [L]
wA, wB = lateral deflection in the upper plastic zone and lower
elastic zone, respectively [L] (Chapter 9)
wB = pile base displacement at the base level [L] (Chapter 7)
wA
IV, wA′′′,
wA′′, wA′
= fourth, third, second, and first derivatives, respectively,
of deflection w with respect to the depth x
wB
IV, wB′′′,
wB′′, wB′
= fourth, third, second, and first derivatives, respectively,
of deflection w with respect to the depth x
wb, wt = settlement of pile base and head, respectively [L]
we = settlement due to elastic compression of pile [L]
we
c = a reduced limiting shaft displacement deduced from
the limiting shaft stress, τf
c
wf = frame (soil) movement during a shear test [L]
wg, wg′ = lateral pile deflection [L], and rotation angle (in
radian) at mudline, respectively, or point O for passive
piles
wg
= wgkλn/AL, normalized mudline deflection
wgi = a lateral pile deflection at point O (sliding level) [L]
wi = settlement or deformation of the i-th pile in a group
of ng piles [L]
wi = initial frame movement during which pile response
is negligible [L]
wp = PtL/(EpAp)
wp = pu/k, lateral deflection at the slip depth of xp [L]
wp
IV, wp′′′,
wp′′, wp′
= values of fourth, third, second, and first derivatives,
respectively, of deflection w with respect to the
depth x at depth xp
ws = lateral (uniform) soil movement, or shaft displacement
[L]
wt = pile-head settlement, or lateral deflection [L]
wt1 = settlement of a single pile under unit head load [L]
x, xp, x, xp = depth below ground level, slip depth of plastic zone,
x = λx, xp = λ xp
xi = depth measured from point O on the sliding interface
[L]
xmax = depth at which maximum bending moment occurs
(xmax = xp + zmax when ψ(x ) p ≥ 0; xmax = xmax when
ψ(x ) p < 0) [L]
xmaxi = depth of maximum bending moment measured from
point O in plastic zone [L]
xp, xpi = slip depth from the elastic to the plastic state; or xp
from the elastic-plastic boundary to point O [L]
xs = thickness of the zone, in which pile deflection
exceeds soil movement [L]
y(z) = pile body displacement at depth z, or simply written
as y [L]
Yi = Bessel functions of the second kinds and of order i
(i = 0, 1)
YRP = yield at rotation point
yo = pile deflection at sand surface [L]
z = depth measured from the mudline [L]
z, z = x − xp, depth measured from the slip depth [L] and
z = λz, respectively
zc = critical depth [L]
zi = xi − xpi, depth measured from the slip depth, xpi [L]
zm = depth of maximum bending moment [L]
zmax2 = depth of Mmax2 measured from the slip depth, xp2 [L]
zmaxi = depth of Mmaxi measured from the sliding interface [L]
zo(z1) = slip depth initiated from mudline (pile-base) [L]
zr = depth of rotation point [L]
zt = an infinite small depth [L]
z* = slip depth zo at the moment of the tip-yield [L]
Greek
α = average pile–soil adhesion factor in terms of total
stress
α, β = stiffness factors for elastic solutions (lateral piles) [L−1]
αc = nondimensional creep parameter for standard linear
model, or ratio of the shear modulus over the undrained
strength, G/su
αE = 0.0231RQD − 1.32 ≥ 0.15, and RQD in percentage
αg = shear modulus factor for ground surface
αij = pile-pile interaction factor between pile i and pile j
αm = shaft friction factor
αn = consolidation factor
αN, βN = α/λ, and β/λ, normalized α and β by λ, respectively
αo(αo) = an equivalent depth to account for ground-level limiting
force with αo
= αoλ
αr = a reduction factor (related to qu) for shaft friction
αsi = a factor correlating maximum shear force to bending
moment (i = 1, 2) of passive piles
αρP, αρM = the interaction factor between the i-th pile and j-th
pile, reflecting increase in deflection due to lateral
αθP, αθM = the interaction factor between the i-th pile and j-th
pile, reflecting increase in rotation due to lateral load
β = average pile–soil adhesion factor in terms of effective
stress
βr = a factor correlated to the discontinuity spacing in the
rock mass
γ (γ) = shear strain (shear strain rate)
γj (γ
.
j) = shear strain (shear strain rate) for elastic spring j
γm = effective unit weight of the rock mass [FL-3]
γrθ, γθz, γrz = shear strain within the r-θ plane, θ-z plane, and the
r-z plane, respectively
γs( s) = unit weight of the overburdened soil (effective γs)
[FL-3]
γw = the unit weight of water [FL-3]
δ = factor used for displacement prediction
δ = interface frictional angle, being consistent with that
measured in simple shear tests
δθ = mean total stress
r (  ) = increments of the effective stress during consolidation
z
= increments of the effective stress during consolidation
in depth direction
εij = strain components within the surrounding soil
εr, εθ, εz = strain in the radial, circumferential, and depth
directions
εv = the volumetric strain
ζ, ζj = shaft load transfer factor, nonlinear measure of the
influence of load transfer for spring j (j = 1, 2) within
the creep models
ζc = a nondimensional creep function
ζg = shaft load transfer factor for a pile in a two-pile
group
η = group efficiency correlated to the ratio ρ of the shaft
(skin) load, Pfs over the total capacity, Pu (i.e., ρ =
Pfs/Pu)
ηs, ηb = the efficiency factors of shaft and base
ηs′ = geometric efficiency
ηγi(η) = shear viscosity for the dash at strain γi (i = 2, 3)
θ = power of the shear stress distribution, nonhomogeneity
factor (Chapter 4)
direction (Chapter 7)
θB = pile rotation angle at base level
θg, θt = rotational angle at groundline (or point O), the angle
θo = pile-head rotation angle, or differential angle between
upper and lower layer at point O (Chapter 12)
θw = rotation angle of pile at the point of ew (Chapter 11)
λ = relative stiffness ratio between pile Young’s modulus
and the initial soil shear modulus at just above the
λ, λi = reciprocal of characteristic length, λ = 4 k / (4EpIp)
λ = factor to correlate shaft friction to mean effective
overburden pressure, and undrained cohesion
λn = the n-th root for the Bessel functions
λs = Lame’s parameter
μ = degree of pile–soil relative slip
νp(νs) = Poisson’s ratio of a pile (soil)
ξ = shaft stress softening factor, when w > we (Chapter 4)
ξ = outward radial displacement of the soil around a pile
[L] (Chapter 5)
ξ = a factor to capture impact of pile-head constraints
and soil resistance, etc., on the resistance zone of a
passive pile (Chapter 12)
ξb = pile base shear modulus nonhomogeneous factor,
GL/Gb
ξmax, ξmin = the maximum and minimum values of the factor ξ
(Chapter 12)
π1, π1* = normalized pile displacement and local limiting of π1
π2 = normalized depth with pile length
π3 = normalized pile–soil relative stiffness factor
π4 = normalized pile–soil relative stiffness for plastic case
π2p = normalized depth with slip length
πv = pile–soil relative stiffness
ρg = ratio of the average soil shear modulus over the pile
embedded depth to the modulus at depth L
σho = horizontal stress [FL−2]
σr, σθ, σz = radial, circumferential, and vertical stress in the surrounding
soil, respectively [FL−2]
σ′v = effective overburden pressure [FL−2]
σ′vb = effective overburden pressure at the toe of the pile
[FL−2]
σ′vs(vs) = effective overburden pressure over the pile shaft
(average σ′vs) [FL−2]
τ(τf) = local shear stress (limiting τ) [FL−2]
τ .
= shear stress rate for spring 1 in the creep model
τf
τj(τfj) = local shear stress on elastic spring j with j = 1, 3
(maximum τj) [FL−2]
τo,τo(t*) = shear stress on pile–soil interface, and τo at the time
of t*[FL−2]
(τo)ave = average shear stress on a pile–soil interface over all
the entire pile length [FL−2]
τoj = shear stress on pile–soil interface at elastic spring j
(j = 1, 2) [FL−2]
τs = (average) shaft friction along a pile shaft [FL−2]
τultj = ultimate (soil) shear stress for spring j (j = 1, 3),
respectively [FL−2]
ϕ(ϕ′) = angle of friction of soil (effective ϕ)
ϕ = ultimate moment reduction factor (Chapter 10)
ϕ(r) = attenuation of soil displacement at r from the pile
axis
ϕ′1 = undisturbed friction angle at the pile toe before pile
installation
ϕ′cv = critical frictional angle at constant volume
ϕp = frictional angle under plane strain conditions
ϕr = residual angle of friction of soil
ϕtr = frictional angle under axisymmetric conditions
χv = a ratio of shaft and base stiffness factors for vertical
ψ = factor to correlate adhesion factor α to unconfined
compressive strength qu
ψj = (τojRfj/τultj), nonlinear stress level on pile–soil interface
for spring j (j = 1, 2)
ω = water content (Chapter 1)
ω = a pile-base shape and depth factor (Chapters 4 and 5)
ω = a rotation angle (in radian) of a lateral pile
ωg = base shape and depth factor for a pile in a two-pile
group
ωh, ωoh = a coefficient for estimating “ω”, accounting for the
effect of H/L and the value of ωh at a ratio of H/L = 4
ων, ωoν = a coefficient for estimating “ω”, accounting for the
effect of νs and the value of ων at a ratio of νs = 0.4
Principal Subscript (Piles Only)
A = upper plastic zone (lateral piles)
B = lower elastic zone (lateral piles)
b = pile-base
max, m = maximum
p = pile