# 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

load

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,

under vertical loading

Cvb = limiting value of the function for the ratio of base

and head loads as z approaches zero

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]

e, eoi = eccentricity or free-length from the loading location

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

head level

FreH = free head, allowing translation and rotation at head

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

and 6) [L], or horizontal load applied on pile-head

(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 = Hλn+1/AL, normalized pile-head load

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

to vertical loading

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,

3 (lateral loading), or a factor representing soil nonlinearity

of elastic spring j (vertical loading)

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

head

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,

the slip part length of a pile under vertical loading

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

FixH = power of the LFP for free-head and fixed-head

respectively. The subscripts s and c refer to sand and

clay, respectively

ne = equivalent nonhomogeneity factor

ng = number of piles in the pile group

nmax = maximum ratio of pile-head load and the ultimate

shaft load

np = ratio of pile-head load and the ultimate shaft load

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

FixH = gradient-correlated soil strength to the pu for freeand

fixed-head piles, respectively

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 = surcharge loading on ground surface

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]

Pt = vertical load acting on pile head [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)

sb = a loading distance between center of a pile (group)

and loading block

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 = loading time

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

(vertical loading)

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

load and moment loading, respectively

αθP, αθM = the interaction factor between the i-th pile and j-th

pile, reflecting increase in rotation due to lateral load

and moment loading, respectively

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)

γb = load transfer factor

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

in radial and circumferential directions

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)

θ = an angle between the interesting point and loading

direction (Chapter 7)

θB = pile rotation angle at base level

θg, θt = rotational angle at groundline (or point O), the angle

at pile-head level

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

base level, Ep/GL (vertical loading)

λ, λi = reciprocal of characteristic length, λ = 4 k / (4EpIp)

(lateral loading), λ for i-th layer [L−1]

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

ξc = a yield stress ratio for cyclic loading

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

ρ = pile-head deformation [L]

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

c = new limiting shaft stress under cyclic loading [FL−2]

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

loading

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

t = pile-head

u = ultimate