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7/28/2019 14_List of Symbols
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List of symbols
This list contains definitions of symbols and an indication of the section in the
book where they first appear. Because of the large number of parameters that are
used, some sym bols represent more than one quantity. To minimise any confusion
this may cause, all symbols are defined in the text, when they are first used.
a , b parameters for hyperbo lic model (5.7.4)
b relative magnitude of the intermediate
principal stress (4.4.3)
b vector which describes the orientation of the
bounding surface in transformed variables (8.7)
c ratio of semi-axes of the bounding surface
ellipsoid in MIT-E3 model
c' soil cohesion
cp' soil cohesion at peak strength
c,.' soil cohesion at residual strength
c7
adjusted coefficient of consolidation
dF vector of element nodal displacements
dllG vector of global nodal displacements
du vector of unknown displacements
dp vector of prescribed displacements
d'p visco-plastic component of displacementd
1*, d
1** vectors of parallel and orthogonal symmetry
displacements respectively
e void ratio
eo initial void ratio
ex , e2 unit vectors in local coordinate system
g{6) gradient of the yield function in J-p1 plane,
as a function of Lo de 's angle (7.5)
gpp(@) gradient of the plastic potential function
in J-p' plane, as a function of Lo de 's angle (7.5)g out of balance vector in iterative solution
procedures (11.4)
h parameter affecting bounding surface
plasticity in MIT-E3 model (8.7)
(8.7)(1.9.
(4.3.
(4.3.
(10.
(2.
(2.
(3.7.
(3.7.
•1 )
.6)
.6)
•9).3)
.3)
•3)
•3)
(9.5.2)
(12.3
(4.3,
(4.4,
( I I I
1)
•1)
•1)
• 1)
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426 / Finite element analysis in geotechnical engineering: Theory
h hydraulic head
iG vector defining the direction of gravity
r,j* unit vectors in global coordinate system
k, spring stiffnessk vector of state parameters for yield function
k permeability matrix
/ length of the failure surface
/ distance along beam element
m vector of state parameters for plastic potential
function
m parameter for plastic expansive strains for
Lade' model
n soil porosityn parameter affecting hysteretic elasticity in
MIT-E3 model
p parameter for plastic collapse strains for
Lade' model
pa atmospheric pressure
p f pore fluid pressure
p' mean effective stress
pc' mean effective stress at current stress state
po' hardening parameter for critical state models
p' fj, plfl the 7
thcosine and sine har mon ic coefficients
respec tivel y, of por e fluid pre ssure
at the j * node
q deviatoric stress
qn infiltration flow rate
f position vector
s natural ordinate for beam element
5 vect or of tra nsfor med deviat oric stresscomponen ts
t parameter fo r plastic expansive strains fo r
Lade ' model
tc critical time step fo r visco-plastic analysis
v specific vo lu me
v, specific vo lu me a t unit mean effective stress
(parameter fo r critical state mo del s) (7.9.1 )
v100 specific volume atp ' = 10 0 k P a
(parameter fo r MIT-E3 model) ( 8 . 7 )vx •>
vy J vz components of pore fluid velocity in Cartesian
coordinate directions (10.3)
u, v, w displacement components in x, y, z
directions respectively (1.5.3)
(10(10
(II. 1
(3.7.
(6(10
(1.9
.3)•3)
•1)
.5)
.8)
• 3 )
•1 )
(3.5.2)
(6.8.
(8,
(3,
(8.
• 3 )
.5)
• 4 )
•7 )
(8.5)
(8. 5)(3.4)
(4.3.2)
(7. 5)
(7.9)
(12.3.
(9.7.
(10.6.
(II.l .
(3.5.
(8.
(8.
(8.5.
(7.9.
7)
2)
4)
1)
4)
7)
5)
3)
1)
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List of symbols / 427
Uj, w, displacements tangential and normal to
a beam element (3.5.2)
u, , Vj displacement components in local
coordinate system (3.6.2)w/
op, v/
opdisplacement components for the top side
of an interface element (3.6.2)
W/bot
, V/bot
displacement components for the bottom side
of an interface element (3.6.2)
x, y, z Cartesian coordinates (1.5.3)
xp, yp point loading axes (3.7.7)
z, r, 6 cylindrical coordinates (1.6.2)
A cross sectional area of a beam element (3.5.3)
A param eter for small strain stiffness model (5.7.5)A elasto-plastic modulus (6.13)
B strain matrix (2.6)
B parameter for small strain stiffness model (5.7.5)
C parameter for small strain stiffness model (5.7.5)
C parameter for plastic collapse strains for
Lade ' model (8.5)
C parameter affecting hysteretic elasticity in
MIT-E3 model (8.7)
Cc compression index, i.e. inclination of the VCLin e-log10crv' plane (4.3.1)
C s swelling index, i.e. inclination of a swelling
line in e-log10crv' plane (4.3.1)
CSL critical state line (7.9.1)
D total stress constitutive matrix (1.5.5)
D' effective stress constitutive matrix (1.5.5)
Dep elasto-plastic constitutive matrix (6.13)
D f pore fluid matrix (1.5.5)
DM diagonal matrix (2.9.2)D dilatancy (7.11.1)
E' drained Yo ung 's modulus (1.5.5)
Eu undrained Young 's modulus (4.3.2)
Eh' Yo ung 's modulus in horizontal direction (4.3.5)
Evf Yo ung 's modulus in vertical direction (4.3.5)
E s' Yo ung 's modulus in the depositional direction (5.6)
Ep Yo ung 's modulus in the plane of deposition (5.6)
E total potential energy (2.6)
F vector of body forces (2.6)F meridional force for beam element (3.5.3)
Fy/ circumferential force for beam element (3.5.3)
F({a},{k}) yield function (6.8)
G elastic shear modulus (5.5)
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428 /Finite element analysis in geotechnical engineering: Theory
Gsec secant shear modulus (5.7.5)
Gkm tangent shear modulus (4.3.3)
Gvh shear modulus in vertical plane (4.5.1)
GPS shear modulus in the plane of the directionof deposition (5.6)
GPP shear modulus in the plane of deposition (5.6)
/ / , , H2 hardening parameters for Lade's model (8.5)
/ cross sectional moment of inertia of a
beam element (3.5.3)
Ip plasticity index (4.5.3)
J Jacobian matrix (2.6)
J deviatoric stress invariant (5.3)
Jc deviatoric stress invariant at current stress state (7.5)KE element stiffness matrix (2.3)
KG global stiffness matrix (2.3)
KH, Kp diagonal components of the global stiffness
matrix, corresponding to unknown and
prescribed displacements respectively (3.7.3)
Kup off diagonal terms of the global stiffness
matrix (3.7.3)
Ka pre-conditioning matrix in iterative
solvers (H-4)Kf bulk modulus of pore fluid (3.4)
K, bulk modulus of the solid soil particles (3.4)
Kskel bulk modulus of the soil skeleton (3.4)
Ke equivalent bulk m odulus (3.4)
Ks elastic shear stiffness of interface element (3.6.2)
Kn elastic normal stiffness of interface element (3.6.2)
Ko coefficient of earth pressure at rest (4.3.2)
KONC coefficient of earth pressure at rest for
normally consolidated soil (7.9.3)K(,
oc coefficient of earth pressure at rest for
overconsolidated soil (7.9.3)
K' effective bulk modulus (5.5)
Ksec secant bulk modulus (5.7.5)
Kkm tangent bulk modulus (4.3.3)
Kn , KF2 parameters for an alternative shape for the
yield function for critical state models (7.11)
KPX , KP2 parameters for an alternative shape for the
plastic potential function for criticalstate models (7.11)
L load on a beam (1.5.2)
L work done by the applied loads (2.6)
L lower triangular matrix (2.9.2)
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List of symbols / 429
Lj area coordinates (II. 1.1)
LER linear elastic region in stress space (5.7.6)
L G off diagonal submatrix in consolidation
stiffness matrix (10.3)M bending moment for beam element (3.5.3)
MX(I circumferential bending moment for beam
element (3.5.3)
M } gradient of the critical state line in J-p' plane
as a constant independent of Lo de 's angle (7.9.1)
M JP yield function parameter (7.6)
MJPPP plastic potential function parameter (7.6)
N elastic parameter for Lad e's model (8.5)
TV matrix of displacement shape orinterpolation functions (2.5)
Nj substitute shape functions (3.5.4)
Np matrix of pore fluid pressure interpolation
functions (10.3)
OCR overconsolidation ratio (4.3)
P({cr},{m}) plastic potential function (6.8)
P parameter for plastic expansive strains for
Lade' model (8.5)
P spherical component of the flow direction (8.7)P
1 spherical component of the flow direction
at the image point (8.7)
P vector of deviatoric components of flow
direction, in transformed variables (8.7)
P!
vector of deviatoric components of flow
direction at the image point, in transformed
variables (8.7)
Q spherical component of the gradient of the
bounding surface (8.7)Q vector of deviatoric components of the
gradient of the bounding surface, in
transformed variables (8.7)
Q1 vector of deviatoric components of the
gradient of the bounding surface at the image point,
in transformed variables (8.7)
Q flow through sources and sinks (10.3)
Q rotation matrix of direction cosines (3.7.2)
R param eter for small strain stiffness model (5.7.5 )R parameter for plastic expansive strains for
Lade' model (8.5)
R parameter for All-Tabbaa & Wood model (8.9)
RE vector of element nodal forces (2.3)
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430 / Finite element analysis in geotechnical engineering: Theory
RG vector of global nodal forces (2.3)
Rp right hand side load vector corresponding to
prescribed displacements (3.7.3)
Ru right hand side load vector corresponding tounknown displacements (3.7.3)
R7*, R
;** parallel and orthogo nal symm etry right hand
side load vectors respectively (12.3.1)
Rlr , R
lz , R
le th e I
thcosine harmonic coefficient of radial,
vertical and circumferential incremental force
respectively (12.3.1)
Rlr , R
lz , RQ the /* sine harmo nic coefficient of radial,
vertical and circumferential incremental force
respectively (12.3.1)
S param eter for small strain stiffness mod el (5.7.5)
S shear force for beam elemen t (3.5.3)
S, T natural coordinates (2.5.1)
S, parameter affecting the degree of strain
softening in MIT -E3 mo del (8.7)
Su undrain ed shear strength (1-9.1)
Srf surface of integra tion (2.6)
SSR small strain region in stress space (5.7.6)
SSTOL substep tolerance in substepping stress
point algorithm (IX. 1)
T param eter for small strain stiffness mod el (5.7.5)
T length of substep in substepping stress
point algorithm (9.6.2)
T adjusted time factor in consolidation
analysis (10.9)
T vector of surface tractions (2.6)
To tensile soil strength (8.3)
U\, Vj , W- the 7th
cosine harmonic coefficients of radial,
vertical and circumferential displacement
respectively, at the fh
node (12.3.1)
UJ, vf , WJ th e Ith
sine harm onic coefficients of radial,
vertical and circumferential displacement
respectively, at the /* node (12.3.1)
Vol volum e of integration (2.6)
VC L virgin consolidation line (7.9.1)W weigh t of a failing block (1.9.1)
W strain energy (2.6)
Wj weigh ts for num erical Gaussian integration (2.6.1)
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List of symbols / 431
YTOL
a
a
a
a
aK, aG
aP, aF
pp
PPa
PP,PF
fxy 5 ixz 5 (yz
?rz > 7rO , izB
S
s
s
s
yield function tolerance for driftinclination of the major principal stress to
the vertical
parameter for small strain stiffness modelparameter for plastic expansive strains for
Lade' model
size of the bounding surface for MIT-E3
model
time step parameter in visco-plastic analysis
parameter for defining the elastic portion
of the stress increment in the substepping
stress point algorithm
parameters for K-G modelparameters for an alternative shape for the
yield and plastic potential functions for
critical state models
inclination of failure surface to the vertical
parameter for plastic expansive strains for
Lade' model
parameter for iterative solver
parameter for K-G model
parameters for an alternative shape for theyield and plastic potential functions for
critical state models
bulk unit weight
shear strain for beam element
parameter for small strain stiffness model
parameter affecting bounding surface
plasticity in MIT-E3 model
bulk unit weight of pore fluid
shear strain components in Cartesiancoordinates
shear strain components in cylindrical
coordinates
parameter for small strain stiffness model
vector of nodal displacements and
rotations for beam element
iterative vector
strain vector
direct strain components in Cartesiancoordinates
direct strain components in cylindrical
coordinatesvolumetric strain
(IX. 1)
(4.3.4)
(5.7.5)
(8.5)
(8.7)
(9.5.3)
(IX. 1)
(5.7.3)
(7.11)
(1.9.1)
(8.5)
(11.4)
(5.7.3)
(7.11)
(1.5.2)
(3.5.2)
(5.7.5)
(8.7)
(10.3)
(1.5.3)
(1.6.2)
(5.7.5)
(3.5.4)
(11.4)
(1.5.5)
(1.5.3)
(1.6.2)
(3.4)
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432 / Finite element analysis in geotechnical engineering: Theory
eve
volu met ric elastic strain (7.9.1)
€ volu met ric plastic strain (7.9.1)
£y axial strain for be am ele men t (3.5.2 )
sw circumferential membrane strain fo r beamelement (3.5.2)
es devia toric strain for triaxial stress space (4.3.3)
ep
component o f plastic strain ( 6 . 3 )
ee
component o f elastic strain (6.13 )
£crack
crack strain ( 8 . 3 )
evp
visco-p lastic strain (9.5.2 )
es elasto-plastic portion of the strain increment in
subs tepp ing stress poin t algor ithm (IX. 1)
£ss substep strains in substepping stress pointalgorithm (9.6.2)
r\ parameter for sma ll strain stiffness mo de l (5.7 .5)
r\ stress ratio (=J/pf) (7.11.2)
rj parameter fo r iterative solver (11.4)
rjP , Y\F parameters for an alternative shape for the
yield and plastic potential functions fo r
critical state mo del s (7.11 )
r\x parameter for plastic expansive strains fo r
Lade ' model ( 8 . 5 )6 inclination of the major principal stress to the
horizontal (1.9.2)
6 Lode's angle ( 5 . 3 )
0c Lode's angle a t current stress state ( 7 . 5 )
0f Lode's angle at failure (7.12)
K inclination o f swelling line in v-\np' plane
(parameter fo r critical state mo del s) (7.9.1)
K0 initial slope of the swelling line in v-lnp' plane,
MIT-E3 model ( 8 . 7 )K inclination o f swelling line in lnv-ln// plane
(parameter for All-Tabbaa & Wood model) ( 8 . 9 )
X inclination o f V C L in v-ln/?' plane (parameter
for critical state mod els ) (7.9.1)
X* inclination o f V C L in lnv-ln// plane
(parameter for All-Tabbaa & Wood model) ( 8 . 9 )
ju' drained Poisson's ratio ( 3 . 3 )
juv undrained Poisson's ratio ( 3 . 3 )
juSP
' Poisson's ratio for straining in the plane of
deposition due to a stress acting in the
direction of deposition (5.6)
ju PSf Poisson's ratio for straining in the direction of
deposition due to a stress acting in the plane
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List of symbols / 433
MPP
jUP,JUF
MV
P
G
G'
Gf
Gx,Gy, Gz
Gz,Gr, G0
G] , G2 , Oj
Oy , Of,'
oa, or
Oa',O r*
Ovc'
Gy
Ot1f
(f
of depositionPoisson 's ratio for straining in the plane of
deposition due to a stress acting in the
same planeparameters for an alternative shape for the
yield and plastic potential functions for
critical state models
elastic parameter for Lade's model
angle of dilation
system of local coordinates coinciding with
two sides of a triangle
parameter for plastic expansive strains for
Lade' modeltotal stress vector
effective stress vector
pore fluid stress vector
direct stress components in Cartesian
coordinates
direct stress components in cylindrical
coordinates
major, intermediate and minor principal stress
vertical and horizontal effective stressaxial and radial total stress
axial and radial effective stress
vertical effective consolidation stress
yield stress
normal effective stress on the failure plane
trial stress in return algorithm
(5.6)
(5.6)
(7.11)
(8.5)
(7.5)
(I I . l . l )
(8.5)(1.5.5)
(1.5.5)
(1.5.5)
(1.5.2)
(1.6.2)
(1.9.2)
(4.3.1)(4.3.2)
(4.3.3)
(4.3.4)
(6.4)
(7.5)
(IX.2)
zxy, rxz, xyz shear stress components in Cartesian
coordinates (1.5.2)
xrz, xr0, xz0 shear stress components in cylindricalcoordinates (1.6.2)
xf shear stress on the failure plane (7.5)
cp ' angle of shearing resistance (1.9.1)
cp j critical state angle of shearing resistance (4.3.2)
(Pp peak angle of shearing resistance (4.3.6)
(p r' residual angle of shearing resistance (4.3.6)
(pTCf
critical state angle of shearing resistance
in triaxial compression (8.7)
cp 1E ' critical state angle of shearing resistancein triaxial extension (8.7)
Xi bending strain for beam element (3.5.2)
Xy/ circumferential bending strain for beam
element (3.5.2)
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434 / Finite element analysis in geotechnical engineering: Theory
V
VCO
CO
¥
Ed
EJEJE
A
parameter affecting rotation of boundingsurface in MIT-E3 model
parameter for All-Tabbaa & W ood m odel
elastic parameter for Lade's modelparameter affecting the hysteretic elasticity
in MIT-E3 m odel
vector of residual load
invariant deviatoric strain
elastic deviatoric strain
plastic deviatoric strain
vector of deviatoric strains in transformed
variables
scalar multiplier for plastic strainspermeability submatrix in consolidation
stiffness matrix
(8.7)
(8.9)
(8.5)
(8.7)
(9.6)
(5.3)
(VII.2)
(VII.2)
(8.7)
(6.8.3)
C10.31