14_List of Symbols

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)



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)



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)



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)




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)




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)




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


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)




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




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


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)




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

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14_List of Symbols