ANALYSIS OF SOIL PENETROMETER USING
EULERIAN FINITE ELEMENT METHOD
BY
SHUANG HU
B.S. IN HYDRAULIC ENGINEERING (1998)
TSINGHUA UNIVERSITY
MASTER OF PHILOSOPHY IN CIVIL ENGINEERING (2000)
HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY
SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL
ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREEE OF
MASTER OF SCIENCE IN CIVIL AND ENVIRONMENTAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2003
©2003 Massachusetts Institute of Technology. All rights reserved.
Signature of Author:
Departme>9fvil and-Epvirorknental Engineering, May 9, 2002
Certified by:
rofessor And/
J. Whittle, Thesis Supervisor
Accepted by:
Professor Oral Buyukozturk,
Chairman, Departmental Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
BARKER
JUN 0 2 2003
LIBRARIES
ANALYSIS OF SOIL PENETROMETER USING
EULERIAN FINITE ELEMENT METHOD
BY
SHUANG HU
Submitted to the Department of Civil and Environmental Engineering
on May 9, 2003 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Civil and Environmental Engineering
ABSTRACT
This thesis investigates the use of an Eulerian Finite Element (EFE) method for
modeling penetration in soils. The formulation decouples material and nodal points
displacements such that soil flows through a fixed finite element mesh. This approach
eliminates problems of mesh distortion associated with conventional Lagrangian
formulations but requires special procedures to convect the soil constitutive law and
prevent numerical diffusion. The current analyses use the program DiekA, developed
at the University of Twente in the Netherlands. Detailed calculations have been
performed to investigate the effects of elements size and load/time step size on the
stability and accuracy of the numerical simulations. Computed results for undrained
penetration in homogeneous clays are similar to prior predictions from approximate
steady state formulations such as the Strain Path Method. Further calculations for
two-layer systems illustrate the potential of the Eulerian formulation to handle
realistic layered soil profiles. A more limited study confirms the complexity of
drained penetration in sands, where the predicted tip resistance is affected by soil
friction and dilation angles, in situ stresses, and moduli. Further study is needed to
establish the role of interface friction and lateral earth pressure. The results in the
thesis present a first step towards implementation of more advanced effective stress
soil models in EFE analyses of penetration in layered soils.
Thesis Supervisor: Andrew J. Whittle
Title: Professor of Civil and Environmental Engineering
2
ACKNOWLEDGMENTS
I wish to express my sincere thanks to all those who helped to bring about the
completion of this research program.
Firstly, special thanks are due to my supervisor, Prof. Andrew Whittle, who
introduced this topic to me and provided constant guidance all the way. He provided
invaluable feedback and assistance during the course of this research. His advice and
comments made during the preparation of this thesis were also very much valued and
greatly appreciated.
I would like to express my great appreciation to Gert-Jan Schotmeyer from
GeoDelft and Ton van den Boogaard from University of Twente, for giving me
constant and patient help on DiekA coding. Thanks for putting up with my long
emails full of questions.
Also, I want to thank my "soil and rock" friends on the third floor of
Buildingl, and friends from other departments. Thanks for the fun we had together,
the nice food you made for me, the happiness we shared, and a sympathetic ear when
I feel frustrated.
Finally, but most importantly, I wish to thank my family for their endless love
and support. Particularly, I want to say 10 thanks to my husband, Shangwen Wei, for
his constant encouragement, looking after me, sharing both my sorrows and joys all
the way, and for the love I treasure forever.
Shuang Hu
3
TABLE OF CONTENTS
ABSTRACT............................................................................................................................................
2
ACK NO W LEDG M ENTS .....................................................................................................................
3
TABLE O F CONTENTS.......................................................................................................................
4
LIST OF FIGURES...............................................................................................................................
7
LIST O F TABLES ................................................................................................................................
12
INTRO DUCTION ...............................................................................................
13
CH APTER 1
1.1
PROBLEM STATEMENT AND SCOPE OF STUDY ...................................................................
13
1.2
ORGANIZATION OF THESIS................................................................................................
15
CH APTER 2
LITERATURE REVIEW ..................................................................................
16
2.1
BEARING CAPACITY THEORY ............................................................................................
16
2.2
CAVITY EXPANSION THEORY ............................................................................................
19
2.3
STEADY STATE FLOW ...........................................................................................................
24
2.4
INCREMENTAL FINITE ELEMENT M ETHOD.......................................................................
26
CHAPTER 3
EULERIAN FORMULATION FOR FINITE ELEMENT ANALYSIS OF
PENETRATIO N PRO BELM S............................................................................................................39
3.1
BASIC EQUATIONS................................................................................................................39
3.2
CONVECTION
3.3
LAYERED DEPOSITS WITH EULERIAN APPROACH .............................................................
49
PENETRATION ANALYSIS IN CLAY...........................................................
53
CHAPTER 4
4.1
ITEM
...............................................................................................................
FINITE ELEMENT M ODEL..................................................................................................
4
45
53
4.2
NUMERICAL ACCURACY AND INTERPRETATION OF PENETRATION SIMULATION .................. 55
4.2.1
NumericalAccuracy....................................................................................................
55
4.2.2
Interpretationof PenetrationSimulation ....................................................................
56
4.3
EFFECT OF SOIL STIFFNESS...............................................................................................
4.4
LAYERED CLAY .........................................................................................................
60
61
........-
PENETRATION ANALYSIS IN SAND ...........................................................
84
5.1
DRAINED SHEAR STRENGTH OF SANDS .............................................................................
84
5.2
FRICTION ANGLE ...............................................................................................................--
89
5.3
NUMERICAL SIMULATION.................................................................................................
89
5.4
EFFECT OF STRENGTH PARAMETERS .................................................................................
93
CHAPTER 5
CHAPTER 6
IMPLEMENTATION OF USER MATERIAL MODEL INTO DIEKA.........117
6.1
INTRODUCTION
6.2
THREE ROUTINES FOR USER SUPPLIED M ATERIAL M ODEL ................................................
...................................................................................................................
117
117
6.2.1
User Supplied InitialRoutine........................................................................................118
6.2.2
User Supplied Stress UpdateRoutine ...........................................................................
119
6.2.3
User Supplied Matrix Routine....................................................................................
121
6.3
VALIDATION OF USER SUPPLIED MATERIAL M ODEL ..........................................................
123
6.4
INITIALIZATION OF STATE VARIABLES ...............................................................................
124
CHAPTER 7
SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FUTURE
STUDIES
127
7.1
SUMMARY AND CONCLUSIONS...........................................................................................127
7.2
SUGGESTIONS FOR FUTURE STUDIES ..................................................................................
7.2.1
Model interfacefriction.................................................................................................129
7.2.2
Initial stress state set up................................................................................................129
5
129
7.2.3
Implementing advanced soil models into DiekA ...........................................................
129
130
APPENDIX A
AN INPUT FILE FOR DIEKA ...................................................................................
APPENDIX B
USER PROVIDED SUBROUTINES FOR A LINEARLY ELASTIC AND PERFECTLY PLASTIC
MATERIAL MODEL...............................................................
.................
....
---.............................
B.]
Initialization routine USRINI........................................................................................155
B.2
Stress UpdateRoutine USRSIG.....................................................................................157
B.3
Stiffness Update Routine USRM TX...............................................................................157
REFERENCES.................................................................................................................--...............-171
6
155
LIST OF FIGURES
Figure 2.1 Assumed failure mechanisms under deep foundations
(Durgunoglu
29
and M itchell, 1975)................................................................................
Figure 2.2 Assumed relationship between cone resistance and cavity limit
. . 30
p ressure ...................................................................................................
Figure 2.3 Relation between Nq and #peak (Vesic, 1975)................................
31
Figure 2.4 Comparison of N, from different methods.....................................
31
Figure 2.5 Predicted deformation pattern around simple pile (Baligh, 1975)..... 32
Figure 2.6 Predicted deformation pattern around a 60-degree cone (Levadoux,
19 8 0 ).....................................................................................................
. . 33
Figure 2.7 Predicted deformation pattern around a FMMG piezoprobe (Sutabutr,
199 9).....................................................................................................
. . 34
Figure 2.8 Steady state behavior (Yu et al., 2000).........................................
35
Figure 2.9(a) Typical contour plot for strain y (Yu et al., 2000).....................
36
Figure 2.9(b) Contour plot for strain y during simple pile penetration (Baligh,
19 8 5).....................................................................................................
. . 36
Figure 2.10(a) Typical contour plot for excess pore pressure (Yu et al., 2000).. 37
Figure 2.10(b) Excess pore pressure due to undrained simple pile penetration
(B aligh , 1985)..........................................................................................
Figure 3.1 Node and material point displacement at one step ........................
37
52
Figure 3.2 Definitions for moving material boundary through fixed FE mesh
(V an den B erg, 1996)..............................................................................
7
52
Figure 4.1 Piezocone tip geom etry..................................................................
66
Figure 4.2 Finite element mesh for penetration analysis.................................
66
Figure 4.3 Rounded cone-cylindar transition part............................................
67
Figure 4.4 504 elements and 5184 elements FE mesh .....................................
68
Figure 4.5 Pressure-displacement curves for 504, 1944 and 5184 elements ...... 69
Figure 4.6 Pressure-displacement curves for varying step sizes......................
69
Figure 4.7 Pressure-displacement curve for basic calculation .........................
70
Figure 4.8(a) Contours of equivalent plastic strain Ep .....................................
71
Figure 4.8(b) Equivalent plastic strain Ep along the penetrometer shaft........... 71
Figure 4.9(a) Octahedral stress increment Au.e, /o
.. .
...............
. . . ..
. . ..
. . . . ..
72
Figure 4.9(b) Au ,,,,/o , along shaft ...............................................................
72
Figure 4.9(c) Radial distribution of Ac,,, /o e at various elevations.............
73
Figure 4.10(a) Contours of shear stress
,,I/o
0
..........
. . .
... .... .... ... .... .
Figure 4.10(b) z,, /a,0 along the penetrometer shaft.....................................
Figure 4.11 Contours of vertical stress
,
/.
Figure 4.12 Contours of radial stress a,/o o
. 74
74
.........................
75
........................
75
.. .. ... ... .
76
/oro along shaft.....................................
76
Figure 4.15 Radial distribution of o /o-0 , at various elevations ...................
77
Figure 4.16 Radial distribution of o-, /o 0 e at various elevations ...................
77
Figure 4.17 Radial distribution of o700 /U-,
78
Figure 4.13 Contours of circumferential stress o,, /o
Figure 4.14
O-,z /oV
o
,,
/0vo ,
o700
8
at various elevations..................
Figure 4.18(a) Contours of normalized cylindrical expansion shear stress qh /UvO
79
.....................................................................................................................
Figure 4.18(b) qhl/
v0
.
along shaft ................................................................
Figure 4.18(c) Radial distribution of qh /v0
at various elevations ...............
79
80
Figure 4.19 Pressure-displacement curves for clays with different Ir................. 81
Figure 4.20 Cone factor comparison of analyszed results with published
so lutio n s ......................................................................................................
Figure 4.21 Layered clay - location of initial layer boundary ........................
81
82
Figure 4.22 Pressure-displacement curve for penetration in layered clay .......... 82
Figure 4.23 Pressure-displacement curves for layered system with different Ir
ratio ..............................................................................................................
83
Figure 5.1 Mohr-Coulomb model .................................................................
98
Figure 5.2 Drucker-Prager m odel....................................................................
98
Figure 5.3 Matching Mohr-Coulomb model and Drucker-Prager model ........... 99
Figure 5.5 Pressure-displacement curves for Drucker-Prager and Mohr-Coulomb
m odels, base case analysis ........................................................................
100
Figure 5.6(a) Contours of mobilized friction angle - Mohr-Coulomb criterion 101
Figure 5.6(b) Contours of mobilized friction angle - Drucker-Prager criterion 101
Figure 5.7(a) Contours of equivalent plastic strain E ..........................
102
Figure 5.7(b) Equivalent plastic strain Ep along the penetrometer shaft............ 102
Figure 5.7(c) Radial distribution of Ep at various elevations............................. 103
Figure 5.8(a) Octahedral stress increment A
Figure 5.8(b) A -', /o
', /o-
.......
.......
along shaft ................................................................
9
104
104
Figure 5.8(c) Radial distribution of Ac', /0o
Figure 5.9(b) r, /o
/
,
Figure 5.9(a) Contours of shear stress
at various elevations..............
along the penetrometer shaft.........................................
/a',
Figure 5.9(c) Radial distribution of z,,
at various elevations...................
Figure 5.11 Contours of radial stress o'
/o
Figure 5.12 Contours of circumferential stress o'
'
/a'o,
Figure 5.14 Radial distribution of a'
/'
107
........................
108
................
109
/
along shaft......................................
a' /a'o
106
108
......................
Figure 5.10 Contours of vertical stress o /a
Figure 5.13 o' / o ,
106
........................
O..
105
at various elevations
..........
109
110
Figure 5.15 Radial distribution of a' /0'0 at various elevations .....................
110
at various elevations.................
111
Figure 5.16 Radial distribution of o' /o-'
Figure 5.17(a) Contours of normalized cylindrical expansion shear stress qh /O
...................................................................................................................
Figure 5.17(b) qh / o along shaft ....................................................................
Figure 5.17(c) Radial distribution of q
1 12
112
/ro at various elevations ................. 113
Figure 5.18 Correlation between cone factor Nq and friction angle
Figure 5.19 Pressure-displacement curves for varying
#
#
............. 114
(Vf = 0)...................
Figure 5.20 Pressure-displacement curves for varying V (# = 45
115
)................ 115
Figure 5.21 Pressure-displacement curves for varying stiffness E ................... 116
Figure 6.1 Fitting Tresca and von Mises models at triaxial compression mode 126
Figure 6.2 Load-displacement curves for different material models ................ 126
10
Figure A. 1 Input file example "init.inv"...........................................................
139
Figure A.2 Finite element mesh represented by the example input file............ 140
Figure A.3 Rounded cone-cylindar transition part............................................
140
Figure A.4 Output file "vorm.dat" for "init.inv"...............................................
154
Figure B. 1 "USRINI" file for linearly elastic and perfectly plastic model with
von M ises yielding ....................................................................................
159
Figure B.2 Statements in input file for user defined material..........................
160
Figure B.3 "USRSIG" file for linearly elastic and perfectly plastic model with
von M ises yielding ....................................................................................
165
Figure B.4 Flow chart of incremental algorithm for linearly elastic and perfectly
plastic model with von Mises yielding......................................................
166
Figure B.5 "USRMTX" file for linearly elastic and perfectly plastic model with
von M ises yielding ....................................................................................
11
170
LIST OF TABLES
Table 4.1 Material parameters for clays with varying rigidity index.............. 64
Table 4.2 Material parameters for clays with varying u,0 ................
Table 4.3 Material parameters and initial stress for
. . . . . .. . . . . .. . .
64
penetration analysis in
layered clays............................................................................................
65
Table 5.1 Material data used in sensitivity analysis.........................................
96
Table 5.2 Material data used by van den Berg (1994) ....................................
97
12
CHAPTER 1
INTRODUCTION
1.1
Problem Statement and Scope of Study
Soil penetration is involved in many practical geotechnical problems, for example,
foundation elements (piles, caissons, etc.), in situ testing devices (piezocone,
dilatometer, etc.), the continuous soil-sampling test, vane test and more. The analysis
of the cone resistance, the evolution of stresses and strains produced in the soil during
the penetration process, are all of relevant interest in geotechnical engineering.
This research focuses on the process of piezocone penetration. The piezocone
penetration testing (PCPT) has become a useful tool for in situ investigation and
geotechnical design. Geotechnical engineers have found that the continuous sounding
cone penetration provides excellent resolution of vertical stratigraphy in soil
exploration projects (e.g. Lunne, et al., 1997). The use of PCPT data for estimating
soil properties requires reliable correlations between PCPT testing results (cone
resistance, sleeve friction ratio and pore pressures measured both during steady
penetration and in subsequent consolidation) and soil engineering properties.
As pointed out by Baligh (1985), the penetration in the soil is particularly
difficult in terms of theoretical analysis, due to the following features: (1) singularities
and high gradients of the field variables (displacements, stresses, strains and pore
water pressures) around the penetrometer; (2) large deformations and strains
developed in the soil; (3) the complex constitutive behavior of soils, including non13
linearity, anisotropy, time-dependency and frictional response; (4) drainage conditions
(controlled by consolidation
characteristics
of the soil);
and
(5)
non-linear
penetrometer-soil interface characteristics.
Due to the complex geometry of the penetration problem, analytical tools can
normally only be used in an approximate manner. Nevertheless, many analytical
solutions have been developed in geotechnical practice, for example, bearing capacity
(Meyerhof, 1961), and cavity expansion theories (Vesic, 1972). Bearing capacity
solutions assume rigid, perfectly plastic soil behavior and are generally solved by
limit equilibrium methods (Vesic, 1977). Cavity expansion theory assumes onedimensional deformation/strain field and most solutions consider elastic, perfectly
plastic material response with Tresca or Mohr-Coulomb yield criteria (Yu, 2000).
Some cavity expansion solutions are developed with Modified Cam Clay model (Yu
et al., 2000)
In principal, numerical techniques, in particular the finite element method can
allow a more complete solution of the penetration problem, by taking into account the
penetrometer geometry, interface properties and soil constitutive behaviors. A largestrain finite element formulation is needed, to carry out the penetration analysis. The
use of conventional Updated Lagrangian methods quickly results in highly distorted
elements, yielding highly inaccurate answers or even divergence of the computational
procedure. In order to address this problem, one option is to use remeshing techniques
(e.g., Hu and Randolph, 1997). Alternatively Eulerian finite element formulations
proposed by van den Berg (1994,
1996) consider the flow of soil around the
penetrometers.
14
Van den Berg's method is followed in this research. A finite element program
"DiekA", developed at University of Twente in Netherlands (DiekA development
group, 2000), is applied for this study to investigate the use of DiekA in penetration
analysis. Penetration analyses on clays and sands, homogeneous and layered soils, are
performed. The detailed stress/strain fields from the FE analysis are explored for a
clear interpretation of the penetration mechanism. A series parametric study is carried
out in each case to evaluate the main factors affecting predictions of penetration tip
resistance.
1.2
Organization Of Thesis
The thesis is organized as follows. Following a brief introduction in this chapter,
Chapter 2 surveys existing methods for penetration analysis. Basic ideas behind each
method are presented together with a comparison of their relative advantages and
disadvantages. Chapter 3 summarizes the formulation of the Finite Element Program
DiekA. Chapter 4 presents results of undrained penetration analyses in homogeneous
clays and layered clay profiles. Chapter 5 explores drained penetration analyses for
sands. In order to develop a more reliable basis for predicting penetration processes,
advanced soil models need to be implemented into DiekA. Implementation of a user
defined material model (linear elastic perfectly plastic model) into DiekA is
demonstrated in Chapter 6. Finally, Chapter 7 presents a summary and conclusions
from this work together with recommendations for future studies.
15
CHAPTER 2
LITERATURE REVIEW
There have been extensive research efforts to develop reliable correlations between
cone resistance and soil properties. For example, Sanglerat (1972) summarizes
correlations between cone resistance and relative density, deformation characteristics
of the soil, etc. Theoretical analyses of cone penetration can be classified into four
main categories: (1)
Bearing capacity theories; (2) cavity expansion methods; (3)
steady state penetration models and (4) incremental finite element analyses. This
chapter presents the basic concepts underlying these approaches together with their
capabilities and limitations.
2.1
Bearing Capacity Theory
Bearing capacity theory is the earliest method used to analyze cone penetration. In
this approach, the cone penetration problem is considered as an incipient failure for a
rigid perfectly plastic soil. Solutions have been based on limit equilibrium (Meyerhof,
1961; Vesic, 1977) and slip-line methods analyses (Sokolovskii, 1965). Well-known
solutions are those published by Meyerhof (1961), Durgunoglu and Mitchell (1975).
Figure 2.1 shows some of the failure patterns that have been used to analyze deep
penetration problems (Durgunoglu and Mitchell, 1975). Bearing capacity theories
have been widely used in soil mechanics due to its simplicity. However, the solutions
from limit equilibrium methods are only approximate, because of the following
limitations: (1) the influence of penetration process on the initial stress states around
the shaft is not considered; (2) the soil stress-strain behavior is completely ignored;
16
and (3) use of shape factor to convert from a planar wedge to axisymmetric cone
penetration introduces uncertainty. It should also be pointed out that the shear
surfaces shown in Figure 2.1 are not generally observed during deep cone penetration
process.
In slip-line method (Sokolovskii, 1965), a set of differential equations is given
by combining the equilibrium equations with a specified yield criterion (usually
Mohr-Coulomb or Tresca). Stress characteristics lines (referred to as slip lines), can
be drawn based on the set of partial differential equations. Collapse load is then
determined after the construction of slip lines. Compared with limit equilibrium
method, the slip-line method is more rigorous, considering that the stress field
obtained through slip-line method satisfies both the yield criterion and the equilibrium
condition everywhere inside the slip-line network region. Nevertheless, the stress
distribution outside this region is not defined and hence, the solution is not actually a
rigorous lower bound solution.
The ultimate bearing capacity of deep foundation qu, is usually presented in
the following form:
ql,(z)= SecN + S,1 YBN,
+ Sq 0Of(z)Nq
where:
Si
shape factors (i = c, q, y)
c
cohesion
Ni
bearing capacity factors (i = c, q, y)
17
(2.1)
width of the foundation (i.e., diameter of cone)
B
initial vertical effective stress at depth z
-'O
unit weight of soil
y
The Nr term is used to include the effect of soil self-weight in the failure
zone. Experimental data from surface model footing tests on dry sand find that the
measured N, value decreases with the footing width B (Meyerhoff, 1953). For
undrained penetration in a purely cohesive soils (#= 0), Nq = 1 and Equation (2.1) can
be reduced to the following total stress format:
(2.2)
q,(z) = NSc + o, (z)
For drained penetration in cohesionless soil, Equation (2.1) becomes:
U
(Z)
qc (z) = NqSqo
(2.3)
+uO
(2.4)
qC =q
uo is the in situ pore pressure.
To be consistent with literature about cone resistance, let Ne represents NcSc
and Nq represents NSq hereafter. Relations (2.2) and (2.3) are widely accepted in the
literature, so that the penetration resistance is characterized by one single factor: the
so-called cone factor Nc for cohesive soils and the so-called bearing capacity factor Nq
for cohesionless soils.
Meyerhof (1961) proposed a cone factor:
18
(2.5)
N, = 1.15* 6.28+ a + cot 2)
where a is the semi-angle of the tip. This solution was based on a limit equilibrium
analysis with the failure mechanism as shown in Figure 2.1(b). This is a plane strain
solution adjusted by an empirical shape factor to account for axisymmetric geometry.
Durgunoglu and Mitchell (1975) derived a bearing capacity factor:
Nq
(2.6)
= 0.194exp(7.629 tan 0')
A major advantage of bearing capacity approach is its simplicity compared
with other methods. However, the bearing capacity theory does not take soil
compressibility into account. Baligh (1975) pointed out that, the penetration process is
controlled by at least two fundamental properties of the soil: (1) the shear strength and
(2) the compressibility of the soil. For example, the horizontal stresses, which are
built up around the shaft and tip of the cone during the penetration process, depend on
the deformation characteristics of the soil.
Since bearing capacity theory ignores
effects of soil compressibility, and the solutions are based on approximate collapse
mechanisms, which do not simulate accurately the mechanism of deep penetration,
researchers have been investigating other approaches since the early 1970's.
2.2
Cavity Expansion Theory
One simple theory which does take soil compressibility into account is the cavity
expansion theory. Cavity expansion theory is the theoretical study of changes in
stresses, pore water pressures and displacements caused by the expansion and
contraction of cylindrical or spherical cavities.
19
Bishop (1945) observed that the pressure to produce a deep hole in an elasticplastic medium is proportional to that necessary to expand a cavity of the same
volume under the same conditions. This shed a light on the analogy between cavity
expansion and cone penetration. In order to use cavity expansion approach to estimate
cone resistance, the theoretical limit pressure solution for cavity expansion must be
developed. Then the cavity expansion limit pressure solution needs to be related to
cone resistance.
Ladanyi and Johnson (1974) assumed a failure mechanism as shown in Figure
2.2(a). They assumed that the normal stress acting on the cone face was equal to that
required to expand a spherical cavity from zero radius. If the cone-soil interface is
perfectly rough, a cone factor for cohesive soil was obtained as:
NC = 3.16+1.331n
G
(2.7)
SU
the cone factor for cohesionless soil is:
Nq
= (1+2K )A 1+ j ta(AO'
(2.8)
3
where A is the ratio of the soil-cone interface friction angle to the soil friction angle.
A = 0 if the penetrometer is smooth; A = 1 if the penetrometer is perfectly rough. The
parameter A, the ratio of the effective spherical cavity limit pressure V/' to the initial
mean effective stress p' , is normally a function of soil strength and stiffness.
Analytical expressions for A are not available for most cavity expansion theories in
sand, although they can be obtained numerically. Collins et al. (1992) found the
following expression for A:
20
A- =
=
m.(p')(IM2+M3vo)
PO
(2.9)
exp(-m 4vO)
where v0 is the initial specific volume of the soil (=1 +void ratio); the constants ml, m2,
m3, m4 depend on the critical state properties. For example, their values for Ticino
sand are: m, = 2.012* 107, m2 = -0.875, m3 = 0.326, m4 = 6.481 (Collins et al., 1992).
Vesic (1977) presented a solution by combining bearing capacity with cavity
expansion method. Both a spherical cavity and a cylindrical cavity expansion in an
infinite soil mass were treated in Vesic's work. A failure mechanism as shown in
Figure 2.2(b) was assumed by Vesic, based on observations of both models and fullsize pile testing. Vesic obtained the following expression for the cone factor for
penetration in cohesive soils:
G
(2.10)
Ne = 3.90 +1.331n SU
and the cone factor for cohesionless soils:
where, K = -ho
-#')tan#']* tan
+-sin ,)exp
=
N
(,,)" (2.11)
45+
is the coefficient of earth pressure at rest;
#'is
soil friction angle;
0 vO
parameter n =
4
3(1+sin#')
rigidity index Ir =
I
pO
G
and I
r"
=
'
+Iv
is the reduced rigidity index, with
and ev denoting the average volumetric strain in the
tan#'
plastically deformed region. Figure 2.3 shows the Nq values for different Ir calculated
by Vesic (1975).
21
The average volumetric strain e, in the plastically deformed region has to be
estimated, by either laboratory testing or empirical correlations. This brings about
uncertainty to the accuracy of the solution. Furthermore, Vesic's solution does not
consider soil dilatancy during the shearing process. As a consequence, this solution is
not suitable for penetration in dense sands where dilation is expected to be significant.
Baligh (1975) considered the total cone resistance as the work required for a
virtual vertical displacement of the tip, plus the work done to expand a cylindrical
cavity around the shaft of the cone in the radial direction. Using this approach, the
cone factor was found to be:
NC =
12 .0
(2.12)
+ 1nG
SU
It should be pointed out that a simple sum of both contributions cannot be entirely
correct for a complex nonlinear problem. As mentioned by Yu (2000), "the
assumption that a renewed cylindrical cavity expansion takes place behind the tip of
the cone from zero radius tends to overestimate the total work required, and this leads
to an overestimation of the cone resistance."
Yu (1993) proposed another solution by assuming the normal pressure along
the shaft after pile penetration is equal to the average of the radial, tangential and axial
stresses. The cone resistance is obtained by combining the cylindrical cavity limit
pressure with a rigorous plasticity solution for the steady penetration of an infinite
rigid cone. For a standard 600 cone, the cone factor for a perfectly smooth cone is:
5I G
N = 4.18+1.1551n 2 2 SU
(2.13)
22
the cone factor for a perfectly rough cone is:
N = 9.4+1.155In
(2.14)
-
2 S,
Salgado (1993) proposed another solution for cone resistance in sands by
combining the ideas of cavity expansion with bearing capacity, or more specifically,
the slip-line method. The failure mechanism assumed by him is illustrated in Figure
2.2(c). The solution by Salgado accounts for the effect of variable friction, dilation
angles, and the dependence of the shear modulus on pressure and void ratio as well. A
numerical procedure needs to be used to obtain the cone factor, since it cannot be
presented analytically.
The relationships between Ne and rigidity index ( I, = G ) from different
su
solutions are plotted in Figure 2.4. Cavity expansion theory is an improvement
compared with bearing capacity theory, in terms of taking soil compressibility into
account, simulating the penetration process as well. However, cone penetration is not
completely identical to expanding a sphere or cylinder inside the soil mass. Cavity
expansion solutions assumes one-dimensional radial displacements in the soil and do
not account for the vertical distortion, which is critical for understanding penetration
(Baligh
and Levadoux,
1980; Whittle,
1992).
Especially
the vertical
soil
displacements around the tip of the cone cannot be neglected. In addition, the
geometric shape of the penetrometer cannot be modeled adequately with cavity
expansion theory. In order to overcome these drawbacks, the strain path method
(Baligh, 1985) has been developed.
23
2.3
Steady State Flow
Both bearing capacity and cavity expansion theory treat cone resistance as a collapse
load problem. In another point of view, for deep cone penetration in an isotropic and
homogeneous soil profile, it can be assumed to occur under a steady-state condition.
In the steady state approach, the penetration process is treated as a steady flow of soil
passing the fixed cone penetrometer.
The Strain Path Method (SPM, Baligh 1985) is the first example of this steady
state approach. Based on observations of penetration in undrained clays, Baligh (1975)
hypothesized that "due to the severe kinematic constraints that exist in deep
penetration problems, soil deformations and strains are, by and large, independent of
the shearing resistance of the soil." Therefore, approximate velocity fields are
estimated and differentiated with respect to the spatial coordinates to obtain strain
rates. Strain rates are then integrated along streamlines, to define the strain paths for
individual soil elements around the cone. A soil constitutive model is used to compute
the stresses, while equilibrium must be imposed on the approximate stress field.
The method of "sources and sinks" from the potential theory (Kellogg, 1929)
is utilized to predict the velocity, strain and deformation fields caused by the deep
steady penetration. The principal advantage of this method is to provide analytic
expressions for the strain rates, everywhere in the soil, which can then be accurately
integrated to obtain strains and deformations. Baligh (1975) originally defined a
"simple pile model", which can be used as initial deformation field for a strain path
method. The velocity field of expanding a spherical cavity is identical to that of a
point source at a fixed location in a fluid at rest. When the point source moved
24
through the fluid with a constant velocity, the material coming from it forms a
cylindrical shape with a rounded tip, as shown in Figure 2.5.
Levadoux (1980) prepared a more detailed model of cone penetration by using
a distribution of 200 sources and sinks along the axis to define the tip geometry
(Figure 2.6). Penetrometer with other geometric shapes can also be obtained by using
different source and sink configurations. For example, analyses and interpretation of
tapered peizoprobe, shown in Figure 2.7, has been carried out by Sutabutr (1999),
using the similar method.
By coupling the Strain Path Method with finite element model, Aubeny (1992)
made detailed prediction of the subsequent behavior during consolidation in cohesive
soils, using a generalized effective stress soil model MIT-E3 (Whittle, 1993; Whittle
and Kavvadas, 1994).
It has been found, however, that the stresses derived from the strain path
method may not satisfy all the equilibrium equations. Baligh (1985) pointed out that
"in more realistic situations (e.g., anisotropic clays) where the strains are "slightly"
dependent on material properties, solutions based on simplified strain fields are
approximate and the effective stresses computed by means of a given constitutive
model will not satisfy all equilibrium requirements."
Baligh (1985) suggested
approaches to satisfy equilibrium requirements, by solving an infinite number of
Poisson equations or solving one Poisson equation and one set of elasticity equations.
However, Whittle (1992) noted, based on research conducted by various authors
[Baligh (1985,
1986), Teh (1987), Teh and Houlsby (1991)] that the iterative
procedure cannot remove all the errors existing in the strain path stress solutions.
25
So far, the application of the strain path method has been restricted to
undrained penetration in clays. The application to sands proves to be much more
difficult, although some work has been done on incompressible sand (Jeng, 1992). For
frictional-dilatant soils it is difficult to obtain a realistic estimate of volumetric strains.
Another application of steady flow approach in penetration analysis is a finite
element procedure proposed by Yu et al. (2000). This method accounts for all
equilibrium equations, which is one advantage over the strain path method. A transfer
from time domain to space domain is made with the steady state assumption; that is to
say, the time dependence of stresses and strains can be expressed as space dependence
in the penetration direction. The key point of this method is that, in the un-deformed
domain, the stress o-at point P(r,z) (Figure 2.8) at a past time (T - dt) is equal to the
stress at point Q(r, z+vodt) at time T, i.e., point
Q is the
image of point P at a previous
time. Therefore, integration over time becomes integration over space, and the stress
at point P is obtained by an integration process considering all points below P until
the initial stress state of undisturbed soil is reached. Yu et al. (2000) applied this
method for penetration analyses in undrained clay, using von Mises and Modified
Cam Clay models, and the results for von Mises (shown in Figure 2.9(a) and 2.10(a))
are similar to those from Stain Path Method (Figure 2.9(b) and 2.10(b)). This
confirms the validity of the basic assumption in the Strain Path Method for undrained
clay, as the coupling between soil deformation and strength parameters is very weak.
2.4
Incremental Finite Element Method
Powerful numerical techniques, such as the finite element method, allow a
more comprehensive solution of the penetration problem. Finite element analyses
26
have been extensively applied for penetration problems in soil. De Borst and Vermeer
(1982), Sloan and Randolph (1982) and Griffiths (1982) carried out the first
computations for penetration analysis, in which a small-strain finite element method
was applied. In the small-strain analysis, the cone is introduced into a "pre-bored"
hole, with the surrounding soil still in its in situ stress state. An incremental plastic
collapse calculation is carried out and the collapse load is assumed to be the cone
resistance. High lateral and vertical stresses are built up around the shaft during the
penetration. The small strain analysis turns to underestimate the cone resistance by
ignoring the change of stress during the penetration process. These analyses yielded a
cone factor that is entirely dependent upon the shape of the penetrometer and the
adhesion between clay and penetrometer, but independent of the soil stiffness.
In order to introduce correct initial conditions, a large penetration distance, at
least several times the diameter of the cone, has to be simulated, in order to
realistically build up the shaft pressure produced by cavity expansion. Large strain
finite element analysis has been introduced into the story. One of the first
developments in large strain finite element analyses was published by Hibbit et al.
(1970).
Improvements were made by Bathe, Ramm
and Wilson (1975)
and
McMeeking and Rice (1975), who introduced Updated Lagarangian approaches.
Budhu and Wu (1991, 1992) and Cividini and Gioda (1988) modeled the frictional
interface between the cone and the soil with Updated Lagarangian large strain
analysis, by adding a thin layer of interface elements between the cone and the soil.
As pointed out by van den Berg (1994), in using these models it is necessary to decide
the new location of the boundary nodes after each calculation step (i.e., logic is
required to decide if a nodal point is on the tip or on the shaft of the cone.) and the
27
boundary conditions have to be modified as necessary. When friction between cone
and soil is taken into account, the procedure is even more complicated. As illustrated
in Figure 2.11(a), a vertical displacement increment 61 is imposed to the cone. The
deformed shape is shown in Figure 2.11(b). Figure 2.11(c) shows that the indices of
the interface elements are updated to avoid numerical problems due to element
distortion. The stress-strain states in the new elements are evaluated based on the
previous step, and the forces acting on the nodes of the new and old sets of interface
elements are determined. The difference between the two force vectors is applied to
the modified mesh, and an additional non-linear analysis is performed to re-establish
equilibrium. Finally the horizontal constraint on the first node below the tip (node #2)
is eliminated.
This technique presents a drawback related to the updating of the interface
indices, and to the subsequent equilibrium iterations. Besides the computation effort
required, the robustness of the whole numerical procedure is not clear.
To avoid frequent re-meshing, van den Berg (1994, 1996) used an Eulerian
formulation to model the penetration in both homogeneous and layered soils. In the
Eulerian framework, the mesh elements are fixed in space while the material convects
through the elements, thus the mesh undergoes no distortion due to material motion.
With this approach, van den Berg developed a general numerical methodology for
analyzing penetration in both homogeneous and layered soil deposits. This thesis
follows the approach presented by van den Berg using a Eulerian finite element code
DiekA, developed at the University of Twente (DiekA development group, 2000).
Details of this formulation are given in Chapter3.
28
a
Ij
Cf
'4H
I1IIII1I1
(b)
(a)
De Beer (1948)
Meyerhof (1951)
Terz aghi (1943)
Ck
a
cis.
(d)
(c)
Biarez etal (1961)
Hu. (1965)
Berezantzev etal (1961)
Vesic (1963)
Figure 2.1 Assumed failure mechanisms under deep foundations
(Durgunoglu and Mitchell, 1975)
29
d
$
/
/
/
8
A
wedge
plastic
zone
C
O
Ld
(a) y&
fntn17)
a
A
T
r
(b). Vesic (1977)
'a
C
(c). Salgado (1993)
Figure 2.2 Assumed relationship between cone resistance and cavity limit pressure
30
C
Q Ve
Nq
~rr~ .5M
600 400-
200 -
Srr
200
Irr
100
Irr
SO
100 -
I 3 ±20
(t0 -
I rr
10
40-
i
IS
2
20"
I
i
:Io
25"
I
1
3"
40"
w
u pc44k
Figure 2.3 Relation between Nq and 0 Peak (Vesic, 1975)
20.00
- -------- -
------------ ~
-----
-
- - - -- ---
-----
--- - -
----- -
18.00
---
16.00
14.00
12.00
----------------- - ---
-----
-U-
Vesic (1972)
-&-
Yu (1993) smooth cone
--
Yu
-4-
Baligh (1985)
Baligh (1975)
- - -- - - - - -- - --- r--- - - - - -- - - - ----------
(1993) rough cone
0
0 10.00
- - - - - - - - -- - - -r-.
8.00 1-
----
-
--
- - - - - - - - - - - -
-----
-
- ~- - -
- ------ + ---------
-+-
Meyerhoff (1961)
-4-
Ladanyi and Johnson
(1974) perfectly rough
------- L-----------------
6.001
4.00
-
---------
---------------------
2.00
0.00.
1.00
'
' '
' ' ' ''' i
100.00
10.00
1000.00
Rigidity index (Ir)
Figure 2.4 Comparison of N, from different methods
31
-1
~
Figure
~~
~
~
deomtoratr
~~
~ ~2. ~ ~rudsml
32
-T
1
-
Irdceie(aih
95
I Ir 1~
~I~i1i~I
If
6'TIP
w
TIPt
1*~
I
~
ii
rhfill
------ ........................
~~4:2U,
4-j
I
~
i
~41~
-
L
Figure 2.6 Predicted deformation pattern around a 60-degree cone (Levadoux, 1980)
33
I
I
K
m 'M
rrT-1
1
m
-5
0
5
10
Horizontal Position, r/R,
Figure 2.7 Predicted deformation pattern around a FMMG piezoprobe (Sutabutr,
1999)
34
D
E
VO
Soil Mass
Position of cone tip
at a past time T-dr
Positionof cone tip
at currenttime T
4
1
I
C
N
N
N
N
N
undeformed location
P(r,Z)
N
N
P'(r',z')
deformed location at time T
N
N
Q(r,z + vodt)
Z 1
A
,
R(r, z + 2vdt)
* S(r, z + 3vodt)
B
Figure 2.8 Steady state behavior (Yu et al., 2000)
35
10
86420-2
Approximately
corresponds to
the limits of the
plastic zone.
.4-6
0*
-8
-10
0
2
4
6
8
r/a
10
12
14
Figure 2.9(a) Typical contour plot for strain y (Yu et al., 2000)
10
a
6
4
2
2
6
4
10
a
IT
10
r
0
8
10
I--
8
-
30*
6
60*
4
S-
4
z
2
SI
z
0
z
z
-
-
----
-
90*
-
0
5000
-
50000
U -2
-2
z
2000
-
w
0
1000
-4
N
E.500%/hr
-4
20
.6
-6
IS
-
-a
#50*
-10
10
a. Strain Rates,
and U=2 cm/sec
8
-
S Gn NT
6
4
180,
*10
2
0
2
RADIAL DISTANCE, r/R
2, for R-1.78 cm
b.
4
6
8
10
Strain Levels; E
Figure 2.9(b) Contour plot for strain y during simple pile penetration (Baligh, 1985)
36
108
t.C-
64
-6
F10
0
4
2
10
8
6
12
14
rna
Figure 2. 10(a) Typical contour plot for excess pore pressure (Yu et al., 2000)
20
15
15
10
5
~~1
0
5
10
PC).
- 20 C
15
2R
Au/
15
0.6
0.8
I0
10
1.0
1.5
5
2 .0
2.0
0
z/R
0
/
1.0 0.8
0.6
5
-5
/
0.4
-lot-
0--.2
-10
-15
-15
Fig. 4b
Fig. 4a
20
15
10
a. Bilinear modelling
0
5
15
10
5 Hyperbolic
modelling
b.
20
Figure 2.10(b) Excess pore pressure due to undrained simple pile penetration
(Baligh, 1985)
37
PILE
INTERFACE ELEMENTS
VtT
1
1
)IL
SR,
S
b
a)
Figure 2.11
Schematization
of Updated Lagarangean
FE model to simulate
penetration including interface friction (Cividini and Gioda, 1988)
38
CHAPTER 3
EULERIAN FORMULATION FOR FINITE ELEMENT
ANALYSIS OF PENE TRATION PROBELMS
An Eulerian framework has been used in this research to model penetration in soils.
The analyses are carried out using DiekA software developed at the University of
Twente (DiekA development group, 2000). The code was originally developed for the
simulation of metal forming process and was adopted to soil penetration by van den
Berg (1994, 1996). The most important features of DiekA include: (1)
the arbitrary
Lagarangian Eulerian method, with which the displacement of the element mesh can
be disconnected from the material displacement; (2) capability to calculate problems
concerning large displacements; and (3) interface elements which describes the
contact conditions.
The Eulerian framework is a special case of the so-called Arbitrary
Lagarangian Eulerian (ALE) method. In the ALE framework, the coupling between
material points and mesh points is released, in order to keep the mesh regular and to
conserve the compatibility of the mesh with the boundary conditions. An Eulerian
analysis represents the special case, where the mesh points (element nodes) are fixed
in space. The Eulerian finite element formulation is briefly outlined in this section.
3.1
Basic Equations
The basis of the method is the equation of virtual work. According to the principle of
virtual displacements, the equilibrium of the body requires that for any compatible,
39
small virtual displacements (which satisfy the essential boundary conditions) imposed
onto the body, the total internal virtual work is equal to the total external virtual work.
This reads:
SW = cY d YIdV - JpF,,vjdV - fTSvjdS = 0
V
V
S
(3.1)
where:
6W
virtual power
V
current volume of the material
o;,
real (Cauchy) stress (stress in current(deformed) configuration)
&d;
virtual rate of deformation, d. = I1vii + v,)
p
mass density
Fi
force per unit mass
Sv
virtual velocity
S
current boundary surface
T,
surface force per unit area
The first term is the internal virtual work, which is equal to the actual stress oju going
through the virtual deformation &di;. The second and third terms in Equation (3.1) give
the external virtual work, done by actual body forces and surface forces.
40
The second basic equation is the constitutive equation. For time independent
and elasto-plastic material properties and isothermal conditions, the constitutive
equations can be written in the rate-type formulation as follows:
&o =-O', +D kldl
P
(3.2)
where:
6
objective stress rate
0
dkl
rate of deformation, dkl =(vk,l +vk)
2
Dikl
a fourth order stiffness tensor depending on material parameters and
current state
A widely used stress rate satisfying objectivity is the Jaumann stress rate. The
Jaumann rate of change is related to the material rate of change by:
0Y
where co, =
2
Y~>
±ikO
v
k]Wik
(3.3)
k]
-vj ) is the skew-symmetric part of the velocity gradient, which is
associated with the rotation of the material.
In the case of an Updated Lagarangian formulation, Equation (3.1)
is
transformed to a known reference configuration and then linearized to obtain
equations that incrementally can be solved. This transformation can be omitted here.
Taking the material time derivative of the virtual work, Equation (3.1) can be
rewritten as follows:
41
dW = da
5d dV
dt dt dt fdt
d
pF vidV-+ JTSvddS = 0
fdt
I34
V
Ss
(3.4)
In a large deformation analysis, the integration area is not constant, so the material
rate of change of an integral with changing integral area has to be considered. This
has the general form as below:
+[fdV] =
J['+fvklv
dt
V
V(3.5)
Substitute this into Equation (3.4):
sw = f1&-id
+9 (sd)±
WY+
9dky
, dkk
V
V
[(p4, v+
-
pFSvj dk
V
V
[tv + T.ijv.a},, S = 0
-
S
(3.6)
The index a denotes the surface components of the velocity. The time derivative of
the virtual rate of deformation and the time derivative of the virtual velocity vanish.
The Cauchy stress rate de can be obtained by combining Equations (3.2) and
(3.3):
doi =Dijkdkl
Uik ok1 + CoikoakJ
-oiidkk
(3.7)
. Substituting Equation (3.7) into Equation (3.4) results into:
where dA = p
42
LDijkldkl&di
6W =
- 2 akIdikSdiy
+ CiVkJjVki
hV
V
f [Plsv
iVi
V
T &i
a,aS
=
(3.8)
S
With finite element method, the velocity field within an element is usually
expressed in the nodal point velocities (represented by superscript "N"):
v1
=Y
N
*VNN
(3.9)
Sv =IV/M*&M
M
where,
iN
(3.10)
are the interpolation or shape functions for velocity component, vi, at
nodal point, N, of the element. Following this, the rate of deformation d,, which is
the symmetric part of the velocity gradient, can be obtained as follows:
BN
dkl=
*VN
(3.11)
N
where:
BN
_L*V
2[
N + L*
N]
(3.12)
BN is a third order tensor. L is a matrix that contains differential operators that relates
the displacement field to the strain field. Similarly, the virtual rate of deformation is
defined as:
SdkI = jBm *
M
M
(3.13)
43
Applying Equations (3.12) and (3.13) to Equation (3.8) leads to the following
discretized form of the equilibrium equation:
SW
[1vM(SMN +SS2
=
Mk
NvN
N,M
M
(3.14)
M
where:
S 1M N
((Bm )(D,
=
- 2o - )BNdV±
f[(L V,1 M
N
(Ly,1 N)}V
S
=m
(3.15)
V
V
(3.16)
f(vimpft'iv + fJ(Vli
V
S
(3.17)
Equation (3.14) should equal zero for any value of the virtual nodal velocities
according to the virtual work principle. This yields the following equation:
S MN *
m
N
(3.18)
where:
S
M
N
S
M
N + S
M
N
The matrix S2mN is non-symmetric and vanishes if there is no surface force. The
matrix SjmN is symmetric for associated plasticity and non-symmetric for nonassociated plasticity.
If the state of stress and boundary conditions are known at time t, the nodal
point velocities can be solved from Equation (3.18).
44
After that, the material
displacement increments of the nodal points within a time increment At are
approximated by:
(3.20)
N =VN * At
Au
AU 1.
3.2
Convection Item
The formulation presented in the previous section is identical to the updated
Lagarangian method. For the Lagarangian procedure, the nodal points are coupled
with the material points. Therefore, the new coordinates and total displacements are
obtained by adding Au/N to the initial coordinates and total displacements respectively
at the beginning of the step.
Uncoupling the material and nodal point displacements
implies that
convection has to be taken into account in order to update the state (stress,
displacement, etc.) at the nodal points. To calculate the convection item, Huetink
(1982, 1986) presented an approach, which basically is to introduce an additional
continuous stress and strain fields by interpolating nodal point stresses and strains.
The convective terms are obtained as a product of gradients of those additional fields
and the displacement increments.
The formulation will be derived in a more general ALE framework, for which
the Eulerian method is a special case with fixed nodal points. As illustrated in Figure
3.1, the displacement of nodal point N is given by Ax/N. The displacement of the
material point that coincides with nodal point N at the beginning of the step is
represented by AuaN. The difference between the new locations of the nodal point N
and the new location of the material point is:
45
AyN
(3.21)
=AX7-Au
The new location (at time t + At) of a material point B, which coincides with
an integration point A of an element at time t, is found by:
x,(B)= x,(A)+
V/ N (Xi(A))*AU
N
The new location of the corresponding integration point C, is:
x, (C)
= x,
(A)+
VN
(x, (A))*
(3.23)
A
Now the stresses and strains at the new integration points (points like C) need
to be determined. The stresses at material point B can be calculated in the same way
as in the Lagarangian procedure:
t+Az
au (B)=
W+f,(A)+
(3.24)
fdt
where 6. is determined from Equation (3.7). The stresses at the new integration point
C are found by:
(3.25)
CU (C)= 0 (B)+ 0%k (B)Ayk + O(Ayk )2
The difference between stresses at point B and point A is of order AuaN. And,
AyN and Au/N are of the same order. Therefore, Equation (3.25) can be written as:
U, (C)= 0 (B) + -,k
(3.26)
(A)Ayk + O(Ayk )2
To calculate the covariant derivative of stress,
0%,k'
in Equation (3.26), the
following approach is adopted (Huetink, 1982). Firstly the stresses at the nodal points
46
are calculated as an average from all elements that are connected to the nodal point.
After that, an additional stress field at time t within an element is defined as:
Ui* (X'k)=NX
k
(3.27)
ti
where, the superscript "t" is the current time step, the asterisk at the left side means
these stresses are introduced and do not exactly coincide with stresses at the
integration points.
U';N
i stress at nodal point at time t.
With Equation (3.27), the derivative of the stresses is 1 st order approximated
by:
a -ij,k
=
V
(3.28)
Xi)k U
Substituting Equation (3.28) into (3.26) and neglecting higher order terms of
Ayk, results in the stresses at the new integration points:
o>(C)
j-(C) = UY (B) +
Vf N
vM I()yNxAk
(XA)AyMf
AX,k CtiMAk
(3.29)
The strains at the new integration points can be solved in the same way. If the nodal
point displacements are chosen to be equal to the material displacement, i.e., Ayi
N
_
0, Equation (3.29) yields:
-ij (C)= o-(B)
(3.30)
It reduces to a Lagarangian procedure, which has element nodes and material points
coupled.
For Eulerian procedure, the nodes are fixed in space, i.e., AyN = -AUN. If this
relation is substituted into Equation (3.29), the result reads:
47
,(C)= U4 (B)-
aNNV
(3.31)
M
The second item in the right side of Equation (3.31) is the so-called convection item.
In the Eulerian formulation, point C is the same as point A, since the nodal points do
not move. Combining with Equation (3.24), Equation (3.31) can also be written in a
more general format as:
t+At
07 (x,t + At)= a (x,t) + f&,dt_V/N(X),ko7
NV/M(x)Au
M
(3.32)
where x is the coordinate for A, or the fixed nodal point.
Huetink (1986) observed that this formulation gave numerical instabilities
depending on the size of the displacement increments. In order to avoid instabilities,
the following equation is introduced by Huetink (1986):
0'jj(x, t+ At)=07y(x -Au~t)+
t+A
fddt
(3.33)
where Au is the displacement of the material particle, which originally coincides with
the nodal point, during time increment At. It should be pointed out that Equation
(3.32) is a first-order spatial Taylor series expansion of Equation (3.33). Huetink
(1986) observed that using Equation (3.33) results in numerical diffusion (i.e., over
smoothing). Equation (3.32) is an element solution, while Equation (3.33) is a nodal
solution. If the continuous field is made by averaging in the nodes, the peaks in the
solution will be averaged out. This is numerical diffusion. On the other hand, if only
the element gradient is taken and no average on the nodes, the solution becomes
48
unstable and starts to oscillate. Therefore, in the present implementation, a weighted
sum of Equation (3.32) and (3.33) is adopted, as follows:
t+At
ai-(x, t + At)= (1 - a a-(x, t) + fd-,jdt + au-,(x - Au, t)
-(-a)[VN
X,k
t' N
xAu
(3.k4
As Van den Berg (1996) suggested, "a reasonable range for a appears to be":
Au < a
le
2
(3.35)
le
where le represents the element length. In the finite element program DiekA, the
weight factor a is automatically taken into account at the element level. The weight
factor depends on the convective displacements, compared to the element size (the
Courant number), as shown in Equation (3.35). In case of small Courant numbers,
mainly the element solution (Equation (3.32)) is taken. Otherwise, the solution is
more close to the nodal solution (Equation (3.33)).
3.3
Layered Deposits with Eulerian Approach
The geology of the subsoil generally consists of layered deposits. With Eulerian
approach, modeling the penetration process in layered deposits requires that the
material particles stream through the fixed mesh including its constitutive behavior. In
DiekA, an extra state variable, material index, is assigned to each integration point, to
store the material type associated with the node.
For analysis with layered soils, at the end of each increment, following the
step of mapping stresses and strains to nodal points, as mentioned in last section, the
49
similar convection is also applied to the total displacements of the nodal points. The
procedure is illustrated in Figure 3.2 with a one-dimensional example (van den Berg,
1994).
As shown in Figure3.3, the material particle reaches point P (xpN, ypN) at end
of certain increment. To update the material type at point P, the original location of
that material particle (xp', y,") needs to be calculated. This is done by subtracting the
convected total displacements from the co-ordinates of the fixed point P.
oN
x
=x
yo
= y
~
-
4-
nincr Au(i) 1-
Ix
P0~~()1
aAu(i
nincr
1-
-Au-(i)
(3.36)
where nincr is the total number of incremental steps. For an element, the average
values of xj' and yfO of all nodal points connected to the elements are calculated:
r
nnod
nnod
y
x
0
0
( Xelemelem
j=1
j=1
nnod ' nnod
(3.37)
where nnod is the total number of nodes for the element.
Following this calculation (x
y ,,) must be checked to establish which
material domain it belongs to. If the material domain changes after the step, the
material index should be modified to reflect the update. In the next step, the new
material properties, according to the updated material index, will be used for this
element, in the definition of the stiffness matrix and other material related routines.
50
Besides the key Eulerian formulation presented in this Chapter, Appendix A
gives more details about DiekA. A typical input file is presented and explained
including the keywords in DiekA, how to define nodes, elements, boundary
conditions, material type and properties, convergence criterion and output control, etc.
51
B
A
P"
C
P
pN
Figure 3.1 Node and material point displacement at one step
initial domain
Y
of material B
X
P(
XP conv
(P,
Yp Conv
Y
(X dun Y clem)
initial domain
of material A
X
Figure 3.2 Definitions for moving material boundary through fixed FE mesh
(Van den Berg, 1996)
52
CHAPTER 4
PENE TRATION ANALYSIS IN CLAY
This chapter describes applications of the Eulerian finite element program DiekA for
simulated undrained cone penetration in clay. The numerical stability and accuracy of
the finite element model are first established, in terms of the mesh coarseness and step
size. The mechanisms of penetration are interpreted from details of the penetration stress field for one base case analysis. A parametric study is then performed to
establish predicted tip resistance factors. Finally, two examples of penetration in a
layered clay profile are explored to illustrate the capabilities of the Eulerian finite
element method.
4.1
Finite Element Model
2
Figure 4.1 shows a typical cone penetrometer with a cross-section of 10cm , or cone
diameter of 35.7mm, and a tip angle of 600 which is generally accepted as standard
(ISSMFE, 1977 and ASTM, 1979). The penetrometer is pushed into the soil at a
constant rate of 2cm/sec.
Figure 4.2 shows the finite element model for the analysis. The penetrometer
is modeled as a fixed boundary and assumed to be fully smooth (i.e. zero interface
friction is assumed). The dimensions of the finite element mesh are normalized to the
cone radius R (18mm). The elements near penetrometer have the smallest sizes, as the
stress/strain field of this region will be greatly changed during the penetration. The
smallest element has height as 0.43R, width as 0.38R. The penetration process is
53
modeled by applying incremental material displacement at the bottom of the mesh.
Material streams upward through the mesh. The soil upward movement is equivalent
to the downward movement of penetrometer. "Old" material will be gradually pushed
out from the top of the mesh as the penetration proceeds.
A uniform initial stress state is assumed, as the gradient of vertical stress is of
secondary importance for a deep penetration problem (i.e., soil unit weight y= 0). The
uniform initial stress is introduced by adding a vertical distributed load ao, at the top
boundary. The lateral pressure ratio K 0 , defined as the ratio of initial horizontal stress
cho and
0
a,
, is related to the Poisson's ratio v as Ko =
V
1-v
One important factor in the penetration analysis is the width of the calculation
area, or width of the mesh. Unrealistic stress will build up if the diameter of the
axisymmetric model is too small. As reviewed by Yu (1998), the chamber size effect
has been recognized for long, while the effect is difficult to evaluate and varies with
different soils. As shown by Salgado (1993)
with a cavity expansion method,
significant chamber size effect still exists even when the chamber to cone diameter
ratio is up to 100. The other belief is that the chamber size effect can be neglected if
the chamber to cone diameter ratio is above 60 or 70 (Ghionna and Jamiolkowski,
1991; Mayne and Kulhawy, 1991). Based on these previous investigations, the width
of the axisymmetric model is chosen as 100 times of the cone radius, as shown in
Figure 4.2.
In order to achieve a smooth stress state around the cone shoulder, the tip
geometry is slightly altered by introducing a circular arc with a radius of curvature
54
equal to 3 times the cylinder radius at the cone-cylinder transition (Figure 4.3)
following the suggestion of Levadoux (1980), "the effect of the smooth transition
between cone and cylinder is believed to alter the deformation and strains of the soil
in its vicinity but not at some distance away." For nodes along the cone face,
movement is only allowed along the cone surface, as shown by arrows in Figure4.3.
Pore water pressure is not considered in this study. The undrained behavior of
clay is modeled by the linear elastic perfectly plastic Tresca model with parameters
Young's modulus E, Poisson's ratio v and the undrained shear strength s.
4.2
Numerical
Accuracy
and
Interpretation
of
Penetration
Simulation
4.2.1
Numerical Accuracy
Several test runs have been performed to validate the proposed finite element model,
by studying the sensitivity of mesh coarseness and step size.
In order to study the effect of coarseness of the finite element mesh, two
additional calculations have been carried out. The finite element model in Figure 4.2
has 1944 elements. Further analyses have been carried out using a coarser mesh with
504 elements and a finer mesh with 5184 elements (Figure 4.4). For these test runs,
Tresca model is applied with the following parameters: E = 2400kPa, s, = 20kPa, v=
0.499. The initial vertical stress oao is 35kPa. KO= 1.0, therefore,
cho
= 35kPa.
The calculated pressure-displacement curves are shown in Figure 4.5. The
difference between 504 and 1944 elements is about 20%, while the difference
55
between 1944 and 5184 elements is negligible. Based on this, the 1944-element mesh
is used in all the following calculations.
Besides the mesh coarseness, sensitivity of solution to step size is also
investigated by comparing results from analyses with varying step sizes. Numerical
analyses with four step sizes, 0.001m, 0.0005m, 0.0001m and 0.00001m, have been
performed, with Tresca model and the same material parameters as above. The step
sizes 0.001m, 0.0005m, 0.0001m and 0.00001m are 0.14, 0.07, 0.014 and 0.0014
times the smallest element size. Figure 4.6 shows that the cone reaction pressure
converges, as step size gets smaller. The difference between 0.0001m and 0.00001m
is less than 2%. The reaction force for step size 0.001m is about 33% higher than that
of step size 0.00001m. Both the reaction force curves for step size 0.001m and for
step size 0.0005m show some numerical oscillations when penetration distance is
within one cone radius. The initial numerical instability probably explains why these
two cases reach a much higher steady reaction force. The penetration distance needed
to reach a steady state cone reaction force is about 6R, and not affected by the step
size.
As the step size is reduced, more computation time is required to reach the
same penetration distance. In consideration of both accuracy and computation
resource, for following analyses the step size is chosen as 0.000lm.
4.2.2
Interpretation of Penetration Simulation
Detailed results from the numerical penetration analysis are discussed in this section
in order to provide a clear interpretation of penetration mechanisms for the Eulerian
finite element analysis. A base simulation is carried out with the same material
56
parameters described previously (E = 2400kPa, v = 0.499), i.e, linearly elastic
perfectly plastic
I = -
soil model with Tresca yield (s, = 20kPa).
= 40. The initial vertical stress aro is 35kPa, horizontal stress
£
SU
Rigidity index
2(1+v)S,
cho = 35kPa.
From the cone reaction pressure vs. penetration distance curve (Figure 4.7), a
steady state is reached with respect to the reaction pressure, after a penetration
distance of about 6R. However, steady state conditions with respect to the stress/strain
distributions around the shaft of the penetrometer are only reached at
Av
R
=
106R (i.e.,
the penetration distance equals the height of the mesh, such that there is a complete
replacement of material throughout the entire finite element model). Stress/strain
contours are shown for this steady state.
Figure
(
, =
j3£
4.8(a)
dt
presents
,
i
contours
of the
equivalent
plastic
strain .,
is the plastic strain increment). The elastic-plastic
boundary is located approximately r R
=
10 away from the shaft and extends to a
distance ZR ~ 4 ahead of the tip. Plastic deformations occur within a 1 OR cylinder
centered at the penetrometer centerline. Soil particles outside of this region experience
only elastic deformation. As shown in Figure 4.8(b), along the shaft, the plastic strain
increases sharply at the tip, where the soil particles are pushed away to make space for
the penetrometer. Along the cone face, plastic strain starts to reduce and it goes to
approximate uniform for
ZR
> 30.
57
Contours for the normalized octahedral normal stress increments
AO-
"' are
UvO
plotted in Figure 4.9(a). The octahedral normal stress increment along the shaft is
shown in Figure 4.9(b). Figure 4.9(c) shows the radial distribution of
various elevations. At initial stress state, as
P
= 0.499 yields KO = 1, the initial
equal to avo. For soil elements outside of the contour line
at
"r'
UvO
UOC,
is
"u' = 0, the octahedral
UvO
normal stresses remain unchanged from the initial condition. The octahedral stress
increases dramatically as the tip is approached (Figure 4.9(a) and (b)). It reaches the
maximum value of about 4.4 times the initial vertical stress, or initial octahedral stress,
and then decreases behind the cone base. Along the penetrometer shaft, the octahedral
normal stress increment reaches a uniform value of more or less 1.5avo at
>6.
The uniform distribution of Aooc, behind the tip can also be seen in Figure 4.9(c): the
curves of stress distribution along r-direction overlap for z/R = 10, 20, 30. The
octahedral normal stress increment Aqoc, gets smaller when the top boundary is
approached (Figure 4.9(b)). This is due to the boundary condition assumed at the top
(i.e., uniformly distributed pressure).
Figure 4.10(a) shows the contours of the normalized shear stress
UvO
. Figure
4.10(b) shows the distribution of shear stress r,z along the shaft. Ahead of the cone,
negative (sign convention shown in Figure 4.10(a)) shear stresses take place, because
soil elements next to the axis (small r) are pushed downward with respect to the soil
elements further away from the axis (large r), as the penetrometer moves downward.
58
On the other hand, the shear stress direction is reversed behind the cone tip. This is
because the soil elements from the tip move upward, and this forces the soil elements
next to the shaft to be pushed upward, with respect to the outer elements. The
normalized shear stress reaches a minimum r
= 0.3.
= -0.5 and a maximum
v0
Uv0
As the penetrometer is assumed fully smooth, the shear stress r,z should completely
vanish along the shaft. The nonzero values of Trz above the cone shoulder (Figure
4.10(a) and (b), are probably due to extrapolations from the Gaussian integration
points.
The contours for normalized vertical stress
,
radial stress
"
,
and
UvO
UvO
are illustrated in Figure 4.11 through 4.13. The variations
circumferential stress
UvO
of the three stresses along the shaft are shown in Figure 4.14. Figure 4.15 through
4.17 present the distribution of the stresses along r-direction at different elevations.
The stresses go through three distinct phase: (1)
Starting from ZR = -5 , vertical
stress a,, increases sharply, while the radial stress orr and circumferential stress a8
drop down by about 40-50%. The relative order of the three normal stresses is
UZ > coo ~ a, ; (2) Along the cone face, the stresses are approximately uniform
(Figure 4.11 through 4.13). The relative order is changed to U, >
",
> o-"" (Figure
4.14); (3) Behind the cone base, the three stresses decrease sharply with a distance of
5R (Figure 4.14). A steady state stress distribution appears to be approached at
ZR>
5
-
10 (Figure 4.14 through 4.17). From Figure 4.14, the relative order of the
59
three normal stresses is
azz . Soil coming from below the tip of the cone
rr > coo
is radially compressed when arriving along the shaft of the cone.
Another parameter of interest is the cylindrical expansion shear stress qh,
defined as qh
rr -
. Figure 4.18(a) shows contours of
2
shows the variation of the
qhv
. Figure 4.18(b)
avo
along the shaft. The distribution of the stresses along
Cv
r-direction at different elevations is plotted in Figure 4.18(c). The cylindrical
expansion shear stress increases dramatically as the cone tip is approached. A steady
state appears to be reached at
ZR>
10. Figure 4.18(c) shows a maximum cylindrical
expansion shear stress is reached at approximately
R = 5 for all elevations. Contours
of the circumferential stress Uor (Figure 4.13) shows that at r = 5
10 ao8 becomes
smaller than its initial value (initially, aoa = ovo=35kPa). Thus the difference between
q, and ooo increases, or qh goes up at r R = 5.
4.3
Effect of Soil Stiffness
It is well established that the shear modulus can affect prediction of tip resistance
(Vesic, 1972; Baligh, 1985). In this research a series of analyses have been performed
varying the rigidity index I, = -
SU
(after Vesic, 1972). Table 4.1 presents the
characteristic data for six simulations. Only Young's modulus is varied among these
analyses.
60
The reaction pressure vs. penetration distance curves for all six cases are
compared in Figure 4.19. It is clear that the cone resistance increases with rigidity
index. The penetration distance to reach a steady state with respect to the cone
resistance, increases as well for stiffer clay. For rigidity index Ir= 4 and 8, the steady
state in terms of cone resistance is arrived within AR = 2. For Ir = 40 and 83, the
cone resistances reach steady state after about
R = 6 and 10 respectively.
In this research, as soil is assumed weightless (i.e., y = 0 ), there is no effect of
UVO on the cone resistance qc. A series of analyses (Table 4.3) have been performed
varying the initial vertical stress
0ov.
cone resistance. Therefore, NC = !-.
Table 4.3 show that the a,O does not affect the
Values of the resulting cone factor Nc are listed
SU
in Table 4.1. The cone factor decreases from 10.75 to 3, as Ir decreases. Figure 4.20
presents the relation between the cone factor and the rigidity index Ir, as derived from
previous investigations and calculated in this study. The cone factors are lower than
the Strain Path Method solutions for the rounded simple pile (Baligh, 1985). The Ne
value is very close to the cavity expansion solution by Yu (1993) for a smooth cone,
with Ir > 200.
4.4
Layered Clay
The subsurface profile usually consists of layers with differing strength properties.
Piezocone penetration signatures can become quite complex in layered profiles, due to
influence of underlying layers (located below the current penetration depth). This
leads to questions regarding the reliable interpretation of q, values based on
61
simulations for homogeneous soils: what is the penetrating distance into a layer, in
order to arrive the new steady state reaction force for that layer?
In this section, example penetration analyses have been carried out. The finite
element model is similar to the one used for homogeneous clays in previous sections.
The only difference is that a two-layer system is considered. The boundary between
the two layers is located initially at z/R = -11.4 (Figure 4.21). Both clays are modeled
by Tresca model. The strength parameters and initial stress state are summarized in
Table 4.3. For all four layered cases, the top clay is the same (i.e., clay4 in Table 4.1),
while the strength parameters of the top clay varies. For casel through case3, the top
clay and bottom clay have the same undrained strength s,, and the stiffness E varies.
For case4, the bottom clay has a stiffness E and undrained strength s, twice as those
of the top clay, therefore, the rigidity index I, (= G/s) for them are the same.
The pressure-displacement curve for case 1 is presented in Figure 4.22. Besides
the
curve
for
the
two-layer
system,
the
pressure-displacement
curves
for
homogeneous clays (clayl and clay4) are also given. The pressure-displacement curve
starts in the same way as that of clay4, the top clay. Then it starts to deviate from the
clay4 curve at about zIR = 5. Bearing in mind that the boundary is initially located at
z/R = -11.4.
This implies that for this case, the tip senses the existence of the bottom
layer 5-7R in advance. The steady state reaction force is reached after penetrating
more or less 40R. That means a penetration distance of 28.6R (= 40R - 11.4R = 0.5m)
is necessary for the penetrometer to develop the steady state reaction force in the
bottom layer.
62
The pressure-displacement curves for all four layered cases are shown in
Figure 4.23. Comparing the curves for casel through case3, with the decrease of the
bottom clay stiffness, the penetration distance to reach a steady state reaction force in
the bottom layer, is reduced to about 20R and 1OR for the two additional runs. The
pressure-displacement curve for case4 shows that q, (= 260kPa) for the bottom clay is
twice of that for top clay. The two layers actually have the same N, value. Case3 and
case4 have the same stiffness for the bottom clay, only the undrained strength s' is
different. The pressure-displacement curves show that the penetration distance to
reach a steady state cone reaction force is more or the less the same for case3 and
case4, zIR = 10. This implies that the penetration distance to reach the steady state is
dependent on the stiffness E of the bottom clay, not the undrained strength s.
63
E
su
UvO
I,
Yield
Cone
Cone
(kPa)
(kPa)
(kPa)
(G/Su)
stain
resistance
factor
Ey (%)
qc (kPa)
Nc
Clayl
24000
20
0.499
35
400
0.1
215
10.75
Caly2
12000
20
0.499
35
200
0.2
196
9.80
Clay3
5000
20
0.499
35
83
0.5
162
8.10
Clay4
2400
20
0.499
35
40
1.0
133
6.65
Clay5
500
20
0.499
35
8
4.9
78
3.90
Clay6
250
20
0.499
35
4
9.8
60
3.00
'-
-
Note:E -
Table 4.1 Material parameters for clays with varying rigidity index
E
Su
(kPa)
(kPa)
1'
VO
I
Yield
Cone
Cone
stain
resistance
factor
(G/Su)
E (%)
qc (kPa)
Nc
(kPa)
Clay4
2400
20
0.499
35
40
1.0
133
6.65
Clay42
2400
20
0.499
0
40
1.0
129
6.45
Clay42
2400
20
0.499
70
40
1.0
136
6.8
Clay43
2400
20
0.499
140
40
1.0
138
6.9
Table 4.2 Material parameters for clays with varying avo
64
Casel
E (kPa)
v
su (kPa) uos (kPa) Ko Irtop/Irbottom
Top (clay4)
2400
0.499
20
35
1
Bottom (clayl)
24000
0.499
20
35
1
Top (clay4)
2400
0.499
20
35
1
Bottom (clay2)
12000
0.499
20
35
1
Top (clay4)
2400
0.499
20
35
1
Bottom
4800
0.499
20
35
1
Top (clay4)
2400
0.499
20
35
1
Bottom
4800
0.499
40
35
1
40
400
Case2
40
200
Case3
40
80
Case4
40
40
Table 4.3 Material parameters and initial stress for
penetration analysis in layered clays
65
IN
,35.7
E
E
mm
friction sleeve
W",
t')
porous element
£1
I
L
60c
cons
tip
Figure 4.1 Piezocone tip geometry
D
44C
II1T1TIF~rF-iZZIZIIZTIZIZZ]ZZI
44444
40
30
20
10
0
N
-10
-20
-30
-40
-50A 0
25
50
r/R
75
10( B
prescribed incremental displacement
Figure 4.2 Finite element mesh for penetration analysis
66
4
2
N
0
= = c
p
1
2
r/R
3
4
Figure 4.3 Rounded cone-cylindar transition part
67
I
50
40
30
20
10
S0
-10
-20
-30
-40
-50
-
0
I
-,
25
II
II
50
r/R
~
75
100
(a) Coarse mesh
50
40
30
20
10
0
-10
-20
-30
-40
-50
r/R
(b) Fine mesh
Figure 4.4 504 elements and 5184 elements FE mesh
68
I
140
--------
--------
130
e---
110
go
- -------------------- ---------- ------
---
- -
S 1 10
------- -
--------- --------------------
-
- -
---
---- ---------- --------------------
--
80
- ------- ---------- ---------- -----------
---
70
--------- -------- +-4 ------------------ 4----------------------- ---------- ----
-----
60
0
I
-------- -------- *- ---
-
----- --- --
5184 eierpents
--- -
---------
---------- --------------
---
S 30
--
U0 20
i i
--- -I ----- - ------ ---i -i -- -------4
2
0O
10
8
6
Normalized penetration depth, z/R
Figure 4.5 Pressure-displacement curves for 504, 1944 and 5184 elements
180
160
--
-
- - -
140
120
0
001
--
-------
- -
--- -------
---
- -----
L
1:100.001m/IATp
0------------I-----------
80
-
S 60
-
-
-
0
4-
4----
20
-=
05nkstep
T
I
- - --
-
-~e
I-
0I 1 INstep
I
fl00t~step - - r--_
--
- - - - - - - -4
0O
2
6
4
8
10
Normalized penetration depth, zIR
Figure 4.6 Pressure-displacement curves for varying step sizes
69
1
--
10
- - -
130
100
-
-
-
-
-
--
- --
-
-
-
--
-
-
--
--
-
S 90
-
-
-
-
-
-
- -
-
--
S 110
.4
- -
- -
120
-
70
- - -+- - - -
60
-
--
50 --
-
3
- -,--
--
- ---
- --
30
- - - - - - - - - - - - -- - - - - - - -
--
---
40
0
0
T-
-
--
- --
--
- --
-
-- I-
0
-
2
-
--
--
-1
4
--
6
-
-
--
8
-
- 10
Normalized penetration depth, z/R
Figure 4.7 Pressure-displacement curve for basic calculation
70
20
0P
1.4
1.2
1
0.8
0.6
0.4
0.2
0
10
iN
0
0
5
10
r/R
15
20
Figure 4.8(a) Contours of equivalent plastic strain e,
50
50
40
40
-------------
--------
30
20
20
------
10
----
----------------
-------------
10
.4
J--.
0
-10
-10
-20
-20
-30
---------30
------ I - ---- ------- I-20 --------
-40
-40
- -------
-----
- -------
------ I
-----
--
--
--
-- -----I --- + --------------
-------- ---------
-
4-----------
-----
----
-
-50
0
0
0.25
0.75
0.5
1
1.25
1.5
Equivalent plastic strain, c,
Figure 4.8(b) Equivalent plastic strain c, along the penetrometer shaft
71
20
$Aoct
/v0
3.4
3
2.5
2
10
1.5
101
0.8
0.6
0.4
0.2
N
0
0
0
5
10
r/R
15
20
Figure 4.9(a) Octahedral stress increment Ao,, /o'
50
50
40
40
30
30
20
20
10
10
-- -
0
-
---- --- ----
--- --
- - -
--
----
--- ---------
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
e- -
-- -- ----------- ---- --
-
- - - - - -- -- -
- + - --
-
- - -- ---
--
-- ---------
-- -- ----
--
- -- - -- ---
- -- -
-
- -- -- --
0
1
2
3
Normalized octahedral stress increment, Aa0 ,,/
Figure 4.9(b) AoT
0 , /or
72
along shaft
-
4
a0
3
I-
I
~
II
--
I
I
1
z
0I--
-0.5
-
05
--
z/R=2.23 c'one shoulder)
z/R=10
-4-~--z/R=20
z/R=30
-~
- I
I
------------ -------------
S 0.5----------co
-
T~
zIR=O (tip)
-
-
-
10
r/R
Figure 4.9(c) Radial distribution of Aa 0 ,, /c00
73
15
20
at various elevations
20
Trz
/
v
0.4
0.3
0.2
0.1
0
10
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
'iy
0
-1
0
5
10
15
20
r/R
7,-z<0
7,- >0
Figure 4.10(a) Contours of shear stress r, /a,0
50
40
40 ----------- + ------------ ------------ + ------------
30
-- -- -- -30 - - -- -- -- ---- ------
20
20
10
10
0
----------- IT-----------------
- -- - +
-10
-20
-20
-30
-30
- - - - -
----- - - - - -- ---
---
-
~-- - - - ~- -
- ~ - - ~-
0
-10
-------- - --
----
- - - ---------- -
- - - - - - -+I
- - --
----------- + ------------------------ + -----------
e----------+-------------------------------------
-40
-50
-50
0
_0. .5
-0.25
U
0.25
Normalized shear stress, T, / aO
Figure 4.10(b) r, /ao
V.5
along the penetrometer shaft
74
20
azz /UO
4.2
3.8
3.4
3
2.6
2.2
10
1.8
1.4
0
-10
r/R
Figure 4.11 Contours of vertical stress o./a0
.
20
orr /v0
5
4.5
4
3.5
3
2.5
2
1.5
10
0.8
0.6
0
-10
'U
r/R
Figure 4.12 Contours of radial stress a,/a,,
75
20
U09/
Uv0
4
3.5
3
2.5
2
1.5
10
0.8
0.6
0.5
0
-10
5
0
-
10
r/R
-
15
--
20
Figure 4.13 Contours of circumferential stress o,, /o a
50
-------
------
------------------
40
30 --------------------
10
4---
-------
---------
--------------------
-------- -
-10 ---------- ---------- -------------------- ---------0
------------
-10---------
----
I
---------- --------------
-20 --------- ----------------------- -------------------- sigma(z)/sigma(vO):
---------i9Or/Sg~0
-30 --------- -----------T------8
sigma(theta)/sigma(v0)
--------
-40
-50 ---------
0
-------
-------
=--------- r---------- I----------- ---------
2
1
zz UOv,
Figure 4.14
--
J--------------
4
3
rr
av,
6w /0
v
, /a,. , or, a'O , orv /or
76
5
along shaft
4
---- - ----
3.5
--
3
---------- - -
-
- ---
(A
--
-------
---
----
---
----- -- -- -
-
-------- --
---- - --- - -- - ---
-z/R=0 (tip)
- z/R=2.23 (cone shoulder)
-- - ------- ---z/R=20
z/R=30
cN.
2.5 ---
-
2
Cd
----------
1.5
z
- ------------- - ----------
-------- - -1 ----
---
- ----
-----
I
0
20
15
10
5
r/R
Figure 4.15 Radial distribution of o./,o at various elevations
4.5
4
-- -------- -
----------- ----------- ------------3.5
z/R=0 (tip):
zIR=2.23 (cone shoulder)
tg
co
3
c)
2.5
2
z
1.5
-.---
-
-
- - -
-
- -- - --
10"
- --- --- ----
z/R=10
z/R=20
z/R=30
-------
-
--------- - -
-- -- --------- - -
-- --
-
- -
- - -- ----------- - - - - - -
±
J.-------------
5
10
15
20
r/R
Figure 4.16 Radial distribution of o,, /oo at various elevations
77
3.5 3.5
I
I
---- ------
- -
I
I
I
----------------------z/R-0 (tip):
z/R=2.23 (cone shoulder)
z/R= 10
-- ---------------
2.5 -
2
I
-z/R =-20- ----
------
- - - - -
z/R=30
1.5
--
-
4-
-- - -- ----
--
-- - ---
-- - ---
-- -- -- - -- ---
N
1 -- -
- - ----- - - -- - - ---- -- - -- -- --- -- - --- --- -
z
0.5L
0
5
10
15
20
r/R
Figure 4.17 Radial distribution of o-v /o,
78
at various elevations
20
(a,.Y-qo) / 2ayo
0.6
0.5
0.4
0.3
10
0.2
0.1
0
0
-10
20
r/R
Figure 4.18(a) Contours of normalized cylindrical expansion shear stress qhlaO
50
40
40
30
30
20
20
10
10
- -----
--+ -- --
-- - - ----
- -- - --
- +
--
----- -- ---
-
-
-
-- - - --- - ---- -- - -- ---
---- -----------------
-10
-10
-20
-20 - -------- - - --- --
-30
-30
-40
-40
9-- -~------ - - - - - - - -
-50
0
-----
- - ----
0
0
-50
---- - ---
----
V
U. I
----- - ---- --
------ - - - - -- --
------V.2
----- -V.3
-----
--------
-- -- V.4
- - - --.
V.5
V.0
Normalized cylindrical expansion shear stress, (q,-6To) / 2ayo
Figure 4.18(b) qh /,g, along shaft
79
t5 0.6
1
0.5
-
S 0.4
------
-
4--
- -
-
-
-
--
----------------
-- -----------
~
4
4
.
---.---
2.--
0
*~0.3------
-
-
-
-
-
-
-
-
-
-
-
-
-- 4 --------
zR =0 (tip)
z/R&-2.23 (cone shoulder)
zIRr'10
4-------------
z/X71R2-43
45
N%0
z
0
10
.
--- --------------
- ---------
20
rIR
30
40
50
Figure 4.18(c) Radial distribution of qh /7v0O at various elevations
80
------- ------------------
------------------- ----
20 0
----------
---
------------
---S- ------15
----- --------- --------- -----------------
0 ----------
0
10
0
OOr=8e'*--
------------------ -----------------
0 ---------------C.)
5
_0
5
15
10
Normalized penetration depth, zIR
Figure 4.19 Pressure-displacement curves for clays with different I,
20.00
-
- -------------18.00 - ------------ - - - -- - - -- - -- - -- - - - -- - - - ---- - - - -
16.00 -
---
----
--
I
- -- -- -- - --- - - I - -- -- -
- - ---
14.00 -
- -- - ---0
- ---- - ---- --- - ---
--
- -- - ---
- -- - -- - -- - --
-------
0
10.00 - - -- -- -- -- -- --- --- - - -- -- --
--------
8.00 - -- - --6.00 4.002.00-
-- -
calculated results
-+0-Vesic (1972)
Baligh (1975)
z
12.00 -
0
U
-
--
---
- ---
--------
-
--
-- --
---
--- -
7- - -- ---
-------
--------------
--------------
---------
-6- Yu (1993) smooth
-- -- - -- -
-----
---
------
- --
cone
-4-Yu (1993) rough cone
--
---
---
-
Baligh (1985)
-+-Van den Berg (1994)
smooth cone
------------
----------
-
0.00
1.00
100.00
10.00
1000.00
Rigidity index, Ir
Figure 4.20 Cone factor comparison of analyszed results with published solutions
81
50
101
10
F 40.l
-20
IrIJ
130
40
-10
-0
5
0
r/R
Figure 4.21 Layered clay - location of initial layer boundary
|homogene us b
- ---- -------
200
--------- ----------- ---------loS'ayer syse
I-
150
-- ----
-- --- -- - -- --- -- -- -- --- - - - --- 4--- - -- -hoZgeneous t
l
Cd
---- ----- --- ---- --- ----------- ---------
100
a)
0
0
a)
0
50
I
-
I
n
nO
I
10
20
30
40
Normalized penetration depth, zIR
50
Figure 4.22 Pressure-displacement curve for penetration in layered clay
82
-4
- -- +
250
"/2=
- - ---
=40/Ir -00
200
--
-
- - - - - - -
-
j
150
I
0
0
- - -- - - -
100
llllllliiiill'llllllI
17
- - - - - -
- - - --
I-
0
I
50
0
10
30
20
40
50
Normalized penetration depth, z/R
Figure 4.23 Pressure-displacement curves for layered system with different I, ratio
83
CHAPTER 5
PENE TRA TION ANALYSIS IN SAND
This chapter describes applications of the Eulerian finite element program DiekA for
simulating drained cone penetration in sand. The mechanisms of penetration in sand
are interpreted from details of the penetration - stress field for one base case analysis.
A parametric study is then performed to establish the relationship between the cone
factors and strength parameters of sand.
5.1
Drained Shear Strength of Sands
Mohr-Coulomb and Drucker-Prager yield criteria have been implemented in DiekA
for axial symmetric elements and plane strain elements. In this section, the soil
constitutive behavior in general and the yield criteria for Mohr-Coulomb and
Drucker-Prager are described.
The constitutive behavior of soil can be written in the rate format:
(5.1)
&Y = Dijkl 41
where
Dikl
is the fourth order stiffness tensor depending on material parameters and
current stress state. The total strain increment
, can be divided into two portions: the
reversible portion, i.e., "elastic" strain increment; the irreversible portion, i.e.,
"plastic" strain increment.
SY = 4 iie
(5.2)
+.'
where ij' is the elastic strain increment, and
84
.' is the plastic strain increment.
For linear and isotropic elasticity, the stiffness tensor can be represented by:
1-2v
in which i,
is the Kronecker delta tensor, G is the elastic shear modulus and v is the
Poisson's ratio. For elasto-plastic behavior, Equation (5.1) can be written as:
i
(5.4)
=Dijkl 1 = (Eikl - XYkl ,
where Y,,, is a fourth order tensor accounting for plasticity effects, including
hardening and/or softening. This term can be determined by introducing the yield
function
f,
which depends on the state of stress and an internal variable H. H is a
function of the plastic strain history.
f = f(o-,j, H)
(5.5)
and,
H =fe
P(5.6)
In DiekA, H is defined as the equivalent plastic strain ,:
j, P'gdt
H =3=
(5.7)
The factor 2/3 is used so that the equivalent plastic strain coincides with the uniaxial
plastic strain in case of isotropic plastic flow.
The plastic strain increments are
defined by a flow rule:
.
=A*
(5.8)
a
aci85
where X is a non-negative multiplier which controls the magnitude of the plastic strain
increment, and g is the plastic potential function.
For plastic yielding an element should be in a plastic state (f = 0) and remain
in a plastic state (.f = 0 ). Thus,
f(-y, H)= 0
(5.9)
_f
(5.10)
af.J=
&. +_
aH
,D-y
Combining Equation (5.4), (5.7) and (5.10) yields the plasticity tensor Yyjkj:
/
E
Ykk
auk&g
=
T
s.
E..
j kl
(5.11)
auJEUkI agak
h+af )'
where,
h =g
(5.12)
aH a.f 8u
And the constitutive equation (5.1) becomes:
T
E
&g = Eyk -
-Eyk
ckl
aa
ky
(5.13)
iki
afg
h+ r
jEk,
au"
y
kl
kl
The yield function and plastic potential function for Mohr-Coulomb model
(Figure 5.1) are defined as follows:
86
fMc
1
2
gMC
where a1 ,
u 2, U3
-(C
3
i
2
1
-U
2
)-I
3
(
2
+U3)sin#-c*cos#
(5.14)
+ C 3 )siny,
(5.15)
are the principal stresses and a,
U-2
C3;
#
is the friction angle,
qf is the dilation angle and c is the cohesion.
The yield function and plastic potential function for Drucker-Prager model
(Figure 5.2) are defined as follows:
(5.16)
3J 2 -ap+k
fDP =
gDP= 3J2 -
(5.17)
P
in which, J 2 is the second invariant of deviatoric stresses S;:
(5.18)
2
(5.19)
3
3
6 sin 0
(5.20)
3- sin #
6 sin yi
(5.21)
3- sin V
k
=
6c*cos#
(5.22)
3 - sin 0
87
The value of a is defined in a way to match the Drucker-Prager yield surface with the
Mohr-Coulomb yield surface at the triaxial compression shear mode (Figure 5.3). For
triaxial compression, a
J
2
=
3
= 03,
thus:
(5.23)
(a 1 -U0)2
p =-(
3
1
(5.24)
+203)
Therefore, the yield function for Drucker-Prager model:
fDP
(
a)
*
3
For element at plastic yielding (i.e., fDP
0-1 =
2a +3
3-a
(5.25)
+2c3
=0),
(5.26)
3
Similarly, for triaxial compression, if fuc = 0:
0-t = (-sinO)
1 -sin#~
(5.27)
Equation (5.26) and (5.27) represent the same relationship if a
=
6 sin#e
,sin
as defined
3 - sin#0
by Equation (5.20). That is to say, the Mohr-Coulomb yield surface and DruckerPrager yield surface coincides in the triaxial compression shear mode.
88
5.2
Friction Angle
The peak friction angle (0' ) for sands is not constant but varies with confining
pressure and density. The constant volume friction angle,
#'
= 30+ 2 is well defined
for most sands. Bolton (1986) gives the following empirical correlations for
estimating the peak triaxial friction angle:
#= #
(5.28)
3I,
Dr
I1 = 10
10 -log,
100%f
a' )-1.0
(5.29)
where Ix is the relative dilatancy index, D, is the relative density, and 0' is the mean
effective stress level at failure.
5.3
Numerical Simulation
The finite element model is the same as that of undrained penetration analyses in
clays (Figure 4.2). The 1944 elements mesh is used for penetration analyses in sand.
The step size is 0.0001m, same as the analyses for clays. The boundary conditions are
the same as that of clay analysis, i.e., zero interface friction, and the initial vertical
stress o' = 35kPa. Initial horizontal stress o' = K * a'
and K =
1-v
=0.43.
A base simulation is carried out with following material parameters:
E = 1OMPa, v = 0.3, cohesion c = 0, peak friction angle
# = 30'
and zero dilation.
The calculations consider a weightless soil (i.e., y = 0). To explore the performance
of both models on penetration analyses for sands, two analyses with the same material
89
parameters have been performed, using Mohr-Coulomb and Drucker-Prager models.
The cone reaction pressure vs. normalized penetration depth curves (Figure 5.5) show
that: (1)
The steady state with respect to the reaction pressure is reached after a
penetration distance about z/R = 10 , for both Drucker-Prager and Mohr-Coulomb
models; (2) The analysis with Drucker-Prager model yields a steady state cone
reaction pressure which is 10% higher than that from Mohr-Coulomb (q, = 755kPa
vs. 686kPa respectively). From Figure 5.3, the Mohr-Coulomb model and DruckerPrager model coincide at the triaxial compression shear mode. However, other than
the triaxial compression mode, the Drucker-Prager criterion allows higher yield
strength than Mohr-Coulomb and over predicts the strength in the triaxial extension
shear mode.
The contours of mobilized friction angle
#0
= sin- 1
from Mohr-
01
+
3
Coulomb and Drucker-Prager criterions are illustrated in Figure 5.6(a) and (b). The
initial stress state is: or', = 35kPa , c'
=
0.3
*35= 15kPa . The vertical and
0
1-0.3
horizontal stresses are the major and minor principal stresses. The mobilized friction
angle at initial stress state:
#', 1o
(35-15
= sin' I3
K35±15)
and Figure 5.6(b), the mobilized friction angle
I = 23.6*. Comparing Figure 5.6(a)
#'ob
reaches 50 for Drucker-Prager
criterion, while Mohr-Coulomb analyses clearly define a zone where
#',::~
30'.
The stress/stain fields for the base case analysis with Mohr-Coulomb criterion
are discussed to make a clear interpretation of the penetration mechanisms. Figure
90
5.7(a) presents the contours of the equivalent plastic strain 6, . Figure 5.7(b) shows
the distribution of equivalent plastic strain along the shaft. The variation of the
equivalent plastic strain at radial direction at different zIR is displayed in Figure
5.7(c). From Figure 5.7(a), the elastic-plastic boundary is located approximately
r/R = 26 away from the shaft and extends to a distance z/R = 13 ahead of the tip.
Soil particles outside of this region experience only elastic deformation. Figure 5.7(b)
shows that along the shaft, the plastic strain increases sharply at the tip, where the soil
particles are pushed away to make space for penetrometer.
Contours for the normalized octahedral normal stress increments
are
Oct
0
plotted in Figure 5.8(a). The octahedral normal stress increment along the shaft is
shown in Figure 5.8(b). Figure 5.8(c) shows the radial distribution of
,"'
0(
at
various elevations. Along the shaft, the octahedral stress starts to increase at z/R = 8
ahead of the tip. There is a very sharp increase at the tip, to a maximum
,
~10.
vO
After passing the tip, the normalized octahedral stress increment starts to decrease and
reaches a uniform state of
2 at z/R > 10. The uniform distribution of
,"c'
behind the tip can be easily observed from Figure 5.8(c): the curves of stress
distribution along r-direction overlap for z/R = 10, 20, 30.
91
Figure 5.9(a) shows the contours of the normalized shear stress -7-.
Figure
0vO
along the shaft. In this study, the cone is
5.9(b) shows the distribution of
UvO
assumed smooth. Therefore, there is no shear stress along the shaft, except around the
cone. Ahead of the tip (z/R = -2), negative (sign convention shown in Figure 4.10(a))
shear stress occurs, because soil elements next to the axis (smaller r/R ) are pushed
downward with respect to the soil elements farther away from the axis (large r/R).
The shear stress direction is reversed behind the cone tip (z > 0). This is because the
soil elements from the tip move upward, and this forces the soil elements next to the
shaft to be pushed upward, with respect to the outer elements. The normalized shear
stress reaches a minimum -
=
-5.5 and a maximum
'r
UvO
=0.5.
vO
The contours for normalized vertical stress
-7
07vO
,
radial stress
,
and
avO
are illustrated in Figure 5.10 through 5.12. The variations
circumferential stress
UvO
of the three stresses along the shaft are shown in Figure 5.13. Figure 5.14 through
5.16 present the distribution of the stresses along r-direction at different elevations. At
initial stress state, the relative order of the three stresses is: 07' > a' = o' . Starting
from z / R = -2, all three stresses increase dramatically around the cone tip. A steady
state stress distribution appears to be approached at z / R > 5 ~10 (Figure 5.13 and
Figure 5.14 through 5.16). From Figure 5.13, the relative order of the three stresses is
92
C',
> o-'
> a' . Soil coming from below the tip of the cone is radially compressed
when arriving along the shaft of the cone.
Figure 5.17(a) shows contours of normalized cylindrical expansion shear
stress
o.
h vO
Figure 5.17(b) shows the variation of the
""i
along the
vO
shaft. The distribution of
"'i
along r-direction at different elevations is plotted in
ovo
Figure 5.17(c). Similarly, the normalized cylindrical expansion shear stress
,
~vO
increases to almost 6 as the tip is approached. A steady state is reached at
is about 2.
z/R > 10 ~ 20, and the steady state value of
UvO
5.4
Effect of Strength Parameters
It is well established that the friction angle O' affects the cone factor Nq for
cohesionless soil (Terzaghi, 1943; Vesic, 1963). Figure 5.18 shows the correlations
between Nq and friction angle O' proposed according to bearing capacity theory.
There are large variations in existing correlations. In this research a series of analyses
have been performed varying the friction angle
#',
dilation angle Vr and elastic
Young's modulus, E. Table 5.1 summarizes the input properties used in simulations.
The reaction pressure vs. normalized penetration depth curves for varying
friction angles are compared in Figure 5.19. The curves for varying dilation angles are
93
compared in Figure 5.20. Those for varying stiffness are shown in Figure 5.21.
results, cone factor Nq = q
0Ov
The
, are summarized in Table 5.1.
The material properties used by van den Berg (1994) for sensitivity analysis
are summarized in Table 5.2 and comparison made between the results from this
research and van den Berg's. Van den Berg also used DiekA for his analyses and
Mohr-Coulomb yield criterion. Van den Berg modeled the interface friction with zero
thickness four-noded interface elements. 6 is the interface friction angle. The 5 value
used by van den Berg varies from 5 to 20. In this research, the penetrometer is
assumed smooth (i.e., 5 = 0). The other difference is the lateral earth pressure
The KO value is related to Poisson's ratio
P
by KO =
1-v
uOa-.
in this research. v = 0.3
for all simulations, so KO = 0.43. The KO used in van den Berg's analyses are 1.0 or
0.67. The lateral earth pressure
O-h0
in his analyses is higher than the analyses
conducted here. Comparing results in Table 5.1 and Table 5.2, the cone factors
obtained are lower than van den Berg's results. For example, comparing"sand_8" in
Table 5.2 with "sandlO" in Table 5.1: Although "sand_8" is softer than "sandlO" (E =
5MPa and lOMPa respectively), the Nq value is 49, 145% higher than Nq = 20 for
"sand10". This can be explained by the smooth interface and the lower lateral earth
pressure used in this research. Van den Berg's results show the contribution of
interface friction on the cone resistance,
(5 = 5'
fi, is less than 3% for all simulations
20*). This may imply that the horizontal earth pressure plays an important
role in the prediction of cone resistance. Similar effects of horizontal earth pressure on
the cone resistance in undrained clay have been reported by Teh and Houlsby (1991).
94
In spite of the lower cone factor Nq due to the smooth cone assumption and
low horizontal earth pressure, the results show some agreements: (1)
The cone factor
Nq increases with the friction angle 0. The results show that Nq increases from 20 to
29 as the friction angle 0 increases from 30 to 45.
increases from 38 to 59 as the friction angle
#
Van den Berg's results show Nq
increases from 25 to 35. (2) The cone
factor Nq increases with the dilation angle yr. (3) Nq increases from 20 to 30 as the
soil stiffness E increases from 1OMPa to 20MPa (Table 5.1). Similarly, van den Berg
has Nq increased from 88 to 107, comparing "sand_6" and "sand_7" in Table 5.2. (4)
is not a constant, or, it is not only dependent on the sand
The cone factor Nq =
UvO
strength parameters, but also the initial stress condition. Comparing "SandlO" and
"Sandl1 " in Table 5.1, both have the same material parameters, the only difference is
that "Sandl 1"
has an initial vertical stress
ro = 5kPa , while "SandlO"
has
So0 = 35kPa. The cone factor Nq = 90 for "Sandl 1", 4.5 times higher than Nq = 20
for "SandlO". Comparing "Sand_5" and "Sand_12" in Table 5.2, the cone factor Nq
increases from 12 to 58 (by 4.8 times) when the initial vertical stresses reduces from
350kPa to 35kPa.
Further study is needed to establish the role of interface friction and lateral
earth pressure.
95
E
v
#
(MPa)
VI
c
(")
(kPa)
U'
(kPa)
Cone
Cone
resistance
factor
qc (kPa)
N
q
SandlO
10
0.3
30
0
2
35
688
20
Sand11
10
0.3
30
0
2
5
450
90
Sand12
20
0.3
30
0
2
35
1065
30
Sand20
10
0.3
35
0
2
35
822
23
Sand2l
10
0.3
35
5
2
35
1100
31
Sand30
10
0.3
40
0
2
35
909
26
Sand3l
10
0.3
40
5
2
35
1210
35
Sand32
10
0.3
40
10
2
35
1610
46
Sand4O
10
0.3
45
0
2
35
1006
29
Sand4l
10
0.3
45
5
2
35
1300
37
Sand42
10
0.3
45
10
2
35
1710
49
Sand43
10
0.3
45
15
2
35
2153
62
Table 5.1 Material data used in sensitivity analysis
96
E
'
p
Vt
c
o'Vo
Ko
f
qc
q
q
(MPa)
(kPa)
(kPa)
(o)(kPa)
0
(kPa)
Sand_5
5
0.3
30
10
2
35
1.0
10
2035
51
58
Sand_6
10
0.3
30
10
2
35
1.0
10
3070
49
88
Sand_7
20
0.3
30
10
2
35
1.0
10
3760
17
107
Sand_8
5
0.3
30
0
2
35
1.0
10
1710
39
49
Sand_9
5
0.3
25
0
2
35
1.0
10
1340
31
38
Sand_10
5
0.3
35
0
2
35
1.0
10
2080
44
59
Sand_11
5
0.3
30
10
2
35
0.67
10
1860
46
53
Sand_12
5
0.3
30
10
2
350
1.0
10
4380
139
12
Sand_13
5
0.3
30
10
2
35
1.0
20
2900
61
83
Sand_14
5
0.3
30
10
2
35
1.0
5
1660
33
47
Table 5.2 Material data used by van den Berg (1994)
97
Vo
a
p
/
0 "1 r-
ua e
2
Mohr Coulomb yield surface
Figure 5.1 Mohr-Coulomb model
03
T
C
) 0
2
=
3
C2
Drucker-Prager yield surface
Figure 5.2 Drucker-Prager model
98
0'
A
Mohr-Coufomb
Prager
F
/
Figure 5.3 Matching Mohr-Coulomb model and Drucker-Prager model
99
800'''
800
1::
1
''['Drucker-Prdge
700 - - - - - -
nMohr-Coub
or
- r- - - - - -
;300 - - -500 - -
300 -
-
S100
0O
-
- - - - - -
-
- - - - - - - - - - - - - - - - - - -- - - - --- -
-- - - - --
- - - - - - r - - - - - -
-- - - - - -
- - - - - -
200
- - - - - -aly
5
10
15
Normalized penetration depth, z/R
Figure 5.5 Pressure-displacement curves for Drucker-Prager and Mohr-Coulomb
models, base case analysis
100
50
40
(D'mob
35
30
25
20
301
20
N
1
-10
-20*
0
25
50
r/R
Figure 5.6(a) Contours of mobilized friction angle - Mohr-Coulomb criterion
50
40
(mob
50
45
40
35
30
25
20
20
N
10
0
-10
-201
r/R
4U
Figure 5.6(b) Contours of mobilized friction angle - Drucker-Prager criterion
101
40
r
30
1.5
1
0.5
0.1
0.01
0.001
0
20
Ix
10
0
-10
U
30
20
'IU
r/R
Figure 5.7(a) Contours of equivalent plastic strain e,
50
50
40
40
30
30
20
20
10
10
0
S0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
II
T
.1
-
L --
-I
I
---
I
I
I
I
-
I
I
I
II
I-
-
-1-
0.25
I
I
- -t
T-
~ ~ -
- --
I
I
-
- -
l
-
~
_
S-
0
:
- - -
I
1- I
I
I
-I
I
- - -
L --
. .
! ! ! 1 1I .! :. -. -. I,...
II.,IwI
-~
0
1-I
--
- - - - - - --
-I
--
---
-
I
I
- - ---
-
-
-
I
I
- - -I
I
I
II
----------------
1
0.75
0. 5
Equivalent plastic strain, cp
-
---
-I
. . . . . . .. ,
1.25
-
I.
1.5
Figure 5.7(b) Equivalent plastic strain e, along the penetrometer shaft
102
1.3
I
I
I
I
I
1.1
-
_
I
_
-
-
-
-
- -
0.9
-
-
-
-
-
I
I
1
rA
-
-
-
1.2
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
--
0.8
-
-
0.7
C13
--
0.6
_
2/42/IL0.tip)-----_
_
--z/R=2.23 (cone shoulder)
0.5
-
0.4
T
0.3
0.2
-
-- 4
-
-- - --
--
z/Rnl1
-----
_
S z/R=20_ _
-u---- z/R=30
-
--
---------I
I
-
0.1
- ----
:
0
- . 0 .
0 1
i
i
i
i
5
.
.
I
10
I
I
.
.
I
15
-
20
r/R
Figure 5.7(c) Radial distribution of c, at various elevations
103
20
Ac
'oct /
6
vO
10
8
6
4
2
1
0.5
0.2
0.1
0
10
N
0
20
15
10
5
-100
r/R
Figure 5.8(a) Octahedral stress increment Aa', /o-'
50
50
40
40
----
30
30 ------
20
20
20
10
10
I
-L
I
--
r- - r - r - r - r - - - I I
I
I
I
I
I
I
- - - - - - - - - - |-- |-- |-- -L
I
I
I
I
L
_
L
I
I
I
I
I
L
I
I
I
I
I
L
-
I
I
0
20 - -
-10
--
f-
f
-20
-30
-50
-40
- L
F
I
-50
0
-
- L
-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
-L ---
-
-
10
5
Normalized octahedral stress increment, A '00 / u'vo
Figure 5.8(b) Aor,
/oro
104
along shaft
6, 6,
I
I
I
I
I
0-(tip)
S-
-
-
.z/R: =2.23 (cone shouder)
-
I
z/R:=10
z/R =20
=30
R
N
z
oe
- - -
0
10
20
r/R
Figure 5.8(c) Radial distribution of Ao', /ao at various elevations
105
20
r /
',
0.5
0.2
0
-1
10
-2
-4
N
0
-10
0
5
10
15
20
r/R
Figure 5.9(a) Contours of shear stress r,, /Oi(
50
50
40
40
30
30
20
20 -
10
10
I
I
I
- --- -1-
S-
-10
-10
-20
-20
-30
-30 ---
-40
-40 *
- -
- ,
+I
-
- 1---
-
--
I 4--
- -- -- -I
----I,, ,
, ,
-1
T-
-1--
-
I
- - ---
I-
||
SI
0
0
- -- -- - - - - - -
, ,
,_
,
- -
-
- - l-- - - - -
-
-
-- -
- ------ -__, , , ,___I ,
-
-
-
-,
__,_
-- -- -- -
--_
,
-50
-50
0
-6
-5
-4
-3
-2
Normalized shear stress,
r,
-1
/ a'7,O
0
Figure 5.9(b) r,/o O along the penetrometer shaft
106
1
1
-
0
-1
- -
-
- -
-
-
-
-
--
-
-
- -
-
-
-
CO,
-2
-- -
-
-
z/R-0 (tip)
~
z/R=2.23 (cone shbulder)
- - e
---lO-I
---- z/R=20
----z/R=30
-
-
z
-
-3
.
.
.
I . | I
10
||
|I
20
r/R
Figure 5.9(c) Radial distribution of r,/
107
at various elevations
20
O', /',o
8
6
4
2
*1.5
1
10
N
0
-10,
ri k
Figure 5.10 Contours of vertical stress o'
/(TO
20
a',r
20
18
16
14
12
10
8
6
4
2
1
0.8
0.6
0.4
0.2
0
10
N
0
-10
/ U'vo
IU
r/R
Figure 5.11 Contours of radial stress a'./O
108
20
0'80/ E',O
7
6
4
2
1
10
0.5
0.4
N
0
5
-100
10
15
20
r/R
Figure 5.12 Contours of circumferential stress o' /,'o
50
50
40
40
30
30
J1
20
20
10
10
t
- --- -- -L
-
- - -- --
-
-- -- - --
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
Igma(rr)/sigma(vT)
ig~theta)/svO)
-- -J
0
L - - - - 5
10
a',, / O'VO,
Figure 5.13 a"'/'O
',rr / a'VO,
-
-
--
---
15
'e / O',o
, c',/o' , ' /o
109
-
along shaft
20
-5
-
-
-
-- --
-----
0d
'0
z/R =0 (tip)
=2:23-fcone shulder)-z/R =10
1
z/R=20 z/R
=30I
10
-
20
r/R
Figure 5.14 Radial distribution of a'
I
I
'
'
10
9
-- - - - - -
8
/o
'
at various elevations
I
'
'
:
- - - - - - - - - - - --
-J-- - ---------
- ----
'
--
--
7
6
z/R=0 (tip)
5
Cd
4
4
- - -
3 --
-
-
-
-
r--
tr
-
-/R--223-c-fe
-
--
shTdet)--
z/R=10
--- 2
---- - ---z/R=30
----------
--- ---
2
-j-------
1
0.
0
-
I
I
10
20
-
r/R
Figure 5.15 Radial distribution of a',./,
110
at various elevations
4
I
- -- -
- -
-
- - - -
-
-I
--
3
42C.)
-z/Lt=0.4tip)
z/R=2.23 (cone shbulder)
1
z/R= 10
2
- -:
C.)
z/R=20
z/R=30
I
I
1
z
0
0
I~~~
I
10
20
r/R
Figure 5.16 Radial distribution of orT /or'
111
at various elevations
20
(ar,-qoo) / 2u',o
6
5
104
3
2
0.8
0.6
';
0.4
0.2
0.1
0
0
-100
5
10
15
20
r/R
Figure 5.17(a) Contours of normalized cylindrical expansion shear stress qh /o
50
50 _
40
40
-
-
-
30
30 - -
-
-
-
- -
-
-
-
-
20
20
-
-
- I
I
- --
10
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
-
-
I
-
- - -
0 - --
0
-10
-20
-
-
-30
-20 - --
-40
-30 --
-50
-50
0
-
-
- -
II
- - y-- -
- T-t- - - -- - - T I
I
- - - - - - - - -~
-1-I
- --
- -
-
1-
--
L
-J -
4
5
6
Normalized cylindrical expansion shear stress, (q,-roi7)
/2a'vO
- - - --
0
1
2
Figure 5.17(b) qh /'o
112
-
-|
3
along shaft
4
- - -
& 3
- - - - - - - -
---
----
rjA
-
-
2 - -
rA
Ce
--. ltip)
z/R=2.23 (cone shulder)
z/R=10
I
z/R=20
z/R=30
Ce
0
00
~1
10
z
20
r/R
Figure 5.17(c) Radial distribution of qh /v0
113
at various elevations
In
De Beer.
J~ky
Meyerhot
1000-
40
3F
100-
Skenipton.
Yassin.
Bishop
Teraghi
5
30
5
40
45
50
Friction angle ep
Figure 5.18 Correlation between cone factor Nq and friction angle
114
1000
Sand40:
900
Ph1
Sand36:
phi-40
Sand20:
phi=30
800
j
700
-
600
cj
- - -
500
-
400
-~~
-
L - - - - - -
~ ~~~~~ -
-- - - - - -
- - - - - - - t--
- - - - --
300
200
100
n
5
10
Normalized penetration depth, z/R
0
15
Figure 5.19 Pressure-displacement curves for varying # (V/= 0)
L
I Sand42: 001
2000
-
I-
-
-
--
-
I-
-
-
--
II
ISand42:
- --
1500
pi=10
- - -- 4 - - - - - - -
0~
Sand4 1: si-=
C,,
0
0~
1000
I
0
Sand40:
dsi=0
I
U
0
-I
I
i
I
T
I
I - -
I
2
4
6
.2.
0
0
500
A
0
10
8
Normalized penetration depth, zIR
Figure 5.20 Pressure-displacement curves for varying y
115
(=
45*)
i i-
1100
1000
- -
-
-
00 - - -
-
- -
800
andl:E-2OMP
-
- - - - - - - - - - -
-
-
--
-
- r - -
-
-
-
-
-
- -
- - - -- - - --
-
lSandI0: E=1Ma
600
-
4-
-
-
-
-
-
-
-
-
o
-
-
I-100
---
- -
,
0O
- - -
-- - - - - -
lii, i
5
,
ii
- - -
-
- - - - -- - - - --
- - - - - - - - -
5 00
-
i
10
I
--
--
I
15
20
Normalized penetration depth, z/R
Figure 5.21 Pressure-displacement curves for varying stiffness E
116
CHAPTER 6
IMPLEMENTATION OF USER MATERIAL MODEL
INTO DIEKA
6.1
Introduction
DiekA allows the implementation of user defined material models. This is done
through providing material type related subroutines and connecting the new routines
to the existing DiekA program. The concepts of user provided material subroutines
are illustrated in this chapter, through the example of implementing a linearly elastic
and perfectly plastic model with von Mises yielding. Besides that, comments are
given on state variables initialization, which is necessary to implement more advanced
soil models, for example, Modified Cam Clay (Roscoe & Burland, 1968), MIT-Si
(Pestana & Whittle, 1999).
6.2
Three Routines for User Supplied Material Model
Three subroutines must be provided to work with the user supplied material model:
(1) USRINI, initializing control data and sizes of storage arrays; (2) USRSIG, which
defines the user material constitutive behavior by updating the stresses and state
variables for each increment; and (3) USRMTX, providing the material stiffness
matrix at integration point level.
The input and output of these routines and examples with linearly elastic
perfectly plastic model are illustrated in following sections.
117
6.2.1
User Supplied Initial Routine
Each material requires different number and type (integer or double precision) of
material parameters and state variables. This kind of information is provided to DiekA
through the subroutine "USRINI", under subfolder "SRC/MATER". For example, a
linearly elastic, perfectly plastic material model with von Mises yielding has three
material parameters: Young's modulus E, Poisson's ratio v, and undrained yielding
stress, oy. It needs one integer state variable, which represents the relative position to
the yield surface, or, whether plastic deformation occurs. A double precision state
variable is required to store the plastic multiplier. These two state variables are
discussed in more details in next section of stress update routine "USRSIG".
Assuming all those three material parameters are double precision, the user should
specify in "USRINI" that this material model requires three double precision material
parameters, zero integer material parameters, one double precision state variables and
one integer state variable. It should be pointed out that "USRINI" does not define the
values of those parameters, but only tells DiekA how many there are, so that DiekA
will anticipate how many parameters to read from input file.
The number of stresses and strains in the stress and strain vector should also
be defined in "USRINI". Usually, for three-dimensional problem, there are six
stress/strain components.
For example, ux, uyy, ozz, rxy, ryz,
-zx, are the six stress
components. For two-dimensional problem, there are four stress/strain components.
The axisymmetric problem is also two-dimensional. The four stress components are
normal stresses az, ar-, u88 and shear stress -rz.
118
Besides that, "USRINI" also specifies whether the stiffness matrix is
symmetric or not for the user supplied material model. DiekA will choose the
symmetric solvers if the stiffness matrix is symmetric.
More details about the coding are illustrated in Appendix B. 1. The "USRINI"
file for a linearly elastic and perfectly plastic model with von Mises yielding is
presented in FigureB.1 in Appendix B. An example, which specifies the user-defined
material in the input file, is shown in Figure B.2.
6.2.2
User Supplied Stress Update Routine
At each Gauss point, the constitutive model computes the increments of stress and
state variables given the initial state of material (stresses and state variables) and the
applied strain increments. For user-supplied material, this process is carried out by
subroutine "USRSIG".
The "USRSIG" file for linearly elastic and perfectly plastic material model
with von Mises yielding is presented in Figure B.3 in Appendix B. Flowchart of
incremental algorithm for the linearly elastic and perfectly plastic model with von
Mises yielding is shown in Figure B.4.
One issue should be pointed out here. DiekA uses solid mechanics sign
convention for stress and stain, i.e., tension is positive, which is opposite to the soil
mechanics
sign
y
'a
=
e
xi s
convention.
for i # j,
In
addition,
DiekA
uses
engineering
+
instead of tensorial strain, c
xbai
2
ix.
strain,
XJ
.
AxF
Attention should be paid when implementing user material routines into DiekA. For
119
the linearly elastic and perfectly plastic material model with von Mises yielding, it
happens that the yield surface is a function of the second invariant J2, which is not
affected by whatever sign convention used. However, for implementation of other
advanced soil models, it may be necessary to convert the signs of stresses and stains
input to USRSIG routine, then work under soil mechanics sign convention, and finally
change them back to solid mechanics sign convention before returning the values and
exiting the USRSIG routines.
The yield function for von-Mises criterion has the format as:
f = r3J -cY
(6.1)
where J2 is the second invariant of deviatoric stress
yield stress. The relationship between or, and
SuTC
J2 =
2
SU ), a3y is the uniaxial
(undrained strength from triaxial
testing) will be derived in following section.
Firstly the stress increment is obtained by elastic prediction.
oc-
(6.2)
K * vol
(6.3)
S, = 2G*e,
where ouo, is the octahedral stress, Ec0 l is the volumetric strain, SY is the deviatoric
stress, and e0 is the deviatoric strain. Then the yield function is evaluated to see the
relative position of the updated stress point to the yield surface. If the point is still
inside the yield surface, i.e., yield function has a nonpositive value, the elastic
assumption is correct. If the yield function is greater than zero, the elastic prediction
120
results in stress point outside of the yield surface. The stress point should be mapped
back from the elastic prediction to the yield surface, as follows:
S = S (elastic)*
(6.4)
'
3J2elastic)
where S (elastic) means the elastically estimated deviatoric stress from Equation
(6.3). J 2(elastic) is the invariant for the elastic predicted stresses. The integer state
variable will be updated to "1" if the stress point is on the yield surface, and the
double precision state variable, the plastic multiplier, will be updated as:
DSV
-
3J 2 (elastic) - O(6.5)
2Gc-,
This plastic multiplier will be used in the following step of updating the stiffness
matrix.
6.2.3
User Supplied Matrix Routine
In the subroutine "USRMTX", the material stiffness matrix [D] needs to be supplied.
{-l= [D]*{)}
(6.6)
[D] is the stiffness matrix at the integration point. [D] is required to build up the
system stiffness matrix. {y} is the vector of engineering strains.
The stiffness matrix [D] is dependent on whether the stress point is in elastic
or plastic state. For elastic state, the stiffness matrix has the format as (illustrated for
2D):
121
[D]=
4
K+-G
3
2
3
2
3
0
where, bulk modulus K
2
3
2
G
K3
4
K+-G
3
2
3
4
K±+-G
3
2
3
0
0
E
=
0
0
(6.7)
0
G
and shear modulus G =(
-
3(1 -2v)
. This is
E
2(1 + v)
equivalent to the relationships presented by Equation (6.2) and (6.3). As discussed in
last section, DiekA uses engineering strain y instead of E; therefore the fourth item on
the diagonal of the stiffness matrix is G, not 2G.
For plastic state, the stiffness matrix is:
4G
G
K+------S11S1
3 A
22
2G
G
K---S
11S22
2G
K---S
3 Z
G
G
11S3
2
G
2G
K
-
S22S
22
G
U2
4G
3 A
K
2G
-3
G
G
K--3A
3 A IJS2
K---22 S
G
33
A.J
G
-s
2
s
S3J
where A
22
2
elastic
1
.
o7-y
22
1
G
GsIs2
-
2
G
J2
G
4G
K+---32
SS
S22 S33
33S
2
s
S
Gs2sI
-
2212
G
Gs3sI
-
2
G GSs +-G
1
12
A
X is the plastic multiplier, the state variable updated by
"USRSIG" for plastic state. The matrix is symmetric for both elastic and plastic
states.
After implementing those three routines, the Makefile (under subfolder
"dieka\dieka" should be rerun to connect these new routines into DiekA.
122
6.3
Validation of User Supplied Material Model
To validate the implementation of the above user material model, an undrained
penetration analysis in clay is performed with this model and the result compared with
that of Tresca model.
The relationship between the undrained shear strength s, in Tresca model and
the yielding stress ay in von Mises model is derived firstly. At triaxial compression
mode, we have:
3 J2 -) a=(a. -
fvM =
fMc
=-1(
2
-
3)
C3)-
Cy
(6.8)
(6.9)
S.
If the two models are matched at the triaxial compression mode, as shown in Figure
6.1, ,, should be twice of s,.
A penetration analysis using the user provided model is carried out with the
following material parameters: Young's modulus E = 2400kPa, Poisson's ratio v =
0.3, and yielding stress uy = 40kPa. Initial vertical stress qOi = 35kPa. The result is
compared with that of Tresca model with s, = 20kPa, which is the case 'Clay4' in
Table 4.1. Figure 6.2 presents the cone reaction pressure qc versus the normalized
penetration distance
V
-,
R
for user supplied von Mises model, Tresca model and the
existing von Mises model in DiekA. The user supplied von Mises model has the same
performance as the DiekA von Mises model, as the curve from user supplied model is
identical to that of DiekA. The Tresca model and von Mises model converges to the
123
same reaction pressure as the penetration distance increases. This proves the good
performance of the user supplied material model.
6.4
Initialization of State Variables
To implement the advanced soil models, for example, Modified Cam Clay (Roscoe &
Burland, 1968), MIT-Sl (Pestana & Whittle, 1999), the same three subroutines as
illustrated in previous section should be supplied. Compared with the linearly elastic
and perfectly plastic model presented previously, one complexity for Modified Cam
Clay and MIT-Sl models, is to initialize the state variables.
By default, DiekA initializes all state variables as zero at the start of the
analysis. While the zero initial values are correct for most situations, sometimes the
state variable should start from a nonzero value. For example, for Modified Cam
Clay, there is a state variable, say "a", which defines the size of the yield surface. The
initial size of yield surface "a'
depends on the stress history and the initial stress
state. If other than zero initial values should be specified, the routine "STATEO"
under subfolder "SRC/ELE1" should be modified. "STATEO" is intended for
initialization of state variables at the integration points. It loops over element groups
and for each group it checks the material type and see whether special initialization
should be performed or not. User material is assigned a reference number or material
ID number 119 in DiekA. Those data are stored in array RDATEL. The user model
state variables is pointed to by IADRIP(15). IADRIP is a common array. For
example, the following code initializes the elements with user material model to have
an initial value of 16.0 for the first state variable:
124
IF (MATTYP .EQ. 119) THEN
RDATEL(IADRIP(15),IP)
=
16.DO
END IF
If the fourth item of the material data needs to be initialized, it should be
"RDATEL(IADRIP(1 5)+3, IP) = the initial value".
125
AL
Von
Mises
yield
Tresca yield surface
T'as
a
Figure 6.1 Fitting Tresca and von Mises models at triaxial compression mode
140
130
- - - - -
- - - - - - r-- - - - - --
I-,
120
--0
110
-
-
--
User supplied von Mises
Tresca
I
- -Ditv-NMiss
-
100
901
0
5
10
15
Normalized penetration depth, z/R
Figure 6.2 Load-displacement curves for different material models
126
CHAPTER 7
SUMMARY CONCLUSIONS AND SUGGESTIONS
FOR FUTURE STUDIES
7.1
Summary and Conclusions
To obtain a better insight into the penetration mechanism and the correlation between
cone factor and soil strength parameters, an Eulerian finite element model has been
applied for penetration analysis. Comparing with the Updated Lagrangian approach,
the Eulerian formulation avoids the severe mesh distortion occurred during largestrain process, which usually leads to incorrect or divergent solutions. On the other
hand, the convection item needs to be calculated for the Eulerean approach, as the
mesh points are decoupled from the material points.
After validating the finite element model, in terms of mesh coarseness and
step size, the FE model is applied to simulate undrained penetration in homogeneous
clays, layered clays and drained penetration in sands. The penetration mechanism has
been interpreted by studying the stress and stain fields from a base simulation.
The undrained clay behavior is modeled by the total stress Tresca criterion. A
series of analyses have been performed for undrained penetration in clays with
varying rigidity index. The results show that cone resistance increases with rigidity
index, and so does the penetration depth to reach a steady state with respect to the
reaction force. Compared with other published results, the cone factors obtained from
this study are lower than Strain Path Method solutions for a simple pile (Baligh, 1985),
127
but are close to the cavity expansion solution presented by Yu (1993) when the
rigidity index I,.
200 . Penetration in layered clays shows that the penetration
distance to reach a steady state in the new layer is dependent on the stiffness of that
layer.
For penetration in sands, the performance of two models, Mohr-Coulomb and
Drucker-Prager are examined and it is found that the Drucker-Prager predicts a higher
cone resistance than Mohr-Coulomb model. This is due to the unrealistic high friction
angle predicted at triaxial extension mode by Drucker-Prager criterion. The effects of
friction angle, dilation angle, soil stiffness and initial stress state on the cone factor
Nq
are discussed after a sensitivity analysis. It is found that the cone factor Nq is not
a material constant, or, it is not only dependent on the material parameters, but also
the initial stress state. Cone factor increases with the friction angle, dilation angle and
soil stiffness. The cone factors obtained in this study are lower than other published
results. One reason is the cone interface is assumed smooth, i.e., the side friction is
ignored in this study.
Besides applying DiekA for penetration analysis, a user material model has
been implemented into DiekA. The model is linearly elastic and perfectly plastic with
von Mises yield criterion. The performance of the user-implemented model has been
confirmed by comparing the penetration analysis results using the user model and preexisting models.
128
7.2
Suggestions for Future Studies
7.2.1
Model interface friction
In this study, the interface friction is not considered. The friction does occur at the
soil-steel interface. Therefore, to be more realistic, the Eulerian FE model should be
developed to include interface friction. DiekA provides zero thickness contact
elements to model the interface, and also a special contact element material type to
model the interface friction. More details about the contact element could be obtained
at DiekA manual (DiekA development group, 2000).
7.2.2
Initial stress state set up
The initial horizontal stress
u-hO
= KO * -o,
where KO =
1-v
. For soils, due to the
stress history, usually KO is not related to the Poisson's ratio as shown above. DiekA
determines the horizontal stress by the Poisson's ratio. This needs to be overridden to
initialize more realistic stress state.
7.2.3
Implementing advanced soil models into DiekA
Advanced soil models, for example, Modified Cam Clay model, MIT-Sl model
should be implemented into DiekA, for more reliable predictions. To implement user
material, the three routines, as discussed in Chapter 6, should be provided and
compiled with DiekA. The complexity with implementation of these models,
compared with the linearly elastic and perfectly plastic model with von Mises yield
criterion, is the state variable initialization. The routine to initialize state variables is
"state0.f' under subfolder "src/eleI".
129
APPENDIX
Appendix A
An Input file for DiekA
The input file for DiekA contains a complete description of the numerical model. An
input file "init.inv" for soil penetration analysis is shown in Figure A.1. The
corresponding finite element mesh and boundary conditions are shown in Figure A.2.
This example input file is to set up the initial stress condition. A downward
distributive load of 35kPa is applied on the top boundary. The Mohr-Coulomb model
is associated with the elements, with Young's modulus E = 5EIOkPa, Poisson's ratio
P = 0.499, and undrained strength s, = 20kPa. The Young's modulus is intentionally
set to very high to minimize the deformation/strain occurred during this stress
initialization step.
The input file has an intuitive, keyword-based format, as most Finite Element
program. The keyword lines are those started with "*". DiekA is not case sensitive.
"START" claims the beginning, and "STOP" keyword announces the end of the input
file. DiekA ignores any other statements before "START" or after "STOP". DiekA
itself is unit free. User should adopt a set of consistent units. The units used for the
input file in Figure A.1 are meter for length, kPa for stress, second for time. The
example
is an axisymmetric,
"DIMENSION".
or a two-dimensional
"IMAGE" requires
DiekA
to
problem,
output
all
as
nodes,
stated by
elements,
displacements, material characteristics it reads from the input file. This is helpful to
identify possible mistakes in the input file.
130
The nodes and elements are firstly defined. "EOG" means end of definition for
a group. "EOD" stands for end of definition for the keyword. For example, there are
more than one way to define the nodes. Each is considered as a "group". "EOG" tells
DiekA that current node read-in format is over and the next group may be defined by
another option, thus in different format. The nodes in the example in Figure A. 1 are
defined by node number, followed by the coordinates. DiekA provides other options
to define nodes, for example, generating a row of nodes by specify the first, last nodes
and the number of nodes to be generated, or reading from the file generated by a
preprocessor.
"AXI4" is axisymmetric four-node element. The first "1" on the line following
"AXI4" represents the material type, i.e., the element group will be assigned the first
material type defined after "*Material" keyword. The second "1" is the fraction
number, which takes value from 0 to 1. For soil problem, this number should be
always "1". "EOT" is the end of one element type.
After the nodes and elements have been defined, the initial and boundary
conditions should be defined. "SUPPRESS" is to suppress degree of freedom. For the
example input file, the left and right boundary are not allowed for horizontal
movement, therefore all nodes along the left and right boundary (node#1, #37, #38,
......
, #1999,
#2035) have the degree of freedom in the x-direction suppressed. "1"
means the x-direction, and "2" means the y-direction.
Particularly, for nodes along the cone of the penetrometer, only displacements
along the cone are allowed. For each of the five nodes along the cone face, a local
coordinate system is defined, of which the local x-direction is defined as the
131
tangential direction of the cone face at that point. DiekA uses "ROTATED BASIS" to
define the local coordinate system. "Angle" means the local coordinate system is
defined through rotating the global coordinate system by an angle (in radius). As
shown in Figure A.1, node 815 has a local coordinate system 600 (=1.0475 radius)
counterclockwise rotated from the global one.
"PRESCRIBED" is to define prescribed displacements. For the nodes along
the cone face, after defining a local coordinate system, the displacements along the
local y-direction (represented by "2") is prescribed as zero, i.e., the movement can
only occur in the local x-direction, along the cone face. There is no vertical
displacement along the bottom boundary, as described by the "PRECRIBED" for
nodes 1-37.
The input of nodal and element forces should start with the keyword
"LOADS". As shown in the example, "DISLOA" adds a distributed load of -35.OkPa
in the y-direction along the top boundary nodes, node 1999-2035. The downward
direction is considered negative in DiekA.
Gravitational acceleration is 9.8m/s 2 .
Material
type and parameters
are specified in "MATERIAL"
section.
"MOHR" represents Mohr-Coulomb material model. The two numbers on the first
line after "MOHR" is Young's modulus and Poisson's ratio. The second line defines
the fraction, dilation angle, friction angle and cohesion successively.
For this
example, the friction angle and dilation angle are zero, and the Mohr-Coulomb model
actually becomes a Tresca model.
A Eulerian analysis is specified by "EULER" keyword.
132
In DiekA, the equilibrium equation is solved by Newton-Raphson process.
The iteration will be terminated if the following condition is met:
RUNB
"
" is the
=
||LOADS - REACTIONS
||R EACTIONS||
<dforc(
<drc(
A.l)
L 2-norm. "LOADS" represents the prescribed forces and "REACTIONS"
is the reaction forces. "dforc" is a convergence criterion specified in the input file
after "MAXITER" keyword. dforc = 0.0001 for this example. The maximum number
of iterations allowed for one step is defined after "MAXITER", 15 for this example. If
the convergence is still not reached after 15 iterations, DiekA will terminate.
"AUTO" states that the step size will be automatically adjusted by DiekA,
according to the "RUNB" value from previous step. The maximum step size is limited
to 5*(original step size), to avoid extremely large step size due to low "RUNB" value.
On the other hand, if the user specified convergence criterion is too hard to reach,
DiekA will reduce the step size. When the steps become too small, too many steps
will be needed to accomplish the simulation. To prevent this, the maximum number of
steps can be defined after the "AUTO" keyword, as 1000 for the example.
During a simulation, numerous pieces of information can be created. "SAVE"
tells DiekA to store the information necessary for a restart. With "OUTPUT"
keyword, choices are made concerning the output files that need to be generated for
post-processing. With "ELGROUP" option, the element group can be chosen, of
which the nodal data will be stored in the nodal files (*.nod). "NODAL" option
further specifies what data to be stored in the nodal file. DiekA uses a group of
numbers to represent different data, for example, "1" for total displacement, "5" for
133
"nodal stress", "6" for "plastic strain". In the input file, the line after "NODAL"
defines what data to be stored by using these numbers. With "FORCES" option, the
reaction force will be stored in nodal file too. As defined by "FOLLOW" and
"SUMREAC", the vertical reaction forces of nodes 1-37 will be summarized for each
step and so is the vertical displacement for nodel. An output file "forcel.sum" will be
generated, which lists the sum of vertical reaction forces and vertical displacement for
every step.
After "STEP" keyword, DiekA will start the calculation with the data
described prior to this line in the input file. "STEP" can be followed by an integer
number n, which defines the number of steps to be performed. By default n equals 1.
The end of calculation is marked with "STOP". Data after this line in the input file is
ignored.
Typing in the executable file name "dieka.slow", DiekA will prompt input the
file name, "init.inv" for this example. Then DiekA will run by itself and messages
will appear on the screen, showing current step number, iteration number, unbalance
ratios, etc. All the screen output can also be found in the output file "vorm.dat". The
output "vorm.dat" for the "init.inv" is shown in Figure A.4.
134
135
136
137
138
Figure A. 1 Input file example "init.inv"
139
D
1
Nhlmlh1I[T IIII
40
I
T
1
r T
T I
T
r-]
II~IL~
I I I -L
IC
30
20
10
0
-10
-20
-30
-40
-50
11
25
A 0
I 1J11 1 1 1 1
I
I
50
75
10( B
r/R
Figure A.2 Finite element mesh represented by the example input file
4
-------r.........
.
---------.-------------/
--------/
2
/............ ........
------------------7---
--
/-- --- -- -----/----
/- /- /
------
..... ..... .............
N
0
----I--------rri
0
1
2
r/R
3
4
Figure A.3 Rounded cone-cylindar transition part
140
141
142
143
144
145
146
147
8J7 1
149
150
151
152
153
Figure A.4 Output file "vorm.dat" for "init.inv"
154
Appendix B
User Provided Subroutines for a Linearly Elastic
and Perfectly Plastic Material Model
B. 1
Initialization routine USRINI
The "USRINI" file for the linearly elastic and perfectly plastic model with von Mises
yielding is presented in Figure B.1.
There are nine argument variables for "USRINI". Only "ID" is an input
argument, while other eight are outputs. "ID" is the identification number for user
material model. The identification number is by default 1 if not specified in the finite
element model input file. By using different "ID" numbers, more than one user
material models can be integrated into DiekA. "NSTR" is the number of stresses or
strains in the stress and strain vector. As illustrated in Figure B.1, "NSTR" equals 4
for two-dimensional models, for example, plain strain and axisymmetric problems.
"NDP" is the number of double precision material parameters. For linear elastic
perfectly plastic von-Mises model, there are only three material parameters: Young's
modulus E, Poisson's ratio v, and yielding stress oy. All are stored as double
precision. "NIP" is the number of integer material parameters. No integer material
parameter is needed for this specific example. "NDS" and "NIS" specify the number
of state variables in double precision and integer. For this example, there is one
integer state variable, which represents location of the stress point with respect to the
yielding surface. It has the value of "0" if the stress point is inside the yield surface, or
in elastic region. If the stress point is on the yielding surface, it will be set to "1" and
plastic strain occurs for that point. Logical variable "SYSM" indicates if the material
155
stiffness matrix is symmetric or not. DiekA will choose the symmetric solvers if
"SYSM" is set to "TRUE". If "SIGCOR" is true, a correction stress will be calculated
in subroutine "USRMTX" and added to the load vector. The correction stress is useful
for visco-plastic or thermo-mechanical analysis, where there can be a stress change
without a change in strain. This correction is only for a better convergence. The
logical variable "SIGROT" is to define whether the material model needs rotation of
stress or not. The rotation of stress is used for large deformations. If active, a corotational formulation is used. The rotation is applied by a tensor transformation.
From the skew-symmetric part of the incremental deformation gradient, the rotation
angle is estimated and a rotation matrix is set up. First, half of the rotation is applied
on the initial configuration, then the symmetric part of the deformation is added, and
finally the second half of the rotation is applied on the deformed configuration. For a
rigid plastic material, the difference with and without rotation is proportional to the
half rotation and usually very small (personal communication with Ton A.H. van den
Boogaard at University of Twente). The rotation is not applied for the example in
Figure B.1.
The syntax for specifying user defined material model to elements is the same
as that for other existing models in DiekA. For example, the statements in Figure B.2
tell DiekA that the user defined material model will be used, with material parameters
Young's modulus E = 2400kPa, Poisson's ratio v = 0.3, and yielding stress a-
40kPa.
156
=
B.2
Stress Update Routine USRSIG
The "USRSIG" routine (Figure B.3) reads in the stresses at the start of the increment,
state variables at the start of the increment, strain increment and outputs the stresses at
the end of the increment and also updates the state variables. Figure B.4 gives the
flowchart of USRSIG routine.
B.3
Stiffness Update Routine USRMTX
"USRMTX" file is shown in Figure B.5. The input to this routine includes: stresses
and state variables at the start and end of the increment, the applied strain increment.
The output is the stiffness matrix.
157
sa.'s'ala-ais.14.-e-.m.
bl.nalbiA
WeidadPi
al.,44etessai6Mao..ilrdhiPMahEidaMMI
haddW4al-illl.danadhM.MM-r'adafwhame~W4ealia*.'MAa
E~.aladesin'adifi~a'.~1.sill
0.).. s
-.
..
is
...- -..
.
.. . .
.-.. .
. ..
158
Figure B. 1 "USRINI" file for linearly elastic and perfectly plastic model
with von Mises yielding
159
Figure B.2 Statements in input file for user defined material
160
161
162
163
164
Figure B.3 "USRSIG" file for linearly elastic and perfectly plastic model
with von Mises yielding
165
Start
&Oct = Kevol
S = 2Ge
Si+-
+S
Si
S=$*2G'
Figure B.4 Flow chart of incremental algorithm
for linearly elastic and perfectly plastic model with von Mises yielding
166
167
168
169
Figure B.5 "USRMTX" file for linearly elastic and perfectly plastic model
with von Mises yielding
170
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