Large Strain Theory as Applied to Penetration Mechanism in ...
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LSU Historical Dissertations and Theses Graduate School
1985
Large Strain Theory as Applied to Penetration Mechanism in Soils Large Strain Theory as Applied to Penetration Mechanism in Soils
(Plasticity). (Plasticity).
Panagiotis Demetrios Kiousis Louisiana State University and Agricultural & Mechanical College
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UniversityMicrofilms
International300 N.Zeeb Road Ann Arbor, Ml 48106
8526378
Kiousis, Panagiotis Dem etrios
LARGE STRAIN THEORY AS APPLIED TO PENETRATION MECHANISM SOILS
The L o u is ian a S tate U n ive rs ity an d A g ric u ltu ra l an d M e c h a n ic a l Col. Ph.D.
University Microfilms
International 300 N. Zeeb Road, Ann Arbor, Ml 48106
IN
1985
LARGE STRAIN THEORY AS APPLIED TO PENETRATION MECHANISM IN SOILS
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and
Agricultural and Mechanical College in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in
The Department of Civil Engineering
by
Panagiotis Demetrios Kiousis Diploma of Civil Engineering, 1980
Democrition University of Thrace, Greece August 1985
ACKNOWLEDGEMENTS
From these lines I would like to express my appreciation to my advisors, Dr G. Z. Voyiadjis and Dr M. T. Tumay for their careful guidance, patience and continued encouragement during this research project. I would also like to thank Dr Y. Acar for his most useful discussions and advices. Finally, I would like to express special thanks to my wife Marianne, for her help with the proof reading of my dissertation and for her encouragement, love and understanding during the difficult times of the entire research project.
ii
CONTENTS
PAGE
Acknowledgements .......................................... il
Contents........... .■............................................ iii
List of T a b l e s ................................................... v
List of Figures . ............................................ vi
A b s t r a c t .......................................................... viii
CHAPTER
1. Introduction . . . . . ..................... . . . . . 1
2. Quasistatic Cone Penetration Testing ................ 6
2.1 Introduction .................................... 6
2.2 Equipment . . . . . . 7
2.3 Analysis of Cone Penetration Testing ......... 10
3. Finite Deformation Preliminaries ..................... 14
4. Constitutive Equations . . . . . ..................... 20
4.1 von Mises Yield Criterion...................... 20
4.2 Cap Model by DiMaggio and S a n d l e r ............. 25
5. Pore water pressures.................................. 28
6. Elasto-Plastic Stiffness Tensor for UndrainedL o a d i n g ............................................... 29
7. Finite Element Formulation ........................... 30
8. Method of Solution.................................... 38
8.1 Mathematical Formulation ...................... 38
8.2 Realization of the M e t h o d ...................... 40
9. Cone Penetration A n a l y s i s ........................... 43
9.1 I n t r o d u c t i o n .................................... 43
9.2 Penetration Simulation ........................ 44
9.3 Presentation and Discussion of the Results . . 48
9.3.1 I n t r o d u c t i o n ......................... 48
9.3.2 Displacement Field .................. 50
9.3.3 Strain F i e l d s ........... 53
9.3.4 Stress Fields and Pore WaterPressures............................. 68
10. Summary, Conclusions and Recommendations ............ 87
11. References............................................. 92
V i t a ........................................................ 98
iv
LIST OF TABLES
Table Page
2.1 Summary of the existing theories of cone penetration
in clays (from Baligh, 1975) ............................... 12
v
LIST OF FIGURES
Figure Page
2.1 Field results obtained with the Dual-Piezo cone
penetrometer.................... 8
2.2 Prototype Dual Pore pressure penetrometer ............... 9
2.3 Summary of the assumed failure mechanisms for deep
p e n e t r a t i o n ..................................... 11
3.1 Coordinate systems and description of displacements . . . 15
4.1 Modification of Prager's Kinematic Hardening rule . . . . 23
4.2 Cap Model by DiMaggio and S a n d l e r .................. 26
7.1 Bilinear and parabolic quadrilateral elements ........... 31
8.1 Newton-Raphson approximation technique .................... 41
9.1 Consequtive positions of the penetrating cone . . . . . . 46
9.2 Detail of boundary condition change of nodal points
at the lower and upper ends of the cone tip . . ......... 47
9.3 Finite element mesh for the discretization of the
soil medium during penetration ............................. 49
9.4 Load-Penetration relation ................................. 51
9.5 Displacement field around the penetrometer ............... 52
9.6 Schematic representation of the separation of
the soil-penetrometer interface during penetration . . . . 54
9.7 Radial strain (er) contours around the penetrometer
for penetrations of 1 mm, 3 mm, and 6 mm . . ........... 55
vi
9.8 Axial strain (ez ) contours around the penetrometer
for penetrations of 1 mm, 3 mm, and 6 mm . .............. 59
9.9 Tangential strain (e^) contours around the
penetrometer for penetrations of 1 mm, 3 mm, and 6 mm . . 62
9.10 Shear strain (e ) contours around the penetrometer
for penetrations of 1 mm, 3 mm, and 6 m m ................... 65
9.11 Octahedral shear strain (v contours around theoctpenetrometer for penetrations of 1 mm, 3 mm, and 6 mm . . 69
9.12 Radial stress (crr) contours around the penetrometer for
penetration of 6 m m ........................................ 72
9.13 Axial stress (.&z) contours around the penetrometer
for penetration of 6 m m .................................... 74
9.14 Tangential stress (Oq ) contours around the
penetrometer for penetration of 6 m m ....................... 75
9.15 Shear stress Cxrz) contours aroung the penetrometer
for penetration of 6 m m .................................... 76
9.16 Octahedral stress (a .) contours around the oct'penetrometer for penetration of 6 m m ....................... 77
9.17 Octahedral shear stress (x contours
around the penetrometer for penetration of 6 m m ......... 79
9.18 Radial distribution of the octahedral stresses
a and t „ at the lower end of the cone t i p ........... 80oct oct9.19 Radial distribution of the octahedral stresses
a and t t at the upper end of the cone t i p ........... 82oct oct r
9.20 Pore water pressure (u) contours around the
penetrometer for penetrations of 1 mm, 3 mm, and 6 mm . . 84
vii
ABSTRACT
A theoretical analysis of a Quasistatic Cone Penetrometer Test
CQCPT) is presented in this work. A large strain, elasto-plastic formu
lation is developed for this purpose and is implemented into a finite
element method program. The basic relations of the theory are developed
in an Eulerian reference frame, subsequently transformed to a Lagrangian
coordinate system, and through simple time differentiation, the neces
sary rate equations are derived. Both isotropic and kinematic hardening
of the 2iegler type are introduced in this theory. The plasticity
models implemented are the extended von Mises and the cap model by
DiMaggio and Sandler. The pore water pressures are obtained through the
introduction of a bulk modulus of the soil-water system.
The theory is applied to the solution of a cone penetration problem
in a soft cohesive soil (E = 5000 KN/m^, s^ = 50 KN/ra^). The displace
ment strain, stress and pore water pressure fields around the penetrat
ing cone are thus calculated. Interesting conclusions are draw from
this analysis, the most important of which are listed below:
a) The penetration mechanism during the QCPT seems to be a localized
phenomenon for soft clays; that is, the recorded response during
the test is averaged over small regions, which results in more
meaningful and accurate predictions of the soil properties.
b) The kinematic field obtained from the axi-symmetric penetration is
different from the one obtained from the plane strain penetration
problem. No slip zones appear in the axi-symmetric problem; conse-
viii
quently, an analysis based on this concept is inappropriate for
soft cohesive soils.
A separation zone occurs between the shaft and the soil above the
cone tip. Readings of the soil response in this area could there
fore be unreliable.
As the penetration acquires steady state characteristics, the pore
water pressures generated around the cone probe become more uni
form* It can thus be concluded that the position of the pore
pressure transducer is not critical.
1. INTRODUCTION
The determination of accurate parameters describing soil behavior
is of primary interest to geotechnical engineers. These parameters may
be determined by either laboratory tests or in-situ techniques.
Laboratory tests are performed on "undisturbed" soil specimens
under well controlled conditions. The triaxial test and the direct
simple shear tests are two important laboratory tests used to obtain the
shear strength parameters of a soil. Despite the distinguished
advantage of controllability, the laboratory methods have serious dis
advantages. Some of these are:
(a) Even the best quality sample is practically disturbed,
(b) The obtained information is discrete and not continuous,
(c) Only certain stress pathes can be simulated with the routine
laboratory devices.
In-situ tests are often preferred to laboratory experiments because
they can be less expensive and some have the distinct advantage of
dealing with undisturbed soil. In certain situations, an accurate
stress path simulation is obtained (e.g., plate test versus footing
bearing capacity, cone penetration versus pile bearing capacity, etc.)
Certain in-situ procedures, such as the cone penetrometer test, provide
a continuous description of the soil behavior. One could argue that the
most important asset of in-situ testing is the minimization of the
effect of sample disturbance. Nevertheless, all in-situ techniques
suffer from insufficient control due to the fact that the stress paths
1
2
created from such a test are not normally known and cannot be con
trolled. This not only limits our understanding of the techniques, but
is also responsible for inadequate interpretation of their results.
Of all the sounding techniques used in the field, the cone penetra
tion test and the pressuremeter test distinguish themselves as the best
available when considered from the standpoint of experimental simplicity
and reliability. Both tests have been extensively studied. (Acar 1981,
Davidson 1980, de Ruiter 1981, Sanglerat 1972). Theoretical, semi-
theoretical, and empirical methods have been proposed to interpret the
results of these tests and predict the soil strength parameters. The
overall effectiveness of these analyses is inhibited by a number of
simplifying assumptions. Such assumptions include
geometric linearity (i.e., small strain theory),
simplifying constitutive laws (Mohr-Coulomb materials), and
homogeneity and isotropy of the soil.
The advent of high-speed digital computers has made it possible to
implement more complex theories through the use of sophisticated numeri
cal techniques such as the finite element method. Advanced plasticity
models have been developed (DiMaggio and Sandler 1971, Lade and Musante
1976, Prevost 1978, Desai 1980) which provide a more accurate descrip
tion of the soil response to loading. A number of numerical techniques
which incorporate such plasticity models and also account for geometric
nonlinearities such as finite strains (Hofmeister et al. 1974, Fernandez
and Christian 1971, Davidson and Chen 1974, Banerjee and Fathallah 1979,
Desai and Phan 1980) have been proposed.
Two approaches are used for the basic formulation and solution of
finite strain elasto-plastic problems. Typically, these are character-
3
ized by taking either an Eulerian or a Lagrangian frame of reference in
the synthesis of the theory (Bathe et. al. 1975, Hibbitt et. al. 1970,
Osias and Swedlow 1974, McMeeking and Rice 1975, Gadala et al. 1984).
Both methods are based on the incremental theory of plastic flow. The
most commonly used is the Eulerian formulation (Carter et. al. 1977,
Yamada and Wifi 1977, Banerjee and Fathallah 1979), where the spatial
coordinates are used in the solution of the problem. In this formula
tion, the incremental equations are in terras of the spatial strain rate
and the Jaumann stress rate. The second approach to the large strain
problems is the Lagrangian formulation (Hibbitt et. al. 1970). However,
even in this formulation, the flow rule is traditionally expressed in
terms of the Jaumann stress rate and the spatial strain rate. The
Jaumann rate is subsequently converted to the second Piola-Kirchhoff
stress rate.
The use of the Jaumann rate of the stress and shift tensors at
finite deformation creates rotational effectsfor materials that exhibit
kinematic hardening. In examining the response of a kinematically
hardening material subjected to simple shear, Lee et. al. (1983), and
Dafalias (1983) found that for a monotonically increasing load, oscil
lating stresses were predicted when the Jaumann stress rate was used.
This obviously incorrect result has been attributed to the inaccurate
definition of the "spin11 for the case of kinematic hardening. New rate
relationships have been proposed to replace the Jaumann rate. Unfortu
nately, none of the proposed new relationships is of sufficient gener
ality, and the problem of defining the correct stress rate still exists
for materials exhibiting kinematic hardening (Lee et. al. 1983). Conse
4
quently there exists a need for further improvements and refinements of
the present formulations.
The objective of this study is to present a new theory which is
free of the inaccuracies of the present methods that deal with large
strain plasticity, and to demonstrate the applicability of this method
through the analysis of the cone penetation mechanism.
The scope of this work encompasses the following:
A. Presentation of the theory in the Lagrangian reference frame.
B. The incorporation of plasticity models suitable for elasto-plastic
analysis of soils. The models chosen here are the extended von
Mises and the cap model by DiMaggio and Sandler (1971).
C. Transformation of the basic elastoplastic formulation given by
Voyiadjis (1984) so that kinematic hardening of Ziegler type (1959)
to be incorporated.
D. Introduction of pore water pressure effects and incompressibility
due to undrained loading.
E. The use of the finite element method for the numerical implementa
tion of the procedures of this theory.
F. The analysis of a cone penetration test in a soft cohesive soil.
Although the results of this analysis reveal very high strain
rates, incorporation of rate effects is not within the scope of
this work. In QCPT the penetration is realized by imposing a
displacement field at the soil-cone interface. As a result, the
strain field is less sensitive to rate effects than the stress
field. Therefore, the kinematic field presented in this work is
more accurate than the stress field.
5
The theory is developed in the material coordinates and is based on
concepts introduced by Green and Naghdi (1965, 1971), modified by
Voyiadjis (1984), and further improved by Voyiadjis and Kiousis (1985).
The "elastic" material strain rate is postulated to be a linear function
of the second Piola-Kirchhoff stress rate. The flow rule is expressed
in terms of the material stress and strain rates. The rotational
effects introduced by the Jaumann rates are thus eliminated. A finite
element method is presented, which incorporates non-linear geometric
relations and employs more advanced plasticity models.
2. QUASI-STATIC CONE PENETRATION TESTING
2.1 INTRODUCTION
In the last two decades, the quasi-static cone penetration test
(QCPT) has gained considerable popularity in a number of countries,
including the United States. The test has proved valuable for a number
of functions such as soil profiling, determination of in-situ relative
density, friction angle and cohesion intercept. Furthermore, the simi
larities in stress paths induced by the QCPT and pile penetration make
the test very useful in predicting pile bearing capacity as well.
The increased popularity of the QCPT is attributed mainly to three
factors (de Ruiter, 1981). These are
A. the general introduction of the electric penetrometer which pro
vides more precise measurements, and a number of improvements in
the equipment which allow deeper penetrations even in relatively
dense materials.
B. the need for penetrometer testing as an in-situ technique in off
shore foundation investigations due to the difficulties in achiev
ing adequate sample quality in marine environments, and
C. the addition of other simultaneous measurements to the standard
friction penetrometer, such as dynamic pofe water pressure, soil
temperature, and acoustics.
The QCPT is the only routine in-situ test that provides accurate
and continuous soil profile. Layering in the soil stratification can be
easily detected from the changes in sleeve and tip resistance, which
correspond to changes in shear strength of the soil. Since the addition
of pore water pressure measurements to the test, the identification of
layering has become easier. For example, very small dynamic pore pres
sure values indicate permeable soil, i.e sand, while large pressure
build-ups indicate clays. In Figure 2.1 a typical recording of the QCPT
is shown which is comprised of tip resistancej side friction and pore
water pressure measurements. When a number of QCPT's are performed over
one site, useful information can be obtained about the uniformity of the
soil stratigraphy. Based on such information, an optimum exploration
program can be designed to include sampling of specific critical layers
(which otherwise could not have been observed) and possibly other in-
situ measurements. The QCPT is also used to detect possible erroneous
results obtained by other in-situ or laboratory tests (de Ruiter, 1981).
For example, if a clay layer shows a constant tip resistance, the shear
strength should also be constant. This fact should be verified by the
remainder of the tests carried out on the soils of this site. It
becomes apparent that the QCPT is of significant value for both qualita
tive and quantitative use.
2.2 EQUIPMENT
The typical state-of-the-art cone penetrometer (Figure 2.2) con
sists of a probe which carries instrumentation for monitoring tip resis
tance, sleeve friction, and pore pressure measurements, both at the tip
and at the sleeve.
Cone penetrometers are found in a variety of shapes and sizes (Acar
1980, de Ruiter 1981). The different sizes are necessary for testing
different soil types. For instance, in testing of very soft clays,
FRICTION
RESISTANCE
(MPA)
8
DEPTH (M)1r % i\lV\ |
W\i
£ Js
\\— *■1DEPTH (M)
q = TIP RESISTANCE£ = FRICTIONAL RESISTANCEsux = TIP PORE PRESSURE u2 = SHAFT PRESSURE
FIGURE' 2.1 FIELD RESULTS OBTAINED WITH THE DUAL-PIEZO CONE
TIP
RESISTANCE
(MPA)
PORE
PRES
SURE
(BAR
)
a a \ ( S . ib { l > ( 6 ) ( p < 3 ) ( ip (jji) ( j G ^ ^ ,16i ' J I }
fin*! U 0 O i'.ci)(>o *
l lU /F U G J IQ DUAL. M A C N t f t t U A I NUO'COK PUrfJlOitfTU tilts)ic t c * % i . p m *
l- n u m H i H O M M 1 1 ( H U1 M H f l H M H l U lW U t t o 'M e - C I U I t . « U L
4. Q HQ Utt
t, q v A M b o ik. r * u a « a u a u iJ. M T tC M « I t . M W f A l i i T »
ft. K k O M P i r i l M & p U I t P U O W T M i U l U T
6 I t i f t ^ l u I l f H U I Ik , ( U U I
Ik. JBjCTjOH lUllrl It m m u m i
FIGURE 2.2 PROTOTYPE DUAL PORE PRESSURE PENETROMETER
10
larger diameter probes are used. To achieve deeper penetrations with
the available thrust, smaller diameters are used. When buckling of the2probe is a danger, larger diameters are required. The 10 cm cone with
a 60° apex angle is the most commonly used penetrometer.
2.3 ANALYSIS OF CONE PENETRTION TESTING
A number of investigators have analyzed the cone penetration as a
bearing capacity problem (Meyerhof 1951, 1961, Mitchell and Durgunoglu
1973). This approach is an extension of the shallow foundation bearing
capacity theory. The Mohr-Coulomb yield criterion is employed and a
failure mechanism is assumed. The ultimate bearing capacity is calcu
lated from limiting equilibrium. Figure 2.3 presents a summary of the
proposed failure mechanisms.
Another approach to the problem is through the introduction of more
complex soil properties and the simplification of the imposed strain
field to that of a cylindrical cavity expansion in an infinite medium
(Ladanyi 1967, Vesic 1972, Forrestal et. al. 1981).
A summary of bearing capacity formulations for cone penetration
problems is presented in Table 2.1.
Levadoux and Baligh (1980) have obtained theoretical strain and
pore pressure distributions in the soil by estimating the deformation
pattern from the potential field around cones in an incompressible,
inviscid fluid. They have used the method of sources and sinks to solve
for the potential field. The flow of an inviscid fluid around a cone
penetrometer was also studied by Tumay et al. (1985). According to this
approach, the strain rates around the cone penetrometer are calculated
analytically with a conformal mapping technique. The method is intended
to give a first approximation to the response of very soft cohesive
t o) t b ) t o Id)
PrandtiReissnerCaquotBuismanTerzaghi
DeBeerJokyMeyerhof
Berezonfsev and Yaroshenko
Vesic
Bishop Hi t ) and Mott
Shemlon, Yessin and Gibson
FIGURE 2.3 SUMMARY OF THE ASSUMED FAILURE MECHANISMS FOR DEEP PENETRATION
TABLE 2.1 SUMMARY OF EXISTING THEORIES OFCONE PENETRATION IN CLAYS
Type of Approach R e f e r e n c e
q = N s + p c c u ro H for c 2 6 = 60° PoExpression for G /b - 100 u G/s = 400 u
+JTerzaghi (1943) Meverhof (1951)
(shape factor)(depth factor) x 5.14 9.25 Same
aVO<J£3JXC3CJGO
Mitchell and Dorgunoglu (1973)
(shape factor)(depth factor) x (2.57 + 2 6 + cot 6 ) 9.63 Same aVO
cMm0)W
Meyerhof (1961) (1.09 to 1.15) x (6,28 + 2 6 + cot 5 ) 10.2 Same aVO
Bishop et al (1945)
1.33(1 + £n G/s ) u 7.47 9.30 unspecified
cc10
Gibson et al (1950)
1,33(1 + £nG/s ) + cot 6 u 9.21 11.03 aVOcRJa.« Vesic (1975,1977) 1.33(1 + In G/su)+ 2.57 10.04 11.84 aoct4-1♦H>(0O
Al Awkati (1975) (correction factor) x (1 + £n G/su)
10.65 13.28 aact
Stea
dype
netr
ati
on
Baligh (1975) 1.2(5.71 + 3.33 6 + cot 6) + (1 + £n 0/s^)
11.02 + 5.61 =16.63
11.02 + 6.99 = 18.01
°ho
13
clays to the cone penetration test. Although it has not been tested as
yet, this approach may yield good results if the appropriate plasticity
model is invoked.
Sanglerat (1972), Baligh (1975), Baligh et al. (1979), and Acar
(1981) have presented extensive reviews on the state of the art on cone
penetration. The reader is referred to these works for more detailed
information.
3. FINITE DEFORMATION PRELIMINARIES
In this chapter, a brief review of some concepts of continuum mechanics is presented. A detailed presentation of these concepts may be found in the monograph by Truesdell and Toupin (1960). In Figure3 .1 , a body is presented in its initial configuration Bq at time t=0 and at ifs current configuration B at time t. The position vectors of the body are expressed by:
zk = gk^xl’ x2’ x3; ^ k ~ 2 ’ 3or
xA = hA (zp z2 , Z3 ; t) A = 1, 2, 3 (3.2)
The functions g^ and are assumed to be single-valued, continuous, and of Class C'. The deformation Jacobian satisfies the following expression:
3z,° < det = J < « (3.3)A
The displacement fields in the material and the spatial coordinate systems are expressed as:
u^ = uA (xp x2 , t) A = 1, 2, 3 (3.4)and
uk “ uk^zp z2 » z3 > ^ k = 1, 2, 3 (3.5)
respectively.
14
15
X
FIGURE 3.1 COORDINATE SYSTEMS AND DESCRIPTION OF DISPLACEMENTS
16
A Cartesian, coordinate system is used here. According to Figure3.1, the components of the displacement vector u are given by:
Uj = - X p u2 ~ z2 ~ x2* u3 “ z3 -x3* (3-6)
The material description of the velocity vector is
3u.VA ~ 3t~ ^3 -7^
while the spatial expression is
9z,vk = ^3 '8^
The material strain tensor e^g is related to the changes in lengthof a line segment dlQ through the expression:
dJi2 - d£o2 = 2eAB dx^ dxfi (3.9)or
d£2 - dJd 2~ ^eAR (3.10)A0 2 AB oA oB
0
and is defined as:
1eAB = 1 C6kI 53^ 333^ " 6AB> (3-n )
or, 3u. 3un 3ur 3nr
eAB = 1 C53^ + 33^ + 55^ 3 3 ^ (3'12)
Similarly, the spatial strain tensor h ^ is related to the changes in length of a line segment dlQ by:
17
dZ2 - d Z 2 = 2hkJi dzk dz^ (3.13)
or
dJJ2 - dZ 2 0 = 2hin Z, Zg (3.14)
and is defined by:
hkZ = 2 (6k£ " fiAB) (3-15)
or
, 3u, 9urt 3u 3uu - 1 ( k , Z m iru „ r<i iz\hk£ ~ 2 ^5z^ 3z^ “ 3z^ 55 ^ (3-16)
The volumetric strain dV/dVQ is expressed in terms of the material
strain tensor:
dV
and
w r = J <3-17>o
J = (1 + 21 + 411 + 8111)^ (3.18)
where I , II , and III are the first, second, and third strain invari-6 0 6ants, respectively, and are defined as:
Ze = eAA
U e ~ l ^eAA eBB ‘ eCD eCD^
H I e = \ ^2eAB eBC eCA " 3eEF eEF eMM + eNN ePP eQ(p
The material strain rate is expressed as:
q - M (3.19)AB u-isj
18
and
Pt = 2eAB dxA dxB (3.20)
Similarly, the spatial strain rate is expressed by:
1 9vk 9v£dk£ = I + 3if> <3 '21>
and
^ d£2 = 2dkA dzk dz£ (3.22)
The relationship between the material and the spatial strain rates
is given by:
9zk 3z£eAB “ dkA 3x^ C3,23)
The volumetric rate is expressed in material coordinates by
(Kiousis et al., 1985)
J = RCD eCD (3.24)
where RCD - [26CD + 46CI) - 4eCD + S e ^ eRC - 8eCD
46CD e0P e0P + 46CD eLL eI<K^2J (3.25)
The spatial expression for the volume rate is simply:
(&-) ■ dkk (3-26)o
The second Piola-Kirchhoff stress tensor (material stress) s^g is
related to the Cauchy stress tensor (spatial stress) as:
The second Piola-Kirchhoff stress rate is
4. CONSTITUTIVE EQUATIONS
Two plasticity models are used in. this study to describe the soil
behavior; the extended von Mises with kinematic and isotropic hardening
and the cap model by DiMaggio and Sandler (1971). For the developmentk
of the incremental constitutive tensor 3 form of decomposition of
the strain rate is assumed:
eAB = eAB + eA£
The terras el,, and e'.'n are termed "the elastic strain" and "the AB ABplastic strain", respectively. It should be noted that the kinematic
interpretation of these terms is not the usual one. They are simply
mathematical quantities defined by the constitutive law only. Neverthe
less, when the plastic strains are much larger than the elastic ones (a
in most cases in soil mechanics) the decomposition in equation (4.1)
acquires physical meaning.
4.1 VON MISES YIELD CRITERION
The yield function for the extended von Mises flow rule is
expressed in terms of the Cauchy stress tensor:
f = 2 tCTk£-£akjP (crkjT£akJ^ “ clCakk~akkJ tcJkk'akk) C2K " k = 0(A.2)
whe rerepresents shift of the center of the yield surface
*
K = °k£^k£ t*le plastic work rate in spatial description;* * ”1K = s1T1 eVn J is the plastic work rate in material description;AB AB r r j
(j, q “ o. 0-| O 6. „ is the deviatoric component of the Cauchy stress;KJL KJb j mmamm^kA t*le deviatoric component of the shift tensor;
8 = 0 when no kinematic hardening is assumed;
e = l when kinematic hardening is assumed;
and C p c^, and k are material constants.
The corresponding flow rule is expressed by *
di'o = A II— = A(cr, „ - a. 0) - 2c. 6, „(ct - a ) (4.3)kx mm
In this work, the plasticity formulation proposed originally by
Green and Naghdi (1965, 1971), later modified by Voyiadjis (1984), and
further modified by Voyiadjis and Kiousis (1985) is adopted.
According to this theory, the yield function is expressed in
material description and the flow rule satisfies normality on the second
Piola-Kirchhoff stress space.
Substituting Equation (3.27) into Equation (4.2) yields:
22
The flow rule is express by
AB(4.6)
where
A = J A (4.7)
It was proved by Kiousis et al (1985) that the use of Equations (4.4)
and (4.6) not only implies normality in the material description, but
also preserves normality in the spatial coordinates.
The relation between the second Piola-Kirchhoff stress rate and the
’’elastic" component of the material stress rate is postulated to be
linear:
The rate of the yield surface shift tensor (see Figure 4) is
expressed by (Shield and Ziegler, 1959):
SAB " EABCD eCD (4.8)
AAB “ ^SAB " AA B ^ (4.9)
where
(4.10)
where b is a material parameter. The parameter A is calculated from the
consistency equation:
f " £ ^SAB’ AAB» eA B ’ K) “ 0hence
(4.11)
f moves in direction of CP
FIGURE 4.1 MODIFICATION OF PRAGER'S KINEMATIC HARDENING RULE
Following the procedure outlined by Voyiadjis (1984) and Kiousis et al*
(1985), the expression for A is obtained:
0f + 8fJABCD 0sAB 3eCD
0f0J R.CD
"CD (4.12)
where
n _ r- 9f 3f 0f Bf T-1y " ABCD 0sAB 0sCD ‘ 0K SAB 0sAB J
9f 0f
<SAB ' AA B » 3 S m 8 S m m tt-13)8sAB “ AB CSq e -a qrJ I f -
The elastoplastic matrix which corresponds to the loading function
f(s> A, e, K, J) is
9f 8f F 3f 8f 9f 8f pr dsm 3sPQ ^ CD 9eCD 0SPQ 9sPQ 9J CD !
ABCD “ ABCD " ABPQ 1 Q J(4.14)
The incremental elasto-plastic constitutive relation can now be
expressed as:
SAB DABCD eCD (^*15)
The applicability of this formulation has been successfully demonstrated
by Kiousis et al (1985).
25
A.2 CAP MODEL BY DIMAGGIO AND SANDLERThe cap model consists of a fixed yield surface f^ and a hardening
yield cap f2 (Figure 4.2).
The expressions for f^ and f2 are as follows:
fl = ^J2D + Y e"PJl ‘ “ = 0 (4,165
where
^2D = J °kJl^kJI t ie secon<* deviatoric stress invariant;J. = a is the first stress invariant;1 mm '
a, P, and y are material parameters.
f2 = f2(s,e,J,aJ) = R2 J2D + (Jj - C)2 - R2 b2 = 0 (4.17)
Equation (4.17) is an ellipse, where
R b = X - C (4.18)
R is the ratio of the major to the minor axis of the ellipse; X is the
value of at the intersection of the cap with the axis; C is the
value of the at the center of the ellipse. X is the hardening para
meter and depends on the plastic volumetric strain:
£PX = - i In (1 - ^ ) + Z (4.19)
where D, Z, and W are material parameters. Z is the value of at the
initial yield.
Equations (4.16) through (4.19) describe the cap model in the
spatial reference frame. Following a procedure similar to the one given
26
Initial Cap
Xc J
FIGURE 4.2 CAP MODEL BY DIMAGGIO AND SANDLER
27
for the extended von Mises, the elastoplastic stifffness tensor is given
by (Voyiadjis et al., 1985):
3f 9f F 3f 3f 3f 3f nr3sA B 9sCD ABPQ 9sP Q 9sCD 9sC D 9J V
MNPQ ~ MNPQ " MNCD 1 3f 3f r 3f 9f 1
9sef 9sgh efgh " aep GH 3sgh (4.20)
where f can be either f^ or ££■ For the case of the yield function f^.
9flthe expression — „ equals zero, v
5. PORE WATER PRESSURES
The elastoplastic formulations developed in Chapter 4 refer to
effective stresses. In the case of undrained loading, pore water pres
sures should be calculated. To calculate of these pressures, the
following incremental expression is assumed:
= Kb j (5.1)
where ^ is the increment of the pore water pressure and is the un
drained bulk modulus of the soil-water system.
28
6 . ELASTO-PLASTIC STIFFNESS TENSOR FOR UNDRAINED LOADING
The total stress ff is related to the effective stress a 1 through:
°kjt = + 6w * (6a)
Transformation of Equation. 6.1 to material coordinate parameters yields
(Kiousis and Voyiadjis, 1985):
SAB = SAB + ^ J CAB
The rate form of Equation (6.2) is
SAB " SAB + DABCD eCD
where (Voyiadjis et al, 1985)
DABCD = J CAB Kb RCD + ^ CAB RCD " ^CAC CBD + CBC CAD^ ^
Substituting for s^g from (4.15)
SAB = °ABCD eCD (6,5^where
°ABCD = DABCD + DABCD (6‘6^
In Equation (6 .6) D' nTl is given by either expression (4.14) or (4.20),AJj LJJ
29
7. FINITE ELEMENT FORMULATION
The formulation used in this work is based on the principle of
virtual work:
6u is the variation of the displacement field;
fie is the variation of the strains due to 5u;
s is the stress:< v *
f is the body force; and
£ is the surface traction.
The domain of integration V of Equation (7.1) is discretized by a quad
rilateral isoparametric element mesh. Both four node bilinear and eight
node parabolic elements are used (Figure 7.1). In the ensuing discus
sion, the four node quadrilateral element is referred to as the Q4
element, while the eight node element is symbolized by Q8.
For an element "k", the corresponding nodal displacements are
described by
where N is 4 for a Q4 element and 8 for a Q8 element.
The global displacement vector £ and the element displacement
vector are related by
Xv fieT s dV - Jv fiuT f dv - Js fiuT £ dS = 0 (7.1)
where
(7.2)
30
2
3
6
2PARABOLIC QUADRILATERAL
FIGURE 7 .1 BILINEAR AND PARABOLIC QUADRILATERAL ELEMENTS
8 i >
4 »
I O-
BILINEAR QUADRILATERAL
32
3k = Xk a (7.3)
The displacement field within each element is related to the nodal
displacements through the shape functions N :
3k ' Hk ak = Jk ak = a *!• I»2 >- • •• ™ ) I a (7.4)
The procedure is treated extensively in most finite element texts
(Zienkiewicz 1971, Desai and Abel 1972).
The strain Equation (3.12) is written in a matrix formalism for an
axisymmetric problem as:
9u3r
J > 3u 9u 9v 0-
er 9r 9r 9r 0 0
3v 0 0 9u 9vez> <
3z>
9z 9z 0
3v 3u 9u 9v 3u 9v 0^rz 9r + 9z 9z 9z 9r 9r
e u 0 0 0 0 u0 r r
. -
or
3v3r
3u > 3z
3v3zur
= <Sk + £ SE> sk
(7.5)
(7.6)
where
33
3N13r
3Njdz~
3N.
3N„ __zdr
9N2dz
3NX 3N2 3N23z 3r 3z 3r
N1 N2o —r r
53r
3N„
3N3z
3N,
N
N3z 3r
NN — 0 (7.7)
and B" = A • G ~k ~ ~ (7.8)
where
A '=
N 3N. N 3N.I 5 ~ U . , I jjj ~ V .. , 3r l ’ . , 3r i1=1 1=1
0
N 3N. K 3N.
N 3N. N 3N., 2 u. , 2 v. ,’ . . 3z i . , 3z ii=l i=l
N 3N. N 3N.i l i v i^ u4 i ^ v,- > ^ av u,- > ^ av >. . 3z i ’ . 3z ii=l i=l
0
. . 3r i ’ . . 3r i i=l i=l
0
0
N N., I — u.9 . r ii=l
(7.9)
and
34
G =
3N]§ F
aNj
o
N,
0
8^9r
3N]dz~
0
9N,9r
59z
N,
9N,0 9r
9N5 - = ......... o9r
9N.° 9z
N2sf ........ 0
NN0 —
N 0
9N,N9r
0
9N,N9z
(7.10)
Hence, Equation (7.6) can be written as:
(7.11)
The flow rule of plasticity requires the incremental expressions
for the quantities used.
The differential expression of the Equation (7.6) is:
d®k = Sk d9k + J d Sk % + I Sk" d9k (7.12)
It is verified (Zienkiewicz 1971) that
dB (7.13)
Due to (7.13), Equation (7.12) becomes
= CBi + ip d£k = Sk d9kor
(7.14)
dek = Bk Tk da (7.15)
35
Using 7.15, the incremental constitutive relation (6.5) becomes:
ds = D de (7.16)w i v / w
or
dEk = 8k \ dak (7.17)
thus
s. = J* D. B. dq (7.18)~k Jo ~k ~k *
The integration in equation (7.18) is along the deformation path.
Substitution of (7.4) and (7.15) into (7.1) yields:
m f l l m f p m n i m rp
da 2 T 1 s B 1 s . d V - 2 T 1 J V f d V (7.19)k=l K vk K k=l k
rp ni rp rp
- «aT * Ik S£ fikT e ds = ok=l k
where m is the number of the discretizing elements.TEquation (7.19) is valid for any virtual dq , hence,
m r p r p ® r p rp
2 T, 1 Jw B, s . d V - 2 T. 1 /„ N. 1 f dV. , JV, ~k ~k . . ~k JV, ~k ~k=l k k=l k
- 2 T.T L N.T P dS = 0 (7.20), ~k J S ~k ~ k=l x
Note that the first terra in Equation (7.20) is a non-linear function of
since both and s^ are functions of q.
Equation (7.20) is written as:
9 (a) = 5 (7.21)
where
36
m T8Cs> = Ik /V|t \ *k dv (7.22)
and
R =m T T ra T T* Xk fu Sk f dV + I Tk J* Nk E dS
k=l k=l(7.23)
The differential form of (7.22) is:
« C a ) » I T fv (dgk Sk ♦ 8 dg ) dVk=l k
(7.24)
It can be proved that (Zienkiewicz, 1971):
dBk Jk = Sk dawhere
(7.25)
C. = G M G ~k ~ ~ ~and
M =
Let
and
(7.26)
11 0 S12 0 0
0 S11 0 S12 0
12 0 S22 0 0 (7.27)
0 S12 0 S22 0
0 0 0 0 S33
TSk dV = S % (7.28)
ds.~k dV = f \ TVk ~ kD, B, ~k ~k dq„dV = E k d£k (7.29)
37
wheresK. = S C, dV (7.30)~k Jv. ~k
is the initial stress or geometric stiffness matrix, and
K. = /„ b / d . B. dV (7.31)~k J V. ~k ~k ~kk
is the material stiffness matrix.
Equation (7.24) can now be expressed as;
m TdS(fl) = I 4 S k l k da C7-32)k-1
where
Kk = K kS + Sk (7.33)
is the tangent stiffness matrix of the element k.
8 . METHOD OF SOLUTION
8.1 MATHEMATICAL FORMULATION
Expression. (7.21) represents a set of functional equations which
are solved by applying the load incrementally and performing iterations
within each increment (Voyiadjis and Buckner, 1983).
Let be the nodal displacement vector and R ^ be the load
vector at the end of the n ^ loading increment, at which
Let the next loading increment be AR. The corresponding increment of
the displacement vector is A<j. Thus,
Atj, is first approximated by the first term of the Taylor expansion of Q
9CaCn)) - £ Cn) = 9 (8.1)
3(qCn) + Afl) - (R(q) + AR) = 0 (8 .2)
at q ^ . Equation (8.2) becomes
9CflCn)) + Ko Aa - (RCl° + AR) = 0 (8.3)
Equation (8.3), due to (8.1), becomes
K Aq - AR = 0 (8.4)
where
a = a(n)
38
39
is the tangent stiffness matrix at 3 = 3 ^ • Solving for A3 from (8.4),
the first approximation of A3 results in:
Aq,-,-, = Ko * AR (1) ~ ~
Equation (8.2) does not hold for j instead,
+ *9(1)) - (S(n) + *5> = - 4 (8-5)
A second corcection for A£ is obtained by expanding g at £ = £ +
A^fi), which results in
2(S(n) + *3(1)) + K j A£ - <8(n) + AR) = 0 <8.6)
Equation (8 .6), due to (8.5), simplifies to:
Ka A3 = t}; (8.7)
S %where is the tangent stiffness at 3 = 3 + A£^^.
From (8.7), a refinement in the approximation for A3 is obtained:
Ad(2) = K"1 4 (8 .8)and
A£ = a£(i) + ^£(2) (8*9)
The procedure is repeated until A3 is approximated to a desired
accuracy.
The ith correction to A3 is given by
* 3 ( 0 = * 1-1 (8'10)where
Ki-X ” S(£Cn) + a£(i) + A£(2) + *-'+ A£(i-1)5 (8.11)
40
and
4i-l = Sta(n) + *9(1) + *9(2) + *9(1-!)) " (£(n) * (8-12)
The value of Aq after i corrections is
Aa = A% ) + a2 (2) + --*+ A<l(i) (8*13)
The procedure outlined here is known as the Newton-Raphson tech
nique for the solution of a system of non-linear algebraic equations,
and is graphically depicted in Figure (8.1).
8.2 REALIZATION OF THE METHOD
The procedure outlined in the previous section is realized as
follows in the finite element method.
At the end of the nth increment, the load is the displace
ments are the strains are e ^ , and the stresses are s ^ . A load
increment AR is applied.
laic .00
Step 1: Calculation of the tangent stiffness matrix for and
from equation (7.33)
Step 2: Solution for A q ^ from the system of equations: A q ^ j =
AR
Step 3: Aq = Afl(l)
Step 4: Calculation of the strain increments Ae from (7.15) and the
stress increments AS from (7.16). The current total strains
are: e = e^n + Ae and the current total stresses are~c ~ ~s ^ = s(n) + Asi " W Q jV
Step 5: The load Rc ^n which is equilibrated from the new total stress
field is calculated from Equation (7.22)
MODAL
LOAD
Q(q)
41
NODAL DISPLACEMENT q
FIGURE 8.1 NEWTON-RAPHSON APPROXIMATION TECHNIQUE
42
Step 6 : The remaining load which has not been equilibrated (t|j) is
given by: $ = R Ctl) + AR - R ^/ \ / \
Step 7: The tangent stiffness matrix Kj for q. + ^ q ^ an( £ c
from Equation (7.33) is calculated.
Step 8 : ^3.(22) is solved for from the system Aq^^j = &
Step 9: Aq = A q ^ + A q ^
Step 10: Steps 4 - 9 are repeated until j{j approaches 0 to the required
accuracy.
The procedure described in Steps 1 - 1 0 requires very small load
increments for a plastified material. As a result, lengthy and expen
sive computer runs are required. A refinement of Step 4 makes the
method more efficient.
The increment of strain is calculated as As = D Ae instead of As =
J" D de. To keep this approximation to acceptable accuracy, small load
increments are required. An alternate and more accurate procedure is as
follows:
a) Calculation of the strain increment Ae
b) Division of Ae to m subdivisions: Ae = Ae/m
c) Calculation of As. = D. Ae~1 ~1 ~d) s(n) = s ^ + As.~c ~ 1e) Calculation of D„ for s ^ and e ^~2 c cf) Calculation of As„ = D- Ae~2 ~2 ~
g) s(n) = s(n) + As. + As, , e ^ = e ^ + 2AeO' ,-w < £ 2 ~C 'V ^h) Repeat Steps e - g for all m subdivisions of Ae
The procedure, for the correct number of subincrements m, results
in convergence in three or four iterations for relatively large load
increments.
9. CONE PENETRATION ANALYSIS
9.1 INTRODUCTION
A large number of friction and piezo-cone penetrometer tests (QCPT)
have been carried out at L.S.U- in the last 5 years, (Tumay and Y'ilmaz
1981, Tumay et al. 1981) which have contributed to the current state-of-
the-art analysis of the QCPT results.
In this work a simulated calibration of the cone penetrometer is
conducted with a completely different approach. A computer simulated
experiment rather than an actual in-situ experiment is carried out.
This approach has a number of very significant advantages which are
listed below.
The test can be carried out with any kind of soil, ranging from
very simple, homogeneous and isotropic, to very complex, nonhomogeneous,
and orthotropic, elastic or plastic.
The displacement, strain, and stress fields around the cone can be
quite accurately calculated. The whole procedure is similar to a very
densely instrumented calibration experiment. Through this approach,
large costs and any disturbance due to the instrumentation may be
avoided. With such a procedure a better understanding of the stress,
strain and pore pressure fields at failure can be achieved.
The basic theoretical improvements for this computer simulated
experiment are the incorporation of large strains and nonlinear
(plastic) material behavior. These improvements increase the accuracy
of the method. Nevertheless, the problem of the correct plastic consti
43
44
tutive equations is not quite solved as yet. All the existing models
have been developed for strains as high as 5% to 10%. Previous analyti
cal and experimental studies have shown that strains on the order of 50%
or higher exist in the neighborhood of the cone tip. (Rourk 1961, Tumay
et al. 1984) It is quite probable, that the existing models do not
describe the material behavior accurately at this high level of strain
ing. This is a drawback of the method, which can only be overcome with
more experimental studies incorporating large strains.
In addition to the above shortcomings, strain rates as high as 700%
per second have been calculated at the tip of the cone. It is almost
certain that at such high rates the behavior of the soil is influenced
by viscous effects. A viscoplastic model is therefore probably more
appropriate for this problem, and is suggested as a future extension of
the present work.
9.2 PENETRATION SIMULATION
The penetration of the cone penetrometer is simulated based on the
following assumptions:
The penetrometer is infinitely stiff
There is no interface friction between the penetrometer and
the soil
Tensile interface forces are not developed i.e., if the force
at an interface node becomes tensile, the node is released and
allowed to move outward
In the following discussion, the conical surface of the penetro
meter is called "cone tip". This is consistent with the terminology
used in geotechnical engineering.
45
Two consecutive positions of the penetrometer in the soil are shown
in Figure 9.1. The finite element discretization for the area around
the cone tip is shown in Figure 9.2a. The shaded area represents the
rigid indentor, while the heavy line with the attached rollers repre
sents the boundary line of the penetrometer. The dotted lines represent
consecutive new positions of the penetrating cone. The elements and the
nodes shown in Figure 9.2a are used exclusively for the discretization
of the soil medium.
The penetration of the cone is simulated by a uniform vertical
movement of the boundary line of the rigid cone tip.rFor certain critical phases during penetration, certain nodes close
to both ends of the tip change their boundary descriptions. These areas
are shown in the encircled regions 1 and 2 in Figure 9.2a and are en
larged in Figures 9.2b and 9.2c, respectively.
Figures 9.2b and 9.2c are examined separately.
UPPER END OF THE TIP:
As the cone penetrates the soil, the position of the first node A1
immediately below the upper end of the cone is examined. This is shown
in Figure 9.2b, where node A1 is traced to the positions A1-B1-C1-D1
through a penetration length 1. When in positions Al, Bl, and Cl, the
node is within the radius of the penetrometer shaft and is restricted to
move along the cone tip boundary. At position D1 the nodal point has
reached the physical boundaries of the cone shaft and the restrictions
of its movement are changed. If the nodal point tends to move outwards
after position Dl, it should be free to do so and all restrictions are
removed. If, on the other hand, the point shows a tendency to move
inwards, a vertical roller restricts its movement on the penetrometer
POSITION m
POSITION m + n
m ,n : Number of Incremental Penetrations
FIGURE 9.1 CONSEQUTIVE POSITIONS OF THE PENETRATING CONE
i
FIGURE 9.2 DETAIL OF BOUNDARY CONDITION CHANGE OF NODAL POINTS AT THE UPPER AND LOWER ENDS OF THE CONE TIP
-E>
48
shaft. This procedure is then repeated for the next node in line A2.
LOWER END OF THE TIP:
Change of boundary condition for the lower end of the tip during
penetration is also required.
In Figure 9.2c, the consecutive positions of the cone tip are
labeled with the lower case letters a-e, while the upper case letters
A-E show the corresponding positions of the node immediately below the
lower end of the tip.
The first node below the lower end of the cone is originally
restricted to move on the axis of symmetry. When the tensile forces
cause fracture of the soil, the node is freed to follow its own move
ment. (Positions A-B-C-D in Figure 9.2c).
Nevertheless, as penetration continues, the node comes into contact
with the cone surface (position E ) . At this point, new boundary
restrictions are applied so that the node remains in contact with the
cone tip. This procedure is repeated for as many nodes as eventually
come into contact with the cone tip.
9.3 PRESENTATION AND DISCUSSION OF THE RESULTS
9.3.1 INTRODUCTION
The soil chosen for this work is a medium soft clay with undrained2shear strength s = 5 0 KN/m . The modulus of elasticity equals: E =
25000 KN/m , and the Poisson's ratio for the soil skeleton is V = 0.30.
For the solution of the cone penetration problem, the soil is
discretized by a mesh of 8-node (Q8) quadrilateral elements as shown in
Figure 9.3. The soil is assumed to undergo undrained loading in a
region defined by a cylinder of radius 3rQ around the cone, where rQ is
49
\ul
FIGURE 9.3 FINITE ELEMENT MESH FOR THE DISCRETIZATION OF THE SOIL MEDIUM DURING PENETRATION
50
the radius of the penetrometer shaft. This assumption was based on a
number of experimental and analytical data (Rourk 1961, Levadoux and
Baligh 1980) and suggest that the strain rates during penetration are
minor outside of this region, and drainage is therefore not prevented.
The analysis carried out here verifies this assumption.
To make the analysis easier and economically feasible, the penetra
tion is assumed to start at a certain depth and continued until a com
plete failure is achieved (Figure 9.4). For the problem solved here,
failure occurs at a penetration depth of 6 mm. The penetration is
continued up to 10.8 mm to ensure that failure has been realized and to
obtain a penetration close to steady state. Although these displace
ments (6 mm - 10.8 mm) appear to be very small, one should realize that
they are of the same order of magnitude with the diameter of the cone
(d = 35.7 mm), consequently the failure seems natural.
9.3.2 DISPLACEMENT FIELD
The pattern of the displacement field obtained here (Figure 9.5) is
found in agreement with experimental results presented by Davidson
(1980), and Davidson and Boghrat (1983). The displacement field is
found to be almost vertical underneath the cone tip, but,as the radial
distance from the cone increases, the displacements acquire oblique
angles, up to 45° with the horizontal.
This displacement pattern is very different from the one of the
analagous plane strain problem. In that case, zones of soil with upward
movement are observed and in many situations, a clear failure line is
obtained (Griffiths, 1982). None of these appears in the axisymmetric
problem.
PEN
ETR
ATI
ON
RE
SIST
ANCE
qc
(x!O
OK
N/m
51
0.7
0.6
0.5
0.4
0.3
0.2
PENETRATION LENGTH 8 1 (mm)
FIGURE 9.4 LOAD - PENETRATION RELATION
52
FIGURE 9.5 DISPLACEMENT FIELD AROUND THE PENETROMETER
53
A number of investigators have analysed the problem of penetration
by treating it as a plane strain problem. A correcting factor similar
to the one relating the shear strength obtained by axisymmetric and
plane strain tests was used in these analyses (Terzaghi 1943, Meyerhof
1951, 1961, Mitchell and Dorgunoglu 1973). In view of the completely
different displacement fields created in the two problems, it seems that
such a relation cannot be as simple nor as general.
A second interesting point that is observed in the displacement
field is the separation of soil and cone shaft interface for approxi
mately 35 mm above the upper end of the cone tip (Figure 9.6). It can
be argued that soil does not satisfy the assumption of being a continuum
in the region so close to the cone (i.e. it has been subjected to frac
ture) and therefore, the size of the separation zone is not reliably
predicted. Nevertheless, this result gives a strong indication that
pore pressure transducers and sleeve friction gauges in this area cannot
function properly. Readings such as side friction and can be severely
underestimated if their values are based on measurements on the separa
tion zone.
9.3.3 STRAIN FIELDS
The change in strains around the penetrating cone for penetrations
of 1 mm, 3 mm, and 6 mm are presented in Figures 9.7 through 9.11. The
penetrations of 1 mm, 3 mm, and 6 mm were chosen based on Figure 9.4.
The penetration of 3 mm seems to be the major breakdown of the resis
tance of the soil, and at the penetration of 6 mm the soil resistance
becomes almost constant.
In Figures 9.7a, b, and c, the radial strain field (e^) is pre
sented. The lower end of the cone tip is an area of high strain concen-
54
i
FIGURE 9.6 SCHEMATIC REPRESENTATION OF THE SEPARATION OF THE SOIL - PENETROMETER INTERFACE DURING PENETRATION
55
0.5
5- 2
-0.5 - 0.2 - 0.1
FIGURE 9.7 (a) RADIAL STRAIN (er) CONTOURS AROUND THE PENETROMETERFOR PENETRATION OF I mm
56
25
15 -0.8 -0.3
FIGURE 9.7 (b) RADIAL STRAIN (er) CONTOURS AROUND THE PENETROMETERFOR PENETRATION OF 3 mm
57
2030
35
FIGURE 9.7 (c) RADIAL STRAIN (er) CONTOURS AROUND THE PENETROMETERFOR PENETRATION OF 6 ran
58
trations and rates. The radial strains are in general compressive
around the cone, but significant tensile strains appear below the lower
end of the penetrometer. In a region of approximately 3 ram around the
lower end of the tip, the strains vary from +45% to —35%. There is no
doubt that the soil in this region is very plastified (practically
ruptured) and that the constitutive relations used are probably inade
quate to describe the soil behavior there. Nevertheless, important
information can be drawn from these figures. Comparing Figures 9.7a, b,
and c, it becomes apparent that as penetration increases, the plastified
region expands and the radial and axial distributions of strains become
more uniform.
The axial strain increment eg for the three penetrations of 1 mm,
3 mm, and 6 mm are presented in Figures 9.8a, b, and c. The entire area
below the cone is subjected to compressive strains which are as high as
81%. The axial strain increments around the shaft are tensile and of
smaller magnitude. It is interesting to note that for a certain region
below the cone, the compressive strains show larger values away from the
axis of symmetry. This is attributed to the geometry of the cone. As
in the case of radial strains e , the radial- and axial distribution ofr ’e^ becomes more uniform as the penetration increases.
The distribution of the tangential strain increments eg is pre
sented in Figures 9.9a, b, and c. The tangential strains are everywhere
tensile and their distributions, both axially and radially, become more
uniform as penetration increases.
Large rates are also observed for the case of shear strain incre
ments e^z (Figures 9.10a, b, and c). The two ends of the cone tip form
poles of strain concentration. At the lower tip, the shear strains drop
FIGURE 9.8 (a) AXIAL STRAIN (e ) CONTOURS AROUND THE PENETROMETERzFOR PENETRATION OF 1 ram
60
74 0.6
FIGURE 9.8 (b) AXIAL STRAIN (e ) CONTOURS AROUND THE PENETROMETERzFOR PENETRATION OF 3 nnn
61
FIGURE 9.8 (c) AXIAL STRAIN (e ) CONTOURS AROUND THE PENETROMETERzFOR PENETRATION OF 6 mm
FIGURE 9.9 (a) TANGENTIAL STRAIN (e0) CONTOURS AROUND THE PENETROMETER FOR PENETRATION OF 1 mm
63
3Q70,
FIGURE 9.9 (b) TANGENTIAL STRAIN (e0) CONTOURS AROUND THEPENETROMETER FOR PENETRATION OF 3 ram
FIGURE
65
05 0.2
FIGURE 9.10 (a) SHEAR STRAIN (e ) CONTOURS AROUND THErzPENETROMETER FOR PENETRATION OF 1 mm
66
-4 -0.5-70
I
FIGURE 9.10 (b) SHEAR STRAIN (e ) CONTOURS AROUND THErzPENETROMETER FOR PENETRATION OF 3 m
'
FIGURE
_l2d/-50-25-20 -10
9.10 (c) SHEAR STRAIN (e ) CONTOURS AROUND THErzPENETROMETER FOR PENETRATION OF 6 mm
68
from e = -120% (v = -240%) to e = -2% in a very small region, rz rz rzAgain, the strain distributions become more uniform with increasing
penetration. Similar distributions are observed for the octahedral
shear strains increments Yoct (Figures 9.11a, b, and c). Since it
reflects the effect of all strains, the octahedral shear strain is very
important to describe the overall straining of the soil. The amount of
straining reaches the value of 110% (engineering strain of 220%), while
the tendency for uniformity of strain distributions with penetration is
demonstrated once more.
In general, the strain fields created from the penetration of the
cone penetrometer are very large and they should be treated as such if a
valid analytical solution of the problem is to be obtained. It is
important to realize that although very large strains are developed,
their rate of drop with radial and axial distance is very rapid which
makes the failure of the soil around the cone very localized. This is
very fortunate because it implies that the response recorded from the
QCPT is averaged on small soil regions and therefore describes the soil
behavior quite accurately.
9.3.4 STRESS FIELDS AND PORE WATER PRESSURES
The stress increments due to the penetration of 6 mm are presented
in Figure 9.12 through 9.17.
At this stage, the penetration resistance is almost stabilized and
the stress changes exhibit small variations with further penetrations.
In the area around the cone, the radial stress a' (Figure 9.12) is
the dominant one, being responsible for large pore water pressure gener
ation. The stress concentration around the lower end of the cone is
very significant. The stresses drop from the compressive value of
69
0.5 0.2
FIGURE 9,11 (a) OCTAHEDRAL SHEAR STRAIN CYoct) CONTOURS AROUNDTHE PENETROMETER FOR PENETRATION OF 1 mm
70
0.630,
70
FIGURE 9.11 (b) OCTAHEDRAL SHEAR. STRAIN (Y ) CONTOURS AROUNDoctTHE PENETROMETER FOR PENETRATION OF 3 mm
71
25 15
I
FIGURE 9.11 (c) OCTAHEDRAL SHEAR STRAIN (Y ) CONTOURS AROUNDoctTHE PENETROMETER FOR PENETRATION OF 6 mm
72
0.2 (xlOOKN/m )0.52.4>
3.0,
-0.7-0.5 -0.3 - 0.2
FIGURE 9.12 RADIAL STRESS (a ) CONTOURS AROUND THE PENETROMETERrFOR PENETRATION OF 6 mm
73
2 2 460 KN/ra to the tensile value of 20 KN/m in a very small region. The
stress bulbs confirm the localized nature of the problem. There exists
a region of small tensile stresses below the cone. If a negative stress
cut-off were used, the overall distribution of stresses would probably
be affected, but it is believed that the changes should be minor.
The large pore pressures developed around the penetrometer are
responsible for a thin layer of tensile axial stresses o ' (Figure 9.13)
surrounding the cone tip. The compressive nature of the problem is
revealed immediately outside of this zone by the creation of compressive
stress bulbs which originate below the cone tip and extend upwards.
Significant tensile tangential stresses are also developed (Figure
9 -14). Again the concentration of stresses around the ends of the cone
are obvious. The bulbs of stresses Oq ' are extending outwards with a
more uniform way than o ' and &z '• It is very interesting to notice at
this point that if the tangential stresses CTg' are compared with the
pore water pressures u (Figure 9.20c), positive total stresses are
found. This observation indicates the significance of the undrained
loading assumption. It is possible that if some drainage were allowed,
the stress distributions would be completely different. The shear
stress increments x are presented in Figure 9.15, and once more
intense stress concentration is observed at the lower end of the cone2 2where the stresses drop from the value of 360 KN/m to 20 KN/m in a
very small region.
The pressure bulbs are presented in Figure 9.16, where large
compressive zones are shown around the cone, while some tensile stresses
are revealed aroung the shaft.
FIGURE
75
FIGURE 9.
0.2 (xlOOKN/m )1.5 -0.9 0.5
14 TANGENTIAL STRESS (aQ) CONTOURS AROUND THEPENETROMETER FOR PENETRATION OF 6 mm
76
FIGURE
(xlOO KN/m )
0.20.30.5
9.15 SHEAR STRESS (T ) CONTOURS AROUND THE PENETROMETERrzFOR PENETRATION OF 6 mm
77
0.5 0.2
I.Oi3.0
0.3 0.2 CxlOOKN/m )1.0 0.5
1.5,
0.50.3
0.2
FIGURE 9.16 OCTAHEDRAL STRESS (a _) CONTOURS AROUND THEoctPENETROMETER FOR PENETRATION OF 6 ram
78
The octahedral shear stress increments ate shown in Figure
9.17. The octahedral stresses are important in the sense that they
provide a combined effect of the straining to which the soil is sub
jected to. In addition, the octahedral shear stresses are an important
factor of pore pressure generation.
In conclusion, form all the stress diagrams it is confirmed that
the QCPT causes localized failure. Figures 9.18 and 9.19 are plotted to
demonstrate this fact. In Figures 9.18a and b the radial distributions
of a . and T . around the lower end of the tip are shown. The oct oct e
stresses drop severely within a distance of 0.5 rQ and become unimpor
tant at the distance of 3r . Similar distributions of a . and t . areo oct octobserved around the upper end of the cone. (Figures 9.19a and b).
The pore pressure generations for the penetrations of 1.0 mm,
3.0 mm, and 6.0 mm are presented in Figures 9.20a, b, and c. Important
information is obtained from these figures. The pore pressure distri
bution around the cone becomes more uniform as penetration increases.
For example, at the penetration of 1 mm, the ratio of the pore pressures2 7at the lower end, and the middle of the cone is = 5.4. The
ratio becomes R = = 2 . 5 for the penetration of 3 mm and h = =U f a • Z tl j « u2.0 for the penetration of 6 mm. This is a very important observation.
Since the pore pressure distribution tends to become uniform around the
cone, the position of the pore pressure transducer on the cone is
probably not as important as it was thought to be. Nevertheless, if the
optimum position were sought, it is suggested that the region between
the lower third and the middle of the cone measures a representative
average.
pl£j
0.51.0
FIGURE 9.17 OCTAHEDRAL SHEAR STRESS ( t J CONTOURS AROUND THEoctPENETROMETER FOR PENETRATION OF 6 mm
OCTA
HEDR
AL
STRE
SS
<roct
{x 10
0 KN
/
m
80
7 i—
CM
o 2r•o
FIGURE 9.18 (a) RADIAL DISTRIBUTION OF THE OCTAHEDRAL STRESS(O ) AT THE LOWER END OF THE CONE TIP oct
OCT
AHED
RAL
SHEA
R ST
RESS
{x
. 100
K
N/m
8a
CM
4 -1
2Or
FIGURE 9.18 (b) RADIAL DISTRIBUTION OF THE OCTAHEDRAL SHEARSTRESS (T ) AT THE LOWER END OF THE CONE TIP oct
OCTA
HEDR
AL
STRE
SS
<roct
{ x 10
0 K
N/m
82
5
4
3
2
1 2
FIGURE 9.19 Ca) RADIAL DISTRIBUTION OF THE OCTAHEDRAL STRESS(a ) AT THE UPPER END OF THE CONE TIP oct
OC
TAH
ED
RA
L SH
EAR
STR
ESS
roct
(x IO
O K
N/m
83
5
4
3
2
2 3rr0
FIGURE 9.19 (b) RADIAL DISTRIBUTION OF THE OCTAHEDRAL SHEARSTRESS (T ) AT THE UPPER END OF THE CONE TIP oct
84
-0.1-0.05J J
0.05 (xlOO KN/m )
FIGURE 9.20 (a) PORE WATER PRESSURE (u) CONTOURS AROUND THEPENETROMETER FOR PENETRATION OF 1 mm
85
0.3
3 . 0 / / /4.5/72.01.2 o.5
0.2 ( xlOO KN/m )5
FIGURE 9.20 (b) PORE WATER PRESSURE (u) CONTOURS AROUND THEPENETROMETER FOR PENETRATION OF 3 mm
86
- 0.60.4
' - 0.2
6.0j
FIGURE 9.20 (c) PORE WATER PRESSURE (u) CONTOURS AROUND THEPENETROMETER FOR PENETRATION OF 6 mm
10. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
In situ soil testing techniques have gained a significant popu
larity in site investigations and in the determination of the shear
strength and compressibility parameters of soils. The increased impor
tance of the in-situ techniques is attributed mainly to three reasons.
1. The growing cost of the traditional exploration techniques which
are based on sampling through boring and lab testings.
2. The increasing number of off-shore projects and constructions on
regions where sampling becomes difficult and unreliable.
3. The improved analytical and numerical capabilities which are pro
vided through the advance of computer technology, and require a
more detailed soil description.
The electric cone penetrometer is one of the most successful in-
situ testing devices due to its wide applicability, simplicity, and
economy. It has proved to be a very useful device in off-shore site
investigation, and its unique capability to provide continuous soil
profiles makes it preferable over other in-situ and laboratory tech
niques .
The means of analysis of the test have been provided mainly through
traditional deep foundation approaches (bearing capacity, and cavity
expansion equations). An alternative procedure was provided by Baligh
et al. (1980) and Tumay et al. (1984) which assumes a steady state
inviscid flow around the cone penetrometer to computer the kinematic
field created through its penetration. Subsequent implementation of a
87
88
plasticity model yields the stress fields and the cone tip resistance.
A number of unrealistic assumptions involved in the aforementioned
methods of analysis limit their applicability. Incorrect kinematic
fields, small strain assumptions and simplified constitutive relations
are some of the drawbacks of these methods.
In this work, a new large strain elasto-plastic approach is intro
duced. To overcome the inaccuracies of the Green and Naghdi Lagrangian
formulation, and the inability of present Eulerian formulations to
consider anisotropic hardening, an alternate approach is used. The
basic relations are developed in Eulerian formulation to preserve their
physical significance. They are subsequently transformed in lagrangian
space and through simple time differentiation, their rate equations are
introduced. With this approach, the disadvantages of the previous
formulations vanish. A computer program called EPAFI (Elasto-Plastic
Analysis For Indentation), which implements the theory presented in
Chapters 3 through 8 , was developed at LSU during the period of this
work. The method proved to be expensive in the beginning, but, as
improved numerical schemes were incorporated in EPAFI, the cost was
considerably reduced. It is expected that further improvements in the
stress increment calculations and corrections during plastic loading
will turn the method into an inexpensive research and practice tool.
The need for accuracy necessitated the use of very refined finite ele
ment mesh, which created unsurpassed computational and storage problems
for the LSU computer system, which is based on an IBM 3081 machine. A
grant from the National Science Foundation (NSF) was obtained for use of
the supercomputer CDC CYBER 205, at Purdue University. Solving the
89
finite element problem in this system proved to be a very smooth and
rewarding operation.
A number of important conclusions were drawn from the obtained
solution of the penetration problem. These conclusions have only quali
tative value, and are presented with the understanding that a larger
number of tests is needed for a concrete generalization. Also, the
sensitivity analysis of the chosed finite element mesh was examined for
elastic loadings only.
1. The penetration mechanism during the QCPT appears to be a localized
phenomenon for soft cohesive soils. This means that the response
recorded during the test (qc) is averaged on small regions and
consequently describes the soil behavior quite accurately.
2. Soil response measurements in the area extending 3 to 5 cm above
the cone tip could be misleading because of possible soil-penetro-
meter separation. Side friction can be severely underestimated if
it is based on measurements on this area.
3. The pore water pressure distribution around the cone tip becomes
fairly uniform as the penetration acquires steady state character
istics. Consequently, the position of the pore pressure transducer
on the cone tip is not very important and should be dictated from
design needs instead. Nevertheless, the area between the lower
third and the middle of the cone is probably the most representa
tive of the average pore pressure generation during penetration.
4. Although distinct bulbs of failure zones are created around the
cone, no distinct slip lines are generated. It can be concluded
that an ultimate bearing capacity solution for soft cohesive soils,
90
based on the slip line theory, cannot be general nor reliable
despite its mathematical simplicity and elegance.
5. The constitutive assumptions' in this work are not satisfactory
through the whole range of loading. The high strains calculated in
this analysis (y = 24%, efi = 105%, e = 81%, e = 45%) raise the xrz o z rquestion of reliability of the constitutive equations. The vast
majority of constitutive laws available have been developed for
much smaller strains (<20%). It ,is probable that the behavior of
soil at high strains is different than the one predicted by the
constitutive models employed in this work. Also, for a penetrating
rate of 2 cm/sec, shear strain rates as high as 700% per second are
calculated. At these high rates, the viscuous effect on the soil
behavior is probably significant. It seems that a viscoplastic
analysis, rather than the classical plasticity approach, is more
appropriate. Constitutive equations that are developed at large
strains and incorporate rate effects are not included in the scope
of this work, but are suggested as essential future improvements.
The concentration of stress at the cone tip results in very abrupt
changes of their values. Hybrid elements which incorporate the expected
stress distributions in the concentration areas can therefore be more
appropriate. Such elements can be introduced through the use of the
virtual complementary work equation (Pian 1964, Atluri 1975).
The results of stress analyses are usually presented in normalized
forms, which are advantageous because of their generality. This
approach is not applied in this study because there is not enough infor
mation to allow a reliable normalization of stresses and strains in
Figures 9.7 through 9.20. The penetration problems should be simulated
91
with a sufficient variety of soil constitutive parameters before any
normalization is attempted. The high nonlinearity of the problem does
not permit predictions in this direction.
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VITA
Panagiotis D. Kiousis was born on January 16, 1956 in Argostoli of Kefalenia, Greece. He received his Diploma in Civil Engineering from the Democritus University of Thrace, Greece in August 1980.He was admitted to the graduate program of L.S.U. in the Fall . semester of 1981 as a research assistant towards a doctoral degree in Civil Engineering, which he was granted upon completion of this research program in August 1985. He is the son of Filitsa and Dimltrios Kiousis and is married to Marianne Mayard Kiousis,
98
Candidate:
Major Field:
Title of Dissertation:
DOCTORAL EXAMINATION AND DISSERTATION REPORT
Panagiotis Demetrios KIOUSIS
Civil Engineering (Geotechnical)
Large Strain Theory as Applied to Penetration Mechanism in Soils
Approved:
/\ a a a a j l /^ A jMehmet T. Tumay
Major Professor and Chairman
Dean of the Graduate School
EXAMINING COMMITTEE:
George Z. VoviacnTsD-JChairma
al m n R A.car.
David Klein_________
-Chp2-George Cross________
Date of Examination:
July 18, 1985