Louisiana State University Louisiana State University

LSU Digital Commons LSU Digital Commons

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

Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses

Recommended Citation Recommended Citation Kiousis, Panagiotis Demetrios, "Large Strain Theory as Applied to Penetration Mechanism in Soils (Plasticity)." (1985). LSU Historical Dissertations and Theses. 4098. https://digitalcommons.lsu.edu/gradschool_disstheses/4098

This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact gradetd@lsu.edu.

INFORMATION TO USERS

This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted.

The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction.

1. The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)” . I f it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity.

2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. I f copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame.

3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method o f “sectioning” the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. I f necessary, sectioning is continued again—beginning below the first row and continuing on until complete.

4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department.

5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed.

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.

REFERENCES

Acar, Y. (1981), "Piezo-Cone Penetration Testing in Soft Cohesive

Soils," Geotechnical Engineering Report No. GE-81/01, Louisiana

State University, Baton Rouge, Louisiana, 81 pp.

Atluri, S. N. (1975) "On Hybrid Finite Element Models in Solid Mechanis"

Advances in Computer Methods for Partial Differential Equations

(Ed. R. Vichnevetsky), pp. 346-356.

Baligh, M. M. (1975), "Theory of Deep Site Static Cone Penetration

Resistance," Publication No. R75-56, Department of Civil Engineer­

ing, M.I.T., Cambridge, Mass.

Baligh, M. M., Vivatrat, V., and Ladd, C. C. (1979), "Exploration and

Evaluation of Engineering Properties for Foundation Design of

Offshore Structures", Publication No. MITSG 79-8, Department of

Civil Engineering, M.I.T., Cambridge, Mass.

Banerjee, P. K. and Fathallah, R. C. (1979), "An Eulerian Formulation of

the Finite Element Method for Predicting the Stresses and Pore

Water Pressures Around a Driven Pile," Proceedings of the Third

Int. Conf. on Numerical Methods in Geomechanics, Aachen, pp.

1053-1060.

Bathe, K. J., Ramm, E., and Wilson, E. L. (1975), "Finite Element Formu-

,.lations for Large Deformation Dynamic Analysis," International

Journal Numerical Methods in Engineering, Vol. 9, pp. 353-386.

92

93

Carter, J. P., Booker, J. R., and Davis, E. H. (1977), "Finite Deforma­

tion. for an Elasto-Plastic Soil," International Journal for

Numerical and Analytical Methods in Geomechanics, Vol. 1, pp.

25-43.

Dafalias, Y. F. (1983), "Corotational Rates for Kinematic Hardening at

Large Plastic Deformations," Transactions of the ASME, Journal of

Applied Mechanics, Vol. 50, pp. 561-565.

Davidson, J. L. (1980), "A study of deformations in sand around a pene­

trated probe using stereo photography" University of Florida

report, NSF Grand ENG 76-18849.

Davidson, J. L. and Boghrat, A. (1983), "Displacements and Strains

Around Probes in Sand" Geotechnical Practice in Offshore Engineer­

ing, Ed. S. G. Wright.

Davidson, H. L . , and Chen, W. F. (1974), "Elasto-Plastic Large Deforma­

tion Response of Clay to Footing Loads," Lehigh Universigh, Fritz

Engineering Laboratory Report No. 355, 18.

de Ruiter, J. (1981), "Current Penetrometer Practice," Cone Penetration

Testing and Experience, Ed. Norris and Holtz, pp. 1-48.

Desai, C. S. (1980), A General Basis for Yield, Failure and Potential

Functions in Plasticity," International Journal for Numerical and

Analytical Methods in Geomechanics, Vol. 4, pp. 361-375.

Desai, C. S., and Abel, J. F. (1972), Introduction to the Finite Element

Method, Van Nostrand Reinhold, New York.

Desai, C. S., and Phan, H. V. (1980), "Three-Dimensional Finite Element

Analysis Including Material and Geometric Nonlinearities," Computa­

tional Methods in Nonlinear Mechanics, ed. J. T. Oden, pp. 731-747.

94

DiMaggio, F, L., and Sandler, I. S. (1971), "Material Model for Granular

Soils," Journal of the Engineering Mechanics Division, ASCE, Vol.

97, No. EM3, pp. 935-950.

Fernandez, R. M., and Christian, J. T. (1971), "Finite Element Analysis

of Large Strains in Soils," NASA Res. Report R71-37, M.I.T.,

Cambridge, Mass.

Gadala, M. S., Dokainish, M. A., and Oravas, G. AE. (1984), "Formulation

Methods of Geometric and Material Nonlinearity Problems," Inter­

national Journal for Numerical and Analytical Methods in Engineer­

ing, Vol. 20, pp. 887-914.

Green, A. E., and Naghdi, P. M. (1965), "A General Theory of Elastic-

Plastic Continuum," Archives for Rational Mechanics and Analysis,

Vol. 18, No. 4, pp. 251-281.

Green, A. E., and Naghdi, P. M. (1971), "Some Remarks on Elastic-Plastic

Deformation at Finite Strain," International Journal of Engineering

Science, Vol. 9, pp. 1219-1229.

Griffiths, D. V. (1982), " Elasto-Plastic Analyses of Deep Foundations

in Cohesive Soils". International Journal for Numerical and

Analytical Methods in Geomechanics, Vol. 6 , pp. 211-218.

Hibbitt, H. D . , Marcal, P. V., and Rice, J. R. (1970), "A Finite Element

Formulation for Problems for Large Strain and Large Displacement,"

International Journal for Solids and Structures, Vol. 6 , pp. 1069-

1086,

Hofmeister, L. D., Greenbaum, G. A., and Evensen, D. A. (1974), "Large

Strain, Elasto-Plastic Finite Analysis," AIAAJ, Vol. 9, pp. 1248-

1254.

95

Kiousis, P. D., and Voyiadjis, G. Z. (1985), "A Lagrangian Continuum

Theory for Saturated Porous Media," to appear in the Journal of

Engineering Mechanics, ASCE.

Kiousis, P. D., Voyiadjis, G. Z., and Tumay, M. T. (1985), "A Large

Strain Theory for the Two Dimensional Problems in Geomechanics," to

appear in The International Journal for Numerical and Analytical

Methods in Geomechanics.

Ladanyi, B. (1967), "Deep Punching of Sensitive Clays," Proceedings, 3rd

Panamerican Conference on Soil Mechanics and Foundation Engineer­

ing, Caracas, Vol. 1, pp. 533-546.

Lade, P. V., and Musante, H. M. (1976), "Three-Dimensional Behavior of

Normally Consolidated Cohesive Soil," Soil Mechanics Laboratory,

School of Engineering and Applied Science, UCLA, Los Angeles, Cal.

Lee, E. H . , Mallett, R. L., and Wertheimer, T. B. (1983), "Stress

Analysis for Anisotropic Hardening in Finite-Deformation Plas­

ticity," Transactions of the ASME, Journal of Applied Mechanics,

Vol. 50, pp. 554-560.

Levadoux, J. N. and Baligh, M. M (1980), "Pore Pressure During Cone

■ Penetration in Clays," Report No. MITSG 80-12, M.I.T., Cambridge,

Mass.

Meyerhoff, G. G. (1951), "The Ultimate Bearing Capacity of Foundations,"

Geotechnique, Vol. 2, No. 4, pp. 301-332.

Meyerhoff, G. G. (1961), "The Ultimate Bearing Capacity of Wedge-Shaped

Foundations," Proceedings, 5th ICSMFE, Vol. 2, pp. 103-109.

Mitchell, J. K., and Durgunoglu, H. T. (1973), "In Situ Strength by

Static Cone Penetration Test," Proceedings, 8th ICSMFE, Moscow,

Vol. 1, pp. 279-286.

96

McMeeking, R. M., and Rice, J. R. (1975), "Finite-Element Formulations

for Problems of a Large Elastic-Plastic Deformation," International

Journal for Solids and Structures, Vol. 11, pp. 601-616.

Osias, J. R., and Swedlow, J. L. (1974), "Finite Elasto-Plastic Deforma­

tion -I. Theory and Numerical Examples," International Journal for

Solids and Structures, Vol. 10, pp. 321-339.

Pian, T. H. H. (1964), "Derivation of element stiffness matrices by

assumed stress distributions", AIAA Journal, Vol. 2, pp. 1333-1336.

Prevost, J. H., (1978), "Plasticity Theory for Soil Stress-Strain

Behavior," Journal of the Engineering Mechanics Division ASCE, Vol.

104 No. EM5, pp. 1177-1194.

Rourk, T. L. (1961), "Model Studies of a Pile Failure Surface in a

Cohesive Soil," M.S. Thesis, Civil Engineering Dept., Georgia

Institute of Technology, Atlanta, 56 p.

Sanglerat, G. (1972), The Penetrometer and Soil Exploration, American

Elsevier Publishing Company, Inc., New York, N. Y.

Terzaghi, K. (1943), Theoretical Soil Mechanics, John Wiley and Sons,

New York, 1943.

Truesdell, C,, and Toupin, R. (1960), "The Classical Field Theories,"

Handbuch der Physik, ed. S. Flugge, Vol. Ill/1, pp. 226-793,

Springer-Verlag.

Tumay, M. T., Acar, B. A., Cekirge, M. H., and Ramesh, N. (1985) "Flow

Field Around Cones in Steady Penetration." Journal of the

Geotechnical Engineering, ASCE, Vol. Ill, No. 2.

97

Tumay, M. T. and Yilmaz, R. (1981), "In situ Determination of Undrained

Shear Strength of Louisiana Soils by Quasi-Static Cone Penetration

Test", Geotechnical Engineering Report No. GE-82/04, Louisiana

State University, Baton Rouge, Louisiana, 282 pp.

Vesic,. A. S. (1972), "Expansion of Cavities in Infinite Soils Mass,"

Journal of the Soil Mechanics and Foundations Division, ASCE, Vol.

98, No. SM3, pp. 265-90.

Voyiadjis, G. Z. (1984), "Experimental Determination of the Material

Parameters of Elasto-Plastic, Work Hardening Metal Alloys,",

Material Science and Engineering Journal, Vol. 62, No. 1, pp.

99-107.

Voyiadjis, G. Z., and Buckner, N. E. (1983), "Indentation of a Half-

Space with a Rigid Indentor," International Journal of Numerical

Methods in Engineering, Vol. 19, No. 10, pp. 1555-1578.

Voyiadjis, G. Z., and Kousis, P. D. (1985), "Stress Rate and the

Lagrangian Formulation of the Finite-Strain Plasticity for a von

Mises Kinematic Hardening Model," International Journal for Solids

Structures, in review.

Yamada, Y., and Wifi, A. S. (1977), "Large Strain Analysis of Some

Geomechanics Problems by the Finite Element Methods," International

Journal for Numerical and Analytical Methods in Geomechanics, Vol.

1, pp. 299-318.

Ziegler, H. (1959), "A Modification of Prager's Hardening Rule,"

Quarterly Journal of Applied Mathematics, Vol. 17, pp. 55-65.

Zienkiewicz, (1971), The Finite Element Method in Engineering Science,

McGraw-Hill.

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