\ANALYSIS OF THE PRESSUREMETER TEST BY FEM FORMULATION
OF THE ELASTO-PLASTIC CONSOLIDATIOM„by
S. K.\QainÖDissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHYin
Civil Engineering
APPROVED:
R. D. walker D. rederick
May, l985
• _ Blacksburg, Virginia
ANALYSIS OF THE PRESSUREMETER TEST BY FEM FORMULATION
S OF THE ELASTO-PLASTIC CONSOLIDATION
S. K. Jain
y (ABSTRACT)
A decade has passed since the development of the self-boring
pressuremeter (SBPM). Even though the device has been recognized by
the geotechnical engineering profession as having high promise for
v evaluating in-situ stress—strain behavior of soils, its use is limited.
In large part, this is due to the fact that there are important un-
answered questions about the SBPM test. One of the major issues con-
cerns the influence of drainage in the soil as it is sheared. In
clays, the test is assumed to be undrained, but there is no way to
control this other than by the rate of loading and no method has been
put forth heretofore to define the required rate. °
This dissertation addresses the drainage issue by applying a
numerical model capable of simulating the pressuremeter test under
variety of conditions. To develop parameters for the soil model, a
comprehensive laboratory testing effort was needed. The validity of
the numerical model and the soil parameters is established by comparing
it to SBPM tests performed in the field.
The numerical model uses the finite element method in a special
code capable of handling large strains, consolidation effects, and
nonlinear soil behavior. Particular attention is addressed to the
issue of pore pressure development and its dissipation. Relative
influences of important soil parameters such as the permeability are
checked against various rates of loading in the SBPM test. The results
demonstrate that drainage likely occurs in most cases using conven-
tional test procedures, and that this, in turn, leads to an error in
interpretation of SBPM data. Based on the findings in the analyses,
a procedure is proposed which should lead to a more rational method
of performing the SBPM test where nearly undrained conditions are
desired.
ACKNONLEDGEMENTS l
To Professor G. N. Clough, for his continuous encouragement and
guidance. without his assistance this dissertation would not have been
in existence.
To Professor J. M. Duncan, for his useful comments on the draft
of the dissertation.
To fellow graduate students in geotechnical engineering, for
their constant support and friendship.
To the National Science Foundation, for the financial support of
this investigation.
And, to the United States of America, the beautiful and wonderful
blessed land, for having me over here.
iv
TABLE OF CONTENTS
Page
ABSTRACT ........................... ii
ACKNONLEDGEMENTS ....................... iv
CHAPTER
1. INTRODUCTION ..................... 1
2. PRESSUREMETER THEORIES ................ 8
2.1 Menard Pressuremeter Test .........·. . . 81. The Un1oading Phase ............ 102. The Re1oading Phase ............ 103. The E1astic Phase ............. 104. The P1astic Phase ............. 14
2.2 Se1f-Boring Pressuremeter Test .......... 182.3 Stress-Strain Curve from Pressuremeter Data . . . 20
(a) Pa1mer-Bague1in—Ladanyi Method ....... 20(b) Oenby-C1ough Method ............ 23
2.4 Determination of Cvh from the PressuremeterHo1ding Test ................... 30
3. ELASTO-PLASTIC CONSOLIDATION: FORMULATION OF THEFINITE ELEMENT EOUATIONS ............... 33
3.1 Biot's Theory .................. 343.2 E1asto-P1astic Conso1idation ........... 373.3 PEPCO - The Finite E1ement Program ........ 39
4. LABORATORY TESTING OF SAN FRANCISCO BAY MUD ...... AQ
4.1 Samp1ing and Preserving Undisturbed Soi1 ..... 424.2 Unconfined Compression Tests ......... . . . 444.3 Oedometer Tests ................. 52
Samp1e Preparation ............... 57Stress Contro11ed Tests ............ 57Strain Contro11ed Tests ............ 58
4.4 Triaxia1 Tests .................. 61UU Test .................... 65Conso1idated Drained Tests ........... 65Conso1idated Undrained Tests .......... 69The Unconventiona1 Test ............ 76
r V
Page
5. MODELING OF SOIL BEHAVIOR: EVALUATION ANDVERIFICATION OF SOIL CONSTANTS ............ 815.1 Mechanism of Hardening in Meta1s ......... 815.2 Mechanism of Undrained Hardening in C1ays .... 845.3 Cambridge Soi1 Mode1s .............. 87
Mode1 Parameters ................ 93Undrained Strength in Cam C1ay ......... 96
5.4 Verification of Constitutive Mode1 ........ 98(a) Undrained Tests .............. 99(b) Drained Tests ............... 102
5.5 Conc1uding Remarks ................ 102
6. MODELING OF FIELD PRESSUREMETER TESTS: AXISYMMETRICPLANE-STRAIN ANALYSIS ................. 107
6.1 Mode1 Parameters ................. 1076.2 Comparisons: Fie1d vs. Mode1 .......... 1106.3 Simu1ation of Ho1ding Test ............ 125
7. EFFECT OF FINITE PRESSUREMETER LENGTH: THEAXISYMMETRIC ANALYSIS ................. 127
7.1 Simu1ation of Fie1d Tests ............ 1287.2 Observations in the Vertica1 P1ane ........ 1357.3 Cost Ana1ysis: Axisymmetric vs. P1ane—Strain . . 141
8. DRAINAGE DURING SHEAR IN THE PRESSUREMETER TEST .... 144
8.1 Effect of Permeabi1ity in Mode1ing Actua1 Test . . 1448.2 Genera1 Effect of Variation in Permeabi1ity . . . 1458.3 Effect of Rate of Loading ............ 1528.4 Effect of Permeabi1ity and Loading Rate ..... 1618.5 Recommended Procedure for Conducting 4
Pressuremeter Tests ............... 164
9. SUMMARY AND CONCLUSIONS ................ 165
9.1 summary ..................... 1659.2 Conc1usions ................... 166
REFERENCES .......................... 169
APPENDIX1. STRESS-STRAIN RELATIONS ................ 176
A1.1 Undrained Deformations ............. 179
( vi
Page
2. LABORATORY TEST DATA FOR SAN FRANCISCO BAY MUD .... 180
3. A NOTE ON MULTI-SURFACE HARDENING MODELS ....... 199Para11e1 Mode1 ................... 200Series Mode1 .................... 203Integrated Hardening Mode1 ............. 203Discussion ..................... 205
4. CUBICAL TESTS DESIGNED TO SIMULATE PRESSUREMETER‘ LOADING . .'...................... 208
VITA ............................. 214
vii
CHAPTER l
INTRODUCTION
Almost thirty years have passed since the first practical version
of the pressuremeter was introduced by Menard (l956). The basic idea
involved drilling a hole in the ground, inserting a cylindrical probe
into the hole, and expanding a membrane which surrounded the probe into
the sides of the hole. Fundamentally, the pressuremeter represents a
cavity expansion experiment, and soil parameters are obtained from the
pressure — volume curve derived from the test. The basic Menard type
of pressuremeter test has attained a considerable degree of popularity
but remains hampered by the stress relief and disturbance in the pre-
drilling process. Improvements and modifications to the original
approach have been made, with one of the most notable involving the
addition of a self - boring capacity. Developed almost simultaneously
in France and England, the self - boring probe is designed to drill
itself into the ground allowing the body of the instrument to maintain
intimate contact with the ground. This process then avoids the stress —
relaxation effect associated with opening the hole for the Menard
probe and supposedly minimizes disturbance. The test with a self -
boring pressuremeter (SBPM) involves the same principle as the basic
Menard procedure, although the methods of measuring response have been
upgraded and capabilities to monitor pore pressure development have
been added (wroth and Hughes, l974).
l
2
The inherent virtues of the SBPM test have been cited in a
number of publications. According to an MIT report to the Federal
Highway Administration (Ladd et al., l980), "The self—boring test
offers the theoretical potential of making measurements of in-situ
lateral stress and undrained stress-strain properties of saturated
clays in greater detail and more accurately than heretofore possible.
This is an exciting prospect which has received well deserved acclaim."
A similar view was expressed during a l983 National Science Foundation
workshop on experimental soil engineering held at Virginia Tech. As
seen in Table l.l, excepting for the pressuremeter, in-situ test can
yield only a very limited infonnation regarding soil properties and
by-and—large their interpretation remains empirical. Moreover, only
the pressuremeter can be used to assess the strain softening properties
of the soil because, "the test has the special virtue that it is always
stable, in the sense that the pressure necessary to produce monotonic
expansion of the membrane always increases even though the soil may be
responding in a strain-softening manner; this is due to the fact that
the size of the element of soil undergoing the test is 'infinitely'
large" (Clough and Silver, l983).
In spite of the inherent virtues of this test, it has yet to be
truly accepted in geotechnical engineering. Presently, the evidence
from field investigations is mixed. As shown in Table l.2, the un-
drained shear strength which is derived from the SBPM is often con-
siderably higher than that from other conventional tests. Many reasons
for this discrepancy have been put forth, including length to diameter
ratio of the probe, disturbance, rheological characteristics of the
3
Tab1e 1.1* Capabi1ities of In-Situ Test Equipment
Instrument Soi1 Property
Mechanica1 Tests Cu ¢' G CVStandard Penetration Test E E ? E —Vane Shear Test D - ? D -Cone Penetrometer E ? E — -Piezocone · E - ? E ? EP1ate Bearing Test D — D -Di1atometer E E E -Pressuremeter D D/L D D
Geophysica1 TestsVibro-seismic Tests — — D -
Note: D = direct eva1uationD/L = direct eva1uation with supp1ementary Iaboratory dataE = empirica1 eva1uation based on experienceCU = undrained shear strength ·¢' = effective stress friction ang1eG = e1astic shear modu1usCV = coefficient of conso1idation
(*From C1ough.and Si1ver, 1983.)
G4-)-I-) •n
'Q Q fiC 1- G(5 1- rs G z-
GJ (*7 CN (*73 U OO -*~ fi G
4-) (D C CN O LI rx Qä*1- GJ fü 1*- G G 1-U) U .1 -« CN >; OO
-I C 1- .¤ CNC GJ G5 • sz C 1*- •!-I L IG Q)!-x Ca 1-
GJ ··- fu 3 cm fcL W- X fü rx G •CI) GJ I/)°G4-) CI 1- GCN 1- 4-)C G ECG) G CFC fü GJ-4-) Ofü 'C CN G--G X G C IG 4-) G
1--C fü-
.C-I-* GJ CC OI\C O7*1- C+9 '1"I\O C *1- DO 'C C·1·· ECN·1- GJ L CC 'C *1-3 G1-—C 'O O 1—GJ fü CS- "J--G 1.1.1 E um ‘..1 c:)GJ4-) GJGJ CE r¤ 0 cw •- •.oGJ j • • • Q] •
L O. G GI 1- CU I 1- KI I I3 C G 'O • I I I I • I I IU) 4-) (/7 1-- FC I\ G G I IG LO I IU', Q) • • • •GJ C °Y* IG 1- IG
1-·L GJ LI.Q. L4-)C7 (/7 Lcz- fü'GLO L CULG fü COCN GJ (/7Gr- C
I (/7 GJQ- ^ IG1-+-) °U G- 1*- I I I I I I Gq)••—- GJ C E • I I I 1 I I • •(./)O C G *1- FC I I I I I I 1- C\I
C ·1·· (/7 (/7EGJ GOG L 4-*
-L 'C U‘+—‘C C GJC G L
"CG *1-GJ II- GCC O-1-C7 4-) CUG3 I./7 I/7 IG-1->O O 1- (U GGJ- •_Qp-· ••-· fü • •-I—)1- 1-GC.) 4-) E ·1·· CN I 1- I 1-· • I (*7 <' RD
G D. X • I I I IGJ1- (*7 I • •CE 1 G fü G I 1- I CNC CN C\.I 1- 1--I—)O (/> *1- • •CII •GL L 1- QL; QC‘·I·- I-QjxzL
U) 1-U) r- <I' 1- LD
I-I—GJ GJ r- 1-01- ¤. II N cu GJ II
>5 U U') r-C>; I- G II fü G COOL GJ \ «- 'U cm \ f-mo •—¤. .1 Q kO AL .1 LO II
*1--|-J •r->, Ca es Lßfü Q) ~./Lfü O}- ...1 II CO 3 U II GGL (/7 >~1-“ II C7 L >, \Q.O GJ fü G II O G G —JEL L.C} 1- >; \ G'C L1. 1- \~/OG OO U G .I G \C U ..IG.1 ~—-L ¤—
-\ .|fü L -' >a
C GJ U .I -« ·1·· GJ fü<\! —I—)'C 1- GJ GJ >»1- «·- 1- fü U• *1-C O U -I-) *1- GGJ CCC G 1-1— (/)G I·— U U) E GU O U C
*1- GJ fü -I-)II C GJGJ -I-) O -I-) U O7 OC 1- O >5 E1- (/7 4-) (./7 3 G >;•r- •1-G +-) O E.Q GJ L fü O -1-> XG E\ U) U7 füG I- O 3 r- G OE fü...I O C L1- C G G Z I--- I~/ G G G
5
soils, and dissipation of pore pressures in the soil around the probe.
Recently, a number of research studies have begun to focus on the
issues surrounding the SBPM test. Often these consist of theoretical
or field investigations, yet rarely are the two efforts combined.
One of the most persistent questions concerning all forms of
pressuremeter test is the effect of, or existence of, dissipation of
excess pore pressures set up in the soil during the test. This is im-
portant in clays and silts since the test results are analyzed assuming
that the test is undrained. Inasmuch as the only control on the
drainage is through the rate of loading, the most often asked question
in the pressuremeter literature is how rapidly the test should be con-
ducted. "Is partial drainage an important consideration?" asks the
MIT report. "If so, what expansion rates are required to achieve
acceptable results?" The "...theory has not provided guidelines regard-
ing suitable rates as a function of the consolidation characteristics
of various soil types." This is one of the objectives of this dis-
sertation, to guide the engineer concerning the rate at which the test
is to be conducted so it is virtually undrained, and what to expect
should the test be conducted below the prescribed loading rate.
The rate of loading can have substantial influence on the results
of the SBPM test. Under very slow loading rates, the soil can
experience drainage. Since there is no way for the engineer to know
if the drainage occurred, he analyzes the results assuming that no
drainage took place. As shown in this dissertation, this leads to an
underestimate of the shear strength. On the other hand,’very fast
loading rates will overpredict the shear strength of soil due to
6
rheological effects. Since the rheological or time effects are not
accounted for in the interpretation of the pressuremeter test data, an .
optimum loading rate has to be found for each soil at which the
drainage just begins to take place.6
Recently, in a Ph.D. investigation by Benoit (l983) under the
supervision of Dr. G. W. Clough, a series of SBPM field tests were per-
formed in a deposit of soft clay to examine the influence of rate of
loading and disturbance on test results. These tests utilized state-
of-the-art technology in the equipment and provided a solid data base.S
while the information derived was useful, it left questions unanswered
because the pressuremeter measurements can only be made at the face of
the cavity. Response in the soil medium away from the cavity is not
known.
This thesis is directed towards providing the theoretical basis
for the phenomena observed in the Benoit tests. In the process an
analytical tool is developed and verified which can be used to assess
the question of relative effects of pore pressure dissipation and dis-
turbance on pressuremeter test results. The methodology is thereafter
used to provide a means of predicting the rate at which a pressuremeter
test should be performed to achieve suitably undrained conditions to
yield realistic soil parameters.The finite element method is used in this work as it is the only
available technique for analyzing problems where the material is to
be modeled as a porous elasto-plastic medium with a capability of
coupled consolidation. A finite element mesh permits the examination
of the states of stress and strain, and the developed pore pressures
7
in the entire soil medium during all stages of the pressuremeter
test.
Following this introduction, Chapter 2 briefly describes the
Menard and the self-boring pressuremeter tests, and presents a review
of the pressuremeter theories which are needed for deriving the
engineering properties of soils from the pressuremeter test data. The
finite element soil model is described in Chapter 3. Chapter 4 sum-
marizes the laboratory testing of the San Francisco Bay Mud, undertaken
in order to determine the soil constants for the model. Subsequently,
validity of the soil model is examined, against the laboratory tests
in Chapter 5, and against the field pressuremeter tests in Chapter 6.
Soil model is then used in Chapter 7 for a detailed analysis of the
SBPM test. Finally, the Chapterääaddresses the important question of
drainage in the SBPM test.
CHAPTER 2
PRESSUREMETER THEORIES
The problem of expansion of a cylindrical cavity in a semi-
infinite medium has been a subject of study for decades. The
analysis proceeds as a limiting case of the problem of thick walled
tube whereby an expansion curve relating radial strain and the internal
pressure is derived once the material properties of the medium are
specified. In the pressuremeter test, we attempt to solve an inverse
problem. Given the expansion curve of a cylindrical cavity-—derive the
material properties.
2.l MENARD PRESUREMETER TEST
In the conventional pressuremeter test developed by Menard (l956)
a cylindrical probe of approximately 6 cm diameter is inflated under
water pressure in a predrilled hole at a desired depth. The probe, 20 cm
long, is supported by two l0 cm guard cells at its ends which expand
along with the probe under the same pressure, Fig. 2.l. The guard cells
eliminate the end effects during the expansion of the probe, and thus
ensure a uniform plane strain deformation of the soil mass around the
probe. The test is conducted by applying pressure increments at one
minute interval and recording the volume of the water injected into the
probe. An idealized curve obtained from the test is shown in Fig. 2.2.
The test passes through four distinct phases.
8
9
PRESSURE 6110666G6 ®\ -_ Ö“’wasn {
II
[ „ I' I[ I1TUBES |/2 : VOLUMETER
750. . ,.. ... .....,.
2* (V 6uAn0 661.1.1 _ Q \S: §L- E Q MEASURING 661.1.. E —•1 'TEST [ __, _1.. g E {-1
GUARD661.1.Fig.
2.1 Schematic of 60nventi0na1 pressuremeter (Bachus et a1., 1981).
10
1. The Un1oading Phase:This happens during the creation of the cy1indrica1 cavity in
the ground prigr_to insertion of the probe. During this phase, the
pressure at the interface of the cavity decreases from the in-situ
horizonta1 stress 6H to zero.
2. The Re1oading Phase:At the initiation of the pressuremeter test, Fig. 2.2, the pres-
sure at the interface increases to attain the origina1 in-situ hori-
zonta1 stress state, 6H. From this data it is theoretica11y possib1e
to derive the va1ue of the horizonta1 stress in the ground, and thus,
an estimate of the coefficient of earth pressure at rest, K6, can bemade.
3. The E1astic Phase:
This phase, with approximate1y a 1inear segment of the pressure-
meter curve, Fig. 2.2, indicates an approximate e1astic deformation
of the soi1 mass, and is usua11y ana1yzed with e1asticity theory.
Assuming that the soi1 deforms in a radia1 p1ane strain mode, the
radia1 and circumferentia1 components of strain, 6P and 66, due to anincrease in tota1 principa1 stresses, 6r and 66 (Fig. 2.3) are given
as
(2.1)
66 = é [M16 — 'v(AoY_ + M1Z)] (2.2)
II
. PQ}
C: \ Phase 4E
LUx: .
Ecxc;.1 pf I3 IJg Phase 3 ÄI
IOH A',-I-· Phase 2I
Vo VfINJECTED WATER VOLUME, V
Fig. 2.2 Diagram shows a pressure-voIume change cuhvet_y|D'IC&I of 6 M€I'IöY‘CI pY‘€SSUY‘€ITI€'IZ€Y‘ test.
l2
-T x or r4Ih*
OZ (EZ = O)
o@(c6)
1IP*
Fig. 2.3 Stress components in the pressuremeter test.Matched plane indicates the orientation offailed plane originating in the r—n plane.
l3
where, the increase in axial stress Adz is computed from the conditionof radial plane strain:
ez [A0Z — v(A0r + A0o)] = Ü · (2.3)
Substituting Adz from Eq. (2.3) in Eqs. (2.l) and (2.2) we obtain
Er - gg = g ((1 (2.4)
eo = $—= g [(l — v2)A0o - v() + v)AGr] ( (2-5)
where dr = dH + Adrand, 0o = 0H + Aoo
Stress-strain relations (2.4) and (2.5) when combined with the equation
of equilibrium,
d0 G — Gr Y' G =-aF— + r 0 (2.6)
and the boundary conditions,
u —> 0 as r —> <>¤(2.7)
0r=0H+Apwh€nY‘=aO
lead to the following solution (Gibson and Anderson, l96l).
dr = dH + AD (ag/r2) (2 8)0o = 0H — Ap (ag/r2) (2.9)
Adz = 0 (2-l0)and- Mao) = AP [(l + v)/Elao (2-ll)
14
where 6r, 66, and 62 are the tota1 radia1, tangentia1, and axia1
stresses respective1y at a distance r from the center of the cavity,
and u(aO) is the radia1 disp1acement of the surface of the cavity.
Eqs. (2.8) and (2.9) imp1y that as the radia1 stress increases, the
tangentia1 stress decreases by the same amount, i.e.,
A66 = - A6r (2.12)
The vertica1 stress 62 supposed1y remains una1tered. Thus, the tota1stress path during the cavity expansion remains vertica1 to the hydro-
static axis, as shown in Fig. 2.4. Eq. (2.11) can be expressed as,
E - AE‘("T21+6 ‘ V6 AV (2*13)and thus, the e1astic phase of the test can be used to determine the
undrained modu1us of the soi1;
· Z = ABi.e., for v 0.5, EU 3 VO AV (2.14)
4. The P1astic Phase:
The e1astic ana1ysis is va1id on1y if the stresses in the soi1 are
everywhere be1ow yie1d;
i.e., if |6r — 69l < 2 cu (2.15)
Substituting Eqs. (2.8) and (2.9) into (2.15), we find that p1astic
yie1ding wi11 first occur on the cavity wa11 (i.e., r = ao) when the
app1ied pressure p attains a va1ue
p = GH + gu (2.16)8
l5
6 - 6—I¥?——Q- Plastic Phase_•-_ _- I
I ActualElastic Phase
or + 692
Fig. 2.4 Diagram illustrating the total stress path duringa pressuremeter test.
T6
with the increase in applied pressure above 6H + cu a plastic annulus
of increasing radius is formed around the cavity. By applying the
equilibrium condition, Eq. (2.6), along with the yield condition (2.l5),
and maintaining the continuity of stress and deformation fields at the
junction of plastic and elastic regions, the following expression is
established for any moment of the test beyond first yield.
p = 6H + cu + cu ZH
[TwhereV = VO + V. The cavity will expand indefinitely under no
further increase in pressure when the internal pressure attains a
limiting magnitude given as
p = 6 + c [T + Zn EU (2 T8)L H u ÖEETTYÜT °
This expression for limit pressure pL was first obtained for metals by
Bishop, Hill and Mott (T945). The elastic-plastic interpretation of
the Menard pressuremeter test given here is due to Gibson and Anderson
(l96l). Shear strength of soil is obtained by combining Eqs. (2.l7)
and (2.T8); i.e. for v = 0.5 we have
p = pL + cu tn 3
{THowever,in practice, pL can never be reached. It is computed by
plotting the pressuremeter data in the form of p:log(AV/V) curve. The.
plastic phase of the curve is approximately linear, and therefore,
extrapolated value of the cell pressure at AV/V = T is the limit pres-
sure pL. From the coordinates of one additional point on the curve,
l7
the shear strength cu can be computed from Eq. (2.l9).
The determination of E and cu completely defines the stress-
strain curve for the elastic-perfectly plastic material. However,
for soft clays the parameters obtained by the Menard pressuremeter
have been found far from in agreement with other conventional field
and laboratory tests though reasonable modulus values are obtained for
stiff clays.
As mentioned by Ladd et al. (l980) intensive efforts were made by
Menard and co-workers in France to correlate their pressuremeter
values of modulus and strength to those obtained from "conventional"
tests. Results were discouraging. Hence, they decided that the
pressuremeter test be viewed as a "model foundation test," which could
become a basis for designing shallow and deep foundations with the use
of empirical scaling factors. The book by Baguelin et al. (l978)
presents a comprehensive treatment to foundation design by the
pressuremeter.
Ladd et al. (l980) indicate that a practice of employing the
Menard Pressuremeter Test for in-situ measurement of soil properties
for use with conventional theories to predict stability, deformations,
in-situ stresses, etc., is "ill-advised." The reasons include: a
substantial disturbance of soil during the unloading and reloading
phases, the unknown drainage conditions, and the simplifications made
in the methods of interpretation by assuming soil an elastic-perfectly
plastic material.
l8
2.2 SELF-BORING PRESSUREMETER TEST
The unloading-reloading phase of the Menard pressuremeter test
can be eliminated by introducing a self-boring capability in the
pressuremeter. To do so, a sharp cutting shoe is attached to the open
bottom end of the pressuremeter (Fig. 2,5). As the pressuremeter is
slowly pushed into the ground, the soil entering the bottom end is con-
‘tinuously removed by a rotating cutter housed in the shoe. Soil
cuttings.are flushed to the surface by a water jet. water is pumped
jg_through the hollow rods driving the cutter and the slurry is pumped
gut through the annular space between the cutter driving rods and the
inside of the pressuremeter cylinder. The cutting shoe is designed
to precisely match the outside diameter of the pressuremeter so that at
no moment of the self-boring phase do the pressuremeter and the in-situ
soil lose contact. In ideal situations, therefore, the horizontal
stresses in the ground are directly transferred to the pressuremeter
membrane.
However, a price is paid. The concentration of stresses at the
sharp edge of the cutting shoe and the shear stresses developed at the
interface of the soil and the advancing pressuremeter generate excess
pore pressures around the probe which take time to dissipate. There-
fore, after insertion, the pressuremeter has to be left in place for
a period of time for pore pressure stabilization. Typically this
requires 30 minutes to several hours.
Self-boring pressuremeter devices were simultaneously developed
at the Laboratoires des Ponts et Chausees in France by Baquelin et al.
I . 19
i ä! * I} RussenSPRING „ msmannna‘ I ä .
FEELER Fans THREAD1 Q PORE Pnsssuns _‘ E é CELL
ÄI
c1.A~u¤-——- § 1Fgi ,
.2l}<*§>LI·
Fig. 2.5 Schematic of the se1f-boring pressuremeter withmembrane part1y inf1ated (wroth and Hughes, 1972).
20
(l972) and at the Cambridge University by wroth and Hughes (l972,l973).The French version is known as the Pressiometre Autoforeur while theEnglish version is called the Camkometer. The two versions differ inmechanical operations but the underlying principle is the same.
A schematic of the Camkometer is shown in Fig. 2.5. During thetest, the cavity is expanded by applying gas pressure from inside thepressuremeter cylinder. The radial displacement is measured by track-ing the movement of the membrane electronically by means of three
strain feeler arms located l2O degrees apart at mid-height in the
probe. The pore pressures are recorded by transducers which are placed
in the membrane wall to move with the expanding membrane. The detailsof a complete self-boring unit and the test procedures have been
given by Denby (l978) and Benoit (l983).
2.3 STRESS-STRAIN CURVE FROM SBPM DATA
(a) Palmer-Baguelin-Ladanyi Method:
In part, success of a pressuremeter test lies in the interpreta-
tion method used for determining the engineering properties of the soil.
As shown in the preceeding discussion, Gibson and Anderson (l96l)
employed the theory of elasticity to determine the shear modulus, and
the theory of plasticity to derive undrained shear—strength of soil from
the pressuremeter data. Subsequently, an impetus to the pressure-
meter test was given by Baquelin et al. (l972), Ladanyi (l972) and
Palmer (l972) who proposed an analytical technique to derive the
complete stress-strain curve from the pressuremeter data. The soil is
assumed to follow a stress-strain relationship of the form
2l
InitialTExpanded~ A A·[ x\
1\
“ ÄII yI ”
y1 1
Fig. 2.6 Plane-strain expansion of a cylindrical cavity_(Palmer, l972).
22
0;* - 01; = 1°(61__ — 69) (2.20)Making the second important assumption that the soil will shear underundrained conditions, then it follows that
66 = — 61„ (2.2l)
the Eq. (2.20) can be simplified to
01; - 01;) = l°(6G) 1 (2.22)
Combining Eq. (2.22) with the equation of equilibrium (2.6), we get
ä — - l f( ) (2 23)dr ” r E6 ‘
Consider a cavity of unit radius. Under an applied pressure 0r(aO), ifthe cavity radius increases by y1 and the material point A, Fig. 2.6,moves outward by y, then the condition of no change in volume demandsthat
urz - n(r-y)2 = ¤(l+y1)2 - ·¤(l)2 (2-24). 21.e., y = r - [rz — 5/1(2 + 5/1)]]/ (2-25)
1, 5/1 (2+5/1 )1 ]](2Therefore, 69 - Fjy-— - l + [l — ——;;y-——U (2-26)Since or changes from the applied pressure ¤P(a0) at ao = l + y1, to0H at infinity, Eq. (2.23) can be integrated to have
da = -( gf (66) ar (2.27)l+y1 l+y1
·„
1 y1(2+y1)--l/2(2 28)]|•€•, ' " F f ' 1 + ' •
23
The stress-strain relation is determined by solving Eq. (2.28) forthe unknown function f. As shown by Palmer (l972), the following rela-
tion is obtained,
do (61)= r 0fb/1) y](l + y])(2 + xl) d Y1 (2-29)That is, at the wall of the cavity
dcrar - ag = 6r(l + er)(2 + er) EE; (2,30)
For small strains, Eq. (2.30) reduces to a particularly attractive form
0 - 0 dor 0 = __r;———j?——— er d€ (2.3l)r
Eq. 2.3l, regarded a milestone in the development of the self-boringpressuremeter, was independently arrived at by Palmer (l972),Ladanyi (l972), and Baguelin et al. (l972), Eq. 2.3l states that theproduct of the slope of the pressuremeter curve and the correspondingradial strain is the shear stress induced in the soil at the cavity
wall. Therefore, the complete stress-strain curve can be derived
from the pressuremeter curve by a simple graphical procedure. The
procedure, known as the subtangent method, is illustrated in Fig. 2.7.
(b) Denby—Clough Method:
Eq. (2.3l) provides a link between the experimental pressure-
meter curve and a general stress-strain relation reflected in Eq.
(2.22). Given an explicit stress-strain relation, the equation for
24
200 1Self-Boring Pressuremeter Test, HPR—l9Type Soil: Soft Clay (San Francisco Bay Mud)
¤éE Depth: 6.8 M\Z¥1.1.1 _ ICEID3 160 IL1.1QS¤. ' I.1E [ I
PY‘€SSUY‘€ITl€'IZ€Y‘ CUTVE ICZ
_ T I
l00
50I-gg Stress—Strain CurveääI-aniä I
i I1Lu t IE I I
0 I0 2 4 6
RADIAL STRAIN %
Fig. 2.7 Diagram illustrating the application of the sub-tangent method to the field pressuremeter test(test data after Denby, l978).
25
the pressuremeter curve could be derived, or vice-versa. If the soil
is assumed to follow a particular type of stress—strain response, say
hyperbolic or Ramberg—0sgood type, the model parameters could directlybe derived from the pressuremeter curves.
Due to the erratic behavior of in-situ soils, often it becomes
necessary to assume the nature of stress-strain curve apriori. The
method of Palmer-Baquelin-Ladanyi requires that the slope of the
pressuremeter be determined at each point. Owing to the slight
disturbance caused during the insertion of the probe, it sometimes
becomes difficult to determine this slope in the beginning portion of
the curve. Considering that most clays require only l to 3% s-rain
to reach failure, and that the shear modulus is determined from the
initial slope of the pressuremeter curve it may be necessary to define
the beginning portion based on the prefailure stage of the curve. Such
an extension of the Palmer-Baquelin-Ladanyi theory was provided by
Denby and Clough (l980) through a hyperbolic formulation.
Duncan and Chang (l97D) proposed that the stress—strain response
of most soils could possibly be fitted by hyperbolic-plastic equations
where the prefailure response is defined by the hyperbolic equation,
— Ea 2 32)(Ü] ' Ö3) ‘ •Ei lääl?
and the post failure response by a plasticity equation,
(61 — ¤3)f = 2 SU (2-33)where Rf is the ratio of actual soil strength (61 — ¤3)f and the
26
asymptotic strength (6] - G3)fh, see Fig. 2.8. Rf has been found to
vary from 0.9 for soft plastic clays to 0.6 for overconsolidated clays
(Duncan et al. l980). Ei is the initial tangent modulus. For the
pressuremeter problem, this equation must be expressed in radial stress
and strain variables. Equating the engineering shear strain in the
pressuremeter to that of triaxial test, we get for the undrained condi-
tions
-ä2eY_—2ea Ai.e., (2.34)
EI :*%*6a 3 r
Noting that, Ei = 3 Gi, under undrained conditions, (6] - 63) in the
pressuremeter test is (6T - 66), and that (6V - 66)/2 is the undrained
shear strength su, Eq. (2.32) can be expressed as
O 'O El..‘%..2=..._%i.... (2.35)$+62i Su (
Relating this equation to Eq. (2.3l)
de ”Rr - .L LE6; ' 2Gi + su 6r (236)
Similarly, Eq. (2.23) can be combined with the Eq. (2.3l) to yield
de (2.37)r u
Eqs. (2.36) and (2.37) state that if the inverse of the slope of the
pressuremeter curve, dar/ÖOT is plotted against the radial strain Er we
should get two straight lines with slopes Ri/su and l/su respectively.
27
"?~»OI
if(°1 ‘ °2)1= '''""' I"-
Hyperbolic(°1 ' °3)fh '°° '°
R Hyperbolic—Plastic
Er
do..Lde r
l/SU
Hyperbolic-Plastic.//« Pressuremeter
R Cure/ .1i/ SU1 //
2Gi /zEr U
Fig. 2.8 Transformed pressuremeter curve for the assumedhyperbolic—plastic stress—strain behavior (fromDenby and Clough, l980).
28
Their junction defines the failure strain erf. As shown in Fig. 2.8,the first line has an intercept of l/(2 Gi) at er = 0. Therefore, su,
Rf, erf and Gi can readily be determined from one additional plot. Theinitial horizontal stress in the ground can be determined by integratingEq. (2.36) under the boundary conditions, or = oH for Er = O. Thefollowing expression is obtained.
Su RfoH = or - ä;-an [T + E; 2 Gi er] (2.38)
Interestingly enough, this equation represents the experimental
pressuremeter curve if the soil follows a hyperbolic stress—strain
relationship.
Dther interpretations of the pressuremeter test exist in the
literature. The pioneering work of Menard (l956) primarily rests on
empirical correlations. The interpretation given by Prevost and Hoeg
(l975) not only requires an elaborate computational effort but is
based on an assumption that the soil follows a "hyperbolic" response,
and not the "hyperbolic-plastic" as suggested by Duncan and Chang
(l970). The assumption of continuous hyperbolic response leads to high
shear strength for soils. A comparison of various available interpre-
tations is presented in Table 2.l.
Fortunately, during the course of present investigation the
author had to deal only with FEM pressuremeter curves which were so
well defined that the graphical subtangent method was more than
adequate.
-¤§
=
9
:
5
m
:__
__:.’3
.
'U
3
cc:
c
’*
-=>“"
0-·
5-*5
5
·
“=”¤£
-
P
3
-5***
:7
Q
Qägäa
_,
4;
3:0*-
3
_,,__
L.:
-
‘-/7
v•
'°·:7~^
1
31
(.303
>
J)
0
1,
Q
;
'..
3
ti
Ö
—·;
'=
°,;=-ä
,_
Ä:
._
(j)
mg
*•-L
3
22
0"‘""
g\
7:-
..,
L
V41.,
3:70
1:L.
5*)*
n
L
U
0
__,:
3
3
-
3-.:5
'·’
gz
ä
0
;;0
-:7*
5
c
3*=S
Z
5%
ä’1-
—
-C
°
v•°¤i“
>.=•
==;
5
*’
4-)
.;L.
<c>L..*!
3;:3
W
35:
L.
2-
°-
v‘Ü
O
,3;
Z
Z;
"’·;Z,'
3*3:
Ä-
:,,,0
2
_.°§
C
"
*‘
:
2::
5;.
7”:.•g\l
gg
Swan
"nw
P
-l‘•
c';
;
.;,3
4,,,
5g--
5;
3:v-
”",‘f„°
°'^
3***
•·¤
:*>"
5E*=
**3
1:***
5
°“°Ä°
-4->
:g°°
E
'Ü
U-!
<v“·-·
P
"°1
Lw
*¤cn"'
35‘*'
'—'!·E
5
°:
Q-
G
‘°m
·•-
Q)
'
cx
0,:0
25;,
„·-·
3
*!
L';
·-
•J·>cL
com
'Zf"
¤
__
Q1
5
o"-¤·
=
1:-*
EQ.,
..,
4;
vw
2;:m
§L.
SZ!
;,·-
:1
°0
EÄ
Ä
°·°ä··
:3
‘;i'5
Ä.
ät
„
‘
•-ß
"
31
r—
S5!
:23
Ug
j,
°3
3;,
~·ä
SE
'§·^.z
"cu
3
«·°
S'
"’;,·*
—-**2
L/im
::1,
U
.„
za-¤
Q
1;
L
>Q•
q_Q
0
U:
U
,¤
vwo
:1*".1•
r¤
°°c,·•°"
°13¤!
Q
·:1¤
'°
·-*
0
g
.:
3.,
·=
2;-EBZ
1:
°·
C
2-..
[*5
0*;
„·-•
*
"‘
O3
i·*
vb
2-
>·^
¥
LU
¤."'ä,
0g_E
1:
„—§g
3
zg§~
,_§§
Ä,.
3:*
‘°
u
6
Z
‘:
U3.
rz,
=·;,’··
2,*
A
~¤—¥¤
‘
__
U.)
>_‘..•
6
¤
.:1:
<„v•
2
Z50
1-
og
5::~°ä°‘3
äu
S3:
6-
S-,,,,
.
SGU
(J
gc.
"
L.:
~—"‘..1
Oo)
Q,
ää
.%§—
:-*,-5::
;
3
[__
:
H6
.
Z;
3
ß
L.
I
3
;
2E
gg:3
.3
g.
E
°
5
B20
·;·
*5:1
3,
···Z'°
:‘
.:·_
4
1*
S
E
E
"‘3
-2:35
2
;
Ä;
E3
.¤
gg;
..4
=•A
6
•
al
j
A
.n;_*^
A
0
v\·gg'
U
cg-
(5)*
U
•¤•
LV!
J?
1
62
-3
*-0u
A:
¤-5gll
:1,1*
11
6*,
:
E
é~=·ä
E-
t
Zt
W
kb
=,
6
3
„,,,
rg
I
U
,
_
:
..-_
„
6
U
BZ-
2
,,1*
..•-
-.2
iz];
'.;‘
E;
52
_
ä
,_g
i
Ü
¤
U"
L.-‘^
·—
2.
at^
Z-1
“-•
.
:7
A
5
3G./‘
,„"*
:2
2:
««
63
E..=
j
3O
2.4 DETERMINATION OF CVH FROM THE PRESSUREMETER HOLDING TESTThe holding test is performed by holding the expanded membrane
stationary at about l0% radial strain following a regular undrained
pressuremeter test (Clarke, Carter and Wroth, l979). A record of the
ensuing pore pressure decay makes possible the determination of the
horizontal coefficient of consolidation through a theoretical solution,
just as it is done in the laboratory for the consolidation tests.
Randolph and wroth (l979) derived a closed form solution for the
time dependent decay of the excess pore pressures around a cylindrical
cavity. In their solution soil is assumed to behave elastically during
consolidation just as it is done in the Terzaghi's theory of oedometric
consolidation. The solution has been found sufficiently accurate for
an elastoplastic soil by Carter, Randolph and wroth (l979). Fig. 2.9
shows the solution for the theoretical time for 50% consolidation. The
non—dimensional time factor TSO is expressed as,
im = (2.:69)m
where, t50 is the real time required for the pore pressure at the
membrane to decay to one half of its maximum value in a holding test,
and, am is the radius of the membrane after expansion. Thus, by read-
ing the value of T5O from Fig. 2.9, CVH can be computed from Eq.
(2.39). Note that the time factor T50 is a function of the undrainedshear strength SU, Fig. 2.9, and therefore, the computed value of CVH
by this procedure is only as accurate as the value of SU given by
the subtangent method. That the membrane expansion must take place
3]
AU]T]äXSU ‘
6llIIIIH”5IlIIlIIAIIIIIÄI3IlII!IIZIIIIIIIIIIÜIIII0 -4 -2 -2 -1 0 1 2tn (T50)
Fig. 2.9 Time for 50% pore pressure decay atthe membrane—soi] interface (afterRand0]ph and wroth, ]979).
32
under undrained condition, acquires an unusual significance in the
success of a pressuremeter holding test.
In summary, this chapter reviews the pressuremeter theories which
are needed in order to derive the maximum possible information from a
SBPM test. It shows how the coefficient of earth pressure at rest,
KO, the undrained modulus, GV, the complete stress-strain curve, andthe coefficient of consolidation, CV, for the soil can be computedfrom the SBPM data. However, the information so derived is only as
accurate as the assumptions it is based upon. Therefore, emphasis here
is placed on the assumptions inherent in the pressuremeter theories.
The validity of the assumptions will be examined in the subsequent
chapters.
CHAPTER 3
ELASTO—PLASTIC CONSOLIDATION: FORMULATION
OF THE FINITE ELEMENT EQUATIONS
_ It is a common practice in geotechnical engineering to predict
the magnitude and the time rate of settlement from the Terzaghi
theory of consolidation. The magnitude is computed from the compres-
sibility of thesoil and the time rate from a heat conduction type
equation,
2cVä—§=§§§ (3.l)az
The underlying assumption of the Terzaghi theory is that the
vertical total stress increment, generated by the loading, remains
constant while the pore pressures dissipate. Thus, the theory does
not permit any redistribution of total stresses or pore pressures even· if the soil yields at some points during consolidation.
The three dimensional theory of consolidation formulated by Biot
(l94l) provided a coupling between the fluid flow and the deformation
of the soil skeleton. The total stress at any time in the medium is
determined through an interplay between the excess pore pressure and
the stress—strain relationship for the porous medium. Thus, it
is possible through Biot's theory to conduct an integrated analysis
of geotechnical problems in terms of stability and settlement. The
theory is particularly suited for the problem of partial consolidation
that we wish to address in this work.
33
34
3.1 BIOT'S THEORY
The following finite element formulation brings out the central
idea underlying Biot's theory of consolidation (Zienkiewicz, 1977). For
an elastic continuum, a standard discretized equation is expressed as,
[K] {6} = {R} (3.1)
in which [K] is the stiffness matrix and {R} constitutes the vector ofspecified forces. An analogous equation for the laminar fluid flow is
given as
{H1 {q} = {Rp} . . (3-2)where [H] is the fluid flow matrix, {q} the vector of pore pressures
at nodes and {Rp} is the applied flux.In conventional analyses, Eqs. (3.1) and (3.2) are solved
separately. Consequently, the compatibility of strains between the
elastic skeleton and the fluid is not ensured. Biot's theory satisfies
such a compatibility by coupling the motion of the fluid and skeleton
as follows.
It is known from the theory of seepage that in a saturated porous
elastic medium, the forces exerted by the pore fluid on the elastic
skeleton are given as,
_ X ax
2 ,, ÄBY py (3.3)
ällZ az
35
Zor,(xi} {Mi) (3.4)
Discretizing these forces in a finite element manner,
- -.ä_.{ Xi } {pxi } [Np] {Q} (3 5)
where, [Np] are shape functions relating the continuous values of the
pore pressures, p, to nodal values, {q}. Therefore, the nodal forces
contributed by the pore pressures are_ T _ T(F}p — fv [N] (Xi} dV —-[L] {Q} (3-5)
where, [N] are the displacement shape functions for the elastic
skeleton, and [L] the coupling matrix defined as,
T _ T 6(L} - (V (N} (Np} av (3.7)Consequently, for a porous elastic medium, Eq. (3.l) takes the
form
(K] (6} - [LJ] (4} = (R} (3.8)
Now, we wish to modify the Eq. (3.2) for the fluid-skeleton compati-
bility. The continuity equation is expressed as,
- • - ei M M MQ Bt [6x + 6y + 62) (3‘9)
where, the rate of discharge Q is equated to the rate of volume change
éii expressed in terms of the displacement components, u, v, and w of
the skeleton, Discretizing, Eq. (3.9)
6 6 T
36
The contribution of Q to Eq. (3.2) is,
IV [Np]T Q dV =-[L] gg {61 (3.ll)
Hence, for a porous elastic medium, Eq. (3.2) takes the form
[Hl iq) - [1.] ä 131 = (Rp) (3.12)
Suppose the solution (6V,qV) is known at time t1 and we are required to
find the solution (62,q2) at time t2 = tV + At. Employing the finite
difference technique, Eq. (3.l2) can be approximated in the form
[Hl 1qV+ 6<q2 — ql)1 A1; - [L] 132 - 311 = 0 (3.13)where 6 corresponds to the desired integration rule, e.g., B = 0.5
corresponds to the trapezoidal rule. Booker and Small (l975) have
shown that the above integration scheme is unconditionally stable for
B j_0.5. Eqs. (3.8) and (3.l3) can be assembled in a standard form as
V 1= (3.14)-L 6At H q2 —L6V - (l-B) At H qV
Thus, from the known solution at time tV, the solution can bedetermined at time t2.
The formulation of finite element equations incorporating
Biot's theory began with the work of Sandhu and wilson (l969) who
derived these equations by minimizing a Gurtin type energy functional.
A much simpler procedure utilizing the principle of virtual work was
later devised by Small, Booker, and Davis (l976). Following the
similar approach, Carter, Booker, and Small (l979) obtained the
discretized consolidation equations for a soil subject to finite
37
deformations. It is this formulation which is used in this investi-
gation through a finite element program named PEPCO, described sub-
sequently. The final set of equations are of the form shown above,
Eqs. (3.l4).
3.2 ELASTO-PLASTIC CDNSOLIDATION
The equations of Biot's consolidation theory can be used to
model elasto-plastic consolidation by replacing the elastic stress-
strain matrix [De] in the stiffness expression
[K] [Bl dV (3-lB)
by a suitable elasto-plastic constitutive matrix [DBB] for the soil
skeleton. A standard technique for deriving elasto-plastic matrix is
explained in Appendix l. In three dimensions, the matrix is expressed
as follows (Banerjee and Stipho, l978).
{ai = [pgp] {aie T e[DA
+ {al [D ] {al
where {6}] = [aF/aa aF/aa aF/aa . ], X, y, Z ' '
The parameter A is defined in Appendix l. In this work, the Modified
Cam Clay model is used which defines its yield curve as
2 2F=pM
where the mean effective stress p' and the octahedral shear stress q
are given as,
38
0); + o' + oé1>' = ————§—— (3.18)
.. I _ I 2 I I 2 I I 2q — ~/[2 [(¤X ¤y) + (¤y - ¤Z) + (¤Z - ¤X)2 2 2+ 6 Txy + 6 Iyz + 6 Izxil} (3.19)
M is the critical state parameter, defining the slope of the failure
line. The non-zero intersection of the curve with the p'-axis defines
the hardening parameter po. The vector {aF/ao'} can be computed
utilizing the expression
QL = äf.ag' ap' 6c' aq
ao'Foraxisymmetric problem with y-axis as the axis of symmetry, the
rows and columns pertaining to Iyz and IZX are set to zero, and thus,
6 x 6 stress-strain matrix reduces to 4 x 4. The elasticity matrix may
therefore be expressed as,
3K+4G 3K-2G 3K-2G O
Q 1 3K-2G 3K+4G 3K-2G 0[D ] = ä- (3.21)
3K-2G 3K-4G 3K+4G 0O 0 0 3G
The elastic bulk modulus K is taken as pressure dependent. An
expression is given in Appendix 1. The shear modulus G is assumed a
linear function of undrained shear strength.
39
3.3 PEPC0 - THE FINITE ELEMENT PROGRAM
PEPC0 (an acronym for Program for Elasto-Plastic Consolidation)
was developed at Stanford University by Johnston and Clough (l983).
The program has mainly been used in the analysis of tunnel behavior in
soft clays. Elasto-plastic analysis of plane strain, axisymmetric, or
one—dimensional problems can be performed by the program under fully
drained, undrained, or partially drained conditions. The program in-
cludes 40 subroutines incorporating 8 different elements and 5 material
types.
A prime example of structured programming, PEPC0 allocates the
array space dynamically. The array storage is controlled by the dimen-
sion of a single vector in the main program. This feature
assists in optimizing costs where the same program has to be used for
analyzing very small and very large problems. The program solves the
finite element equations using a skyline solution algorithm.
The PEPC0 is well-documented in Johnston and Clough (l983) where
the finite element formulations are extensively tested against the .
available closed—form solutions. Hence, an elaborate description of
the program will not be given in this report.
CHAPTER 4
LABORATORY TESTING 0F SAN FRANCISC0 BAY MUD
A comprehensive laboratory testing program was undertaken during
this project in order to determine engineering properties of the soil
underlying Hamilton Air Force Base in Navoto, California where the
field pressuremeter tests were carried out by Denby (l978) and Benoit
·(l983). The soil, locally known as San Francisco Bay Mud, is a soft
silty clay of medium to dark gray color whose natural water content
often exceeds its liquid limit, and thus, upon remoulding the soil
turns into a thick "mud". San Francisco Bay Mud at the Hamilton Air
Force Base has a saturated unit weight of 90 to 94 pcf (l.44 to l.5l
gm/cm3) and a plasticity index of 40, and is classified a CH soil in
the Unified System.
For over the last two decades, the Bay Mud at the Hamilton Air
Force Base has been a subject of studies at the University of Cali-
fornia, Berkeley and at Stanford University. The available data on
engineering parameters of Bay Mud are given in a report by Bonaparte
and Mitchell (l979) which compiles the laboratory and field investiga-
tions done on Bay Mud until l978. Key strength and consolidation
parameters are presented in Table 4.l.The laboratory effort in this thesis is directed towards the
consolidation behavior of the Bay Mud, especially as to any anisotropic
properties. Both the lateral and vertical response is needed since
drainage from the pressuremeter is influenced by both, especially the
40
4l
Table 4.l Summary of Engineering Properties of S.F. Bay Mud at theUniversity of California Hamilton Air Force Base Test Site(Bonaparte and Mitchell, l979)
Property Value
Thickness of Deposit 55-58 ft
Saturated Unit weight 94 pcf
Natural water Content (%) 90
Liquid Limit (%) 88
Plasticity Index (%) 40
Liquidity Index (%) l
0rganic Content (Total Carbon - %) l.5
Compression Index CCJust past pp' l.2-l.8
2 (l.5~average)For p' > 2-3 kg/cm .8-.9
Recompression Index CV .l0—.l5
Coefficient of Consolidation, CV 8-l0 ft2/year
Effective Stress Friction Angle, ¢'Consolidated undrained triaxial 32.5°-35°
(34° average)Vertical Plane Strain C-U 38°Horizontal Plane Strain C-U 35°Consolidated Drained 3l°
Undrained Strength Ratio SU/pIC-U .34AC-U (K ) .35uu " .32Field Vane .3l-.32Cone Penetrometer .3l-.32Self Boring Pressuremeter .40
42
horizontal response. It appears to be the first time that such an
experimental investigation has been made on Bay Mud.
Also new in the experimental program is an investigation of the
effect of OCR on the stress-strain behavior of Bay Mud. This became
necessary as the OCR in the Bay Mud is believed to vary from l to l.5.
for the depths of l8 to 5O feet where the field pressuremeter tests
were conducted. ·
Drained triaxial tests are included in the testing program as no
such tests with volume change measurements have been reported in the
literature. The volume change data are needed in the constitutive
modeling of Bay Mud by an effective stress model. A summary of the
testing program, including a brief explanation of why each test was
undertaken, is presented in Table 4.2.
4.l SAMPLING AND PRESERVING UNDISTURBED SOIL
Undisturbed piston tube samples of 5 inch diameter were taken at
the Hamilton Air Force Base on January 25, l983. Five tubes, each
approximately l5 inches long, encased tightly in styrofoam boxes with
styrofoam popcorn, arrived in Blacksburg on February 8, l983. As
several months were to pass before the newly purchased triaxial testing
equipment was assembled, and a long duration which is required in the
testing of highly impermeable clays, it was thought necessary to
extrude the samples from the tubes in order to prevent any deterioration
of the soil (which occurs if the soil water reacts with the iron
present in copper tubes). .
WGJ L L•1- GJ GJ
4-* °O 'OL C C U?GJ 3 3 CQ L -1-O W CD W 1-L GJ 'O G) GJQ. ·•·· C •1- 'U4-* 3 4-* EC L L
4-* QJ W CD4-* OWW WQ GJ C. GJan CGJ LJ GJO *1- O >Q) GJIC -1- 1-·L 4-* L -1-I—* LQ Q QQ L Q 4-*-1-E om E GJ :GJ WFG LCD f¤C Q C 4-*L an +-*-1- I/IO O O *1-4-* U O4-* •1- L -1- 4-*•1-I4- WL '4-4-* Q 4-* WQ- Q.O *1-GJ Ofü YU CO O CC. 'OCDC 'OGJO
L>;FGOG)4-*4-* L -I-*1--DOW 1-3
W O•1- GJQ -1-OWC OWLQ an- .: «—mmGJ «.nmOQ ·1-FCS +-*C IUCGJL C('D‘4—L C3 O DOL4-* OL3 ¢'UO” GJ•I- O'OQWQf.UC."OO. 4-*4-* L) GJ(UC fUf'O CQ)OGJOGJO'O
C-- CTO ·1—C·1-C C--GJ-1- (Q •r· •1- t‘O'I- O.°!' O7°f* O.GJE4-> ·•->„- -6->EoE:Eos::LL WO LL~LL·1-LLGJGJ GJm GJGJ+-*<1J>5(IJ4—*fU+-*L) >C L)4-*O4-*L4—*O4-*GJm CQ u1GJmGJfüGJmfüC<[ I-U
IC f'\ Z'\
-1--1- -1- -1- -1- -1-
Q-¤i3U, C OWÖÖ 1-kO C
cn Z'-C-1-4-*WCUI-
'OSö W GJQ 4-* C 'O4-* I/I *1- GJFU GJ fü CL I-' 'O”O L -1-O CDG) 'O FU.Q C 1-1- C Lfü O 1-1- E C..I 4-* -1- OO l IIn In LL ‘¤ 'CJQ- CD um —+->-> GJ GJO I- GJ CC 4-* 4-*
L OO FU füQ Q- Q. I/'IUU 'U 'U„ as 0 E 4-*¤l vv '•* '•—IC O WWC 4-* IC IC¤. GJ LJ GJ•.n·•- uv O O
Q I-—GJfü GJ W W>> "O LL I·— C C
cx: I- GJ L.-1->+-> EO QQ- ·1·· 4-* YU
Q- GJ *1-IC O OVUC ¢”U*‘Ö.O O_Q Q 'QC/C4 -1-CzC4 C4FU C CD L1- D O I- ,
44
A steel extruder was designed and fabricated wherein the samplingtube could be held vertically and the sample extruded with the help ofa hand-operated hydraulic jack. The complete process is shown in
Fig. 4.l. In this essentially one-man operation, a very small amountof force was required to push the entire sample (l2 to l3 inches long)within the tube.
The extruded sample was quickly coated with a layer of melted wax
which was then covered with a single wrapping of gauze. A second layer
of wax was now applied such that the entire gauze layer was completely
soaked in the applied wax. The process was repeated with another
application of gauze and wax coating. Layers of gauze acted as a re-
inforcement providing tensile strength to wax layers.
Thus, three layers of wax reinforced by two layers of gauze
imparted enough stiffness to the cylindrical sample of soft clay so it
could be turned upside—down in hands and carried to a steel closet,incubated with a large dish of water, for storage. The soil samplesstored by this process showed no signs of deterioration or loss in
U
water content over a period of nine months that were spent in thetesting program.
4.2 UNCONFINED COMPRESSION TESTS
Two sets of unconfined compression tests were conducted on
cylindrical samples of l.4-inch (35 mm) dia and 2.8-inch (70 mm) in
height. Each set consisted of two tests: first, a conventional test,
in which the sample is prepared such that its axis lies perpendicularto the direction of the bedding planes, considered horizontal for the
46
San Francisco Bay Mud. Such tests are assigned a prefix V (i.e.,
vertically cut) in this dissertation. Second--a substitute for exten-
sion test——in which the sample is prepared with its axis parallel tothe bedding planes, Fig. 4.2. All such tests are given a prefix H(i.e., horizontally cut). For example, HUC denotes an unconfined
compression test on a horizontally cut sample.
To open a tube of stored bay mud, first a circular cut was madeby piercing the gauze-wax shell by a sharp knife. The tube was then
sliced through with the help of a wire saw, as shown in Fig. 4.3. One
of the halves was immediately sealed by the gauze-wax process, and .stored.
The shell of the other half was removed with the help of a
butcher knife. In most cases, only two cuts were necessary--one
vertical and the other horizontal (on the bottom periphery)-—and the
entire shell could be separated as one unit as shown in Fig. 4.3.
Four samples (2 H and 2 V) were extracted from a tube of ap-
proximately 6 inches long. Trimming operation of the samples is il-lustrated in Fig. 4.4. Table 4.3 presents the results of four tests.A ratio of .8l between the shear strengths of H and V samples is in
approximate agreement with Duncan (l965) who obtained a ratio of .75
for unconsolidated-undrained tests, and a ratio of .78 for plane
strain tests.
It is interesting to note the differing pattern of failure sur-
faces in H and V tests, Fig. 4.5. whereas failure in V-tests occurred· on a well-defined inclined plane in H-tests, the direction of the
progressing failure plane was largely dictated by the silt lenses
47Q
Vertically Cut Sample
Horizontally Cut Sample
=.."'*-..
Bedding Planes(Assumed Horizontal)
Fig. 4.2 Diagram illustrating the difference between vertical andhorizontal test samples.
.C > O3 C\lG : : ¤\ 00U tf) cf) •
·UI-1-UCfüL
LL.C"¤ .U3
O <‘ N (*3 N*4- 3—> (*3 C\1 (*3 C\|O tf) G ····UI**4 .I"“
Q.Efü
C/3'UGJ.Q C\I (*3 ® ® ®g E • • • •: :\ 00 <r r\ <r
-I-* (./3 Z •— r— •— r—U) x••—'UCEGJ.C4-*CO
C\| C\I C\| C\| C\IU) QE • • • •-I-) '>$ <' ÜÜU)
O Z LO LO LO LDGJ Ad,I—
CO••-tl)U)GJLQ.EO U C\lC\|L)O3 Ö3 O3 O3'UGJC
*f'**4-COUC3
*4- .CO 4-* r- r·—· •··
r—Q_E • • . •m‘¤ GJ CO CO CD LD
4-*3 Qr—·E3V*>sGJfüQSI
(*3
GJ •— r—· Q1 C\1•···· -I-J I I I I.D U') C.) L) C.) L)fü GJ E E 3 E|— I— > I > I
52
running in the vertica1 direction.
How do the stress paths compare in two cases? Stree paths for
undrained test on V-samp1es are we11 known (Bur1and, 1971). Shown in
Fig. 4.6, point A represents the in-situ stress state of a soi1 e1ementthrough which passes the current yie1d enve1ope DAD'. The yie1d
enve1ope is necessari1y non-symmetric about the hydrostatic axis,
6; = 6; for unequa1 soi1 strength in V and H tests. During the
samp1ing operation, the soi1 e1ement traverses a path AB to acquire a
hydrostatic stress state at point B.
In a V-test, the stress path rises to meet the yie1d enve1ope at
point C, and it remains on the enve1ope unti1 the fai1ure occurs at
point D. On the other hand, the undrained path for a H-samp1e fo11ows
a pattern BC'D'.
OEDOMETER TESTS
A conventiona1 oedometer test inc1udes a compression test on a
1atera11y confined soi1 samp1e whose axis coincides with the vertica1
direction in the fie1d, i.e. test on a horizonta11y cut samp1e. Such
tests are termed "HCON" test in this dissertation. An HCON test yie1ds
permeabi1ity or conso1idation characteristics in the vertica1 direction
of a stratum. Using the conventiona1 Casagrande interpretation this
test a1so yie1ds an estimate of its maximum past overburden pressure.
It fo11ows that, if the samp1e of an oedometer test is cut
vertica11y such that its axis 1ies para11e1 to the horizonta1 p1anes,
we are ab1e to obtain conso1idation characteristics of the stratum in
the horizonta1 direction. Tests on vertica11y cut samp1es wi11 be
53
FailureEnvelope
Gu KO — Line
D cAYield Surface
Failure 'C' EnvelopeDI
0 GA
Fig. 4.6 Stress paths in undrained tests.
54
called "VCON" tests hereafter. However, there is a pertinent question
in the case of this test--will the Casagrande's method for a VCON test
give the maximum past horizontal pressure? Or, could such a test
determine the present horizontal pressure in a stratum?
The stress path helps explain the nature of the different tests.
In Fig. 4.7, point A represents the maximum stress state to which soil
has ever been subjected. Through this point passes the current yield
surface CAA'C. Point A also represents the current stress state if the
soil is normally consolidated. If there is erosion of overburden
pressure, the current stress state moves to point P but the yield sur-
face remains the same. Point P represents a state of light over-
consolidation. Sampling of the soil brings the soil specimen to point
B (wroth, l975).
In an HCON test, the specimen reloads along path BQA. Casagrande's
method approximately gives the ordinate of point A, which is the
maximum past effective vertical stress. In a VCON test, the specimen
follows the stress path BO'A' and the Casagrande's method will approxi-
mately give us the absissa of point A', oéy, which does ngt_represent
the maximum past effective horizontal stress (abcissa of A). From a
VCON test we are unable to derive any information on horizontal stress
in the soil.A
Tavenas et al. (l975) suggest that the ratio of horizontal to
vertical preconsolidation pressure oéc/oQC is approximately the value
of KO in-situ. It is seen in Fig. 4.7 that such a ratio, given by the
coordinates of point A, defines KO in a normally consolidated state, but
ngt_in a state of preconsolidation such as the one defined by point P.
55
UIV K · Linel o
M AYield
C Surface
AI
Failure| Envelope
C' ;s ' .Ighy
O H H' 0*;
Fig. 4.7 Stress paths in oedometer tests.
GC\l
'
E GP"
LD2:.0 [ °C ' ••'ULJRO3>
'E 4:
4-*+-*>;<./7 Q •FU G) CU GJGQIPG P
LQ QEFUU)• •• O „_FUU'|‘
+->LQ)’ >
(\I FUE• •• • Q äU1
U)->ClJ0 +->
L(U-0-*Q)• •• • "' gUCDOGJ
-6->CU(_) 1---Q_ Q_
° ä...-p--.....
• ·• • LQU
<E
®<l'
• ••O[ LLI
A Q I"'
OO <l' G RO C\1 G 'C“~J C\I öl P Pc9 OILVHGIOA
57
If we compare Figs. 4.6 and 4.7, it is reasonable to postulate
that the ratio of shear strength in extension and compression is nearly
equal to the ratio of yield stresses in VCON and HCON tests.
1.e., (4.1)uc vy
An experimental evaluation of the relation (4.l) will be presented
later in this chapter.
Sample Preparation:
Samples for the oedometer test were prepared with the help of a
knife edged ring cutter having inside dimensions precisely equal to
that of the consolidation ring which was 2.5 inches in diameter and l.0
inch in height. The cutter was initially pushed 5 mm into a 50-mm
thick block of Bay Mud, obtained from the tube cut either horizontally
or vertically. Now a 3-mm thickness of soil was slowly chipped off by
a wire saw. The stepped operation of cutter being pushed 3 mm into the
block and an equal thickness of soil removed from the outside was
continued. Finally a sample of approximately .85 inch was extruded
from the cutter with the help of the top porous stone. The consolida-
tion ring was pushed upside-down around the specimen. Generally, the
ring slid downward almost under its own weight.
Stress Controlled Test:S
A total of 9 stress controlled consolidation tests, including 5
VCON and 4 conventional HCON tests, were performed on undisturbed a
samples. Following standard practice, the load on samples was doubled
58
every 24 hours during the loading phase. An interval of l2 hours was
permitted between successive steps of unloading or reloading. Fig. 4.8
presents a typical e-log p' curve. Curves for the other tests are in
Appendix 2. A summary of material parameters and other useful informa-
tion that could be derived from the tests is given in Table 4.4. An
average value of 0.8 for the ratio of yield strengths in VCON and HCON
test provides an encouraging evidence to the validity of Eq. (4.l).
Fig. 4.9 demonstrates a comparison between the values of the coef- (
ficient of consolidation for Bay Mud in horizontal and vertical direc-
tions. A value of 2.5 is suggested for the ratio Cvh/CV in the
normally consolidated zone of Bay Mud.
Strain Controlled Tests:
A considerable amount of time is required for performing a con-
T ventional stress—controlled test. Even then, only a handful of data
points are obtained for an e-log p' plot, and thus, the test requires
a subjective judgement in the determination of preconsolidation pres-
sure. However, if the specimen in an oedometer is compressed at a
constant rate of strain (CRS) slow enough to dissipate all excess pore
pressures in the specimen, just like in a drained triaxial test, a
continuous plotting of the e-log p' curve can be made within a
relatively short period (l to 2 days) of time. This allows an accurate
definition of the preconsolidation pressures. Tests of this type by.
Denby (l978) led to preconsolidation loads higher than those of con-
ventional oedometer tests on Bay Mud.
Ü Ü Ü ÜU I I I I(U O Ü O Ü O OU) IÜ I IZ I IÜ I-·
> \ X O X O X I X I IR.) C\I Ü IZ LO r- I- F1
E• X • X • •
U LO N (*7 O') LO O')
(*7 (*7 RO LO O (*7 (*7 AI ROU) IZ IZ IZ I- IZ IZ I··— IZ I-
Q • • • • • • • • •
!"'fü O O Ü Ü Lt') Ü LO RO O')c GO GO Ix IL Ix Ix I\ I\ RO ••·
I I I I I I I I
ILI. (UE·L) „,..L) ,.
fü (\I (\I LO Ü IZ N Ü Ü I- C'I" O O Ü RO (\I (\I O) (\I O) O+.7 • • • • • • • • • •|—•I— I- I-- IZ IZ IÜ IÜ IZ fl (/7C U)>-• GJ
LQE.: <I· r\ GO O— -U (Ü N N U C(D slu 3 Q Q I I I I • • I • I O
L/7 U') O II ·I-II -I-*U fü
L7 'O•I-IZ I- (*7 ••* ft
_c (U • • • C O->,¤. I I I I GO <r I GO I 0 InIn Q AC Ü Ü Ü •I- C
-I-* -9 OIn U UCU GJI- O IÜ RO O L '~I--U·> O O C\I •I·- OL Q O I I I • I I • I • 'OQ) FZ Ii IZ -I-—*+-* II IÜ CQ) fü (DE 4-* •I-O C U
'O O7 C\l (*7 O •I-Q) (U • • • ‘ Q-O -U¤. I I I RO I I N I C\.I ·I- I4-
O. A4 Lt') LO RO L (D'O O OG.) .C UIZIÜ Ü Ü Ü Ü Ü (\I GI O) O) C1) IIO . Q (U • • • • • • • • • _C
L —>O- RO RO RO RO RO Ü Ü IÜ IÜ +-* >-I-* O A6 Lt') LO Lt') Lt') LO LC) Lt') Lt') LO OC CO nl- nß
Q,) X• U1 Q}U) U N N G7 N O) RO O C7) U) 'OU) • Ü Ü Ü Ü Ü Ü LO Ü G) CQ) • n • • • • • • | L •I-
L L I- IÜ I- IZ IÜ I- I- IZ -I-*4-* O) U) O')L/7 C
'O •I-RI- I··· IÜO U (D fi
Nä (*7 (\I O LO N RO RO •I·- Q)U) E O) O) O) O) O) O) O) O) N 3-9 U)IÜ II3 IIU) .C .CQ) -I—* RO RO RO RO RO I- IÜ RO RO ··N U)E n • • • • • • • • Q_
UG) RO RO RO RO RO RO RO L.') LF7Oq- „
• CÜ O
L •I·-Q) Q) IÜ C\I Il C\I (*7 Ü (*7 LO Ü -I—*Ii .O Z Z Z Z Z Z Z Z Z fü.Q E O O O O O O O O O -I-*fü 3 LJ O O O C.) LJ C.) C.) (.7 OI- Z > > I I > > I > I Z
60
O an
m 3 3 3 3~ LLQ > > :1: :1: Ä]
‘0 LO2 2 ‘•° ' E4-> I 0 0
>, Q.•— U
¢¤ cu I cuE 2 I
OGJI ‘“‘ =
ä P60‘>‘ §
\ I*1-‘ 05
\ COE
l «¤"ON •‘ E S
\ 3 ¤I cn
AC Q
\ ->U
O 9-
Ö · °-6-*
\ s:GJ
\ ‘*‘U\| ,.. 1;:,L1-mCDG)@|—
_;’ :2FG’/ :0
!*"!*@4-*>L„
ffGJ
0g>4->‘O
CqöföCTO04-*(DC•P¤O
LNFUN-0.LE0
C<E0
OWfi O
C\I
l qr
u-—· p- ° '• Q7
. A A*•·—
(DBS/ZLU3) X L4 UJQ 3 1.1..
6lA total of 4 CRS consolidation tests were run during this testing
program on Bay Mud including a test each on undisturbed HCON and VCON
samples and two tests on remoulded Bay Mud at different strain rates.
Drainage was permitted at the top and bottom of the specimen.
Constant rate of loading was applied through a hydraulic frame. _
A newly acquired electronic data acquisition system was used for con-
tinuous recording of the applied load and the change in height of the
specimen. Load was measured by a 2000 lbs. load cell and displacement
through a Linear Variable Differential Transformer (LVDT). Micro-
computer program was designed in such a way that it recorded data every
5 minutes for the first two hours, and every l5 minutes thereafter.
A complete set-up of the test equipment is demonstrated in Fig. 4.lO.
Fig. 4.ll shows e-log p' curves for the CRS tests. The plot
gives an overconsolidation ratio of l.3 against a value of l.06 noted
for the conventional test for a depth of 20 feet. However, CC values
are similar to those obtained from the conventional test. Table 4.5summarizes the results of the CRS tests. A value of .84 is obtainedfor the ratio of yield stresses in the horizontal and vertical direc-
tions which is again comparable to that obtained from the conventional
test. Variation in strain rate in two CRS tests had no effect on thecompressibility of remoulded Bay Mud, as seen in Table 4.5.4.4 TRIAXIAL TESTS '
In order to determine stress-strain strength parameters, drained
and undrained triaxial tests were performed on undisturbed samples.
During the consolidation phase, careful measurements of volume change
63
C)r-· (*7
Z IO6 SI //U')Z .L.) /
I•
In
ri [IE“° Z I „.E .C Z /
I G)-0-* ® -I-0>„ Q. C.) / I GJFU CU I EQ Q I [ I c
tf) U2 'I I 2. / M I C. I „_
/ I N5 fs?..' 2 2 2// Q) U7x I+-/ <¤j ¤’ / G
-O> GJ/ I- -I->ruQZ L°f E"' EIUj
OUH-OIn, -I->
{ EQ)[ QtI .I F"
I '"ZI <rI - .I .9| L1.I Ä I-
00 <r Q ao cx: oo•
GI (\I GI ·•— s-
6 ‘01Iva GIOA
P(U CO (*7 (*7 C\IC N N LO LO
I I l O
LJ.U
LJPf¤ (O G G•r- P (*7 N I+.7 • • 5°f* IC IC
CI-•
-CGJ U - U Q: Q Q. Q. 00 I Itf) U') •
ll
.O
- >~, UI N I IQ. Q. LO
(*7(.7 I • I IQ P
In RO4-* - L) •I-
~In Q. In I CO I IGJ Q. SDI- •
•I-L U7(U C C G O.4-* ·r· GJ ·¤— LO kD (D Pcu ru -I-> E C> CJ CD c <I·E S- IG \ G G G G •O 4-* M C G G G G P'¤ (f) •y-• • • • • C(U 4-* O .G •'¤ •r-
4-*'O C\| (\I 4-* UCU O ¢‘¤ •
· -C GJP • > Ü. I I G7 LP O X LO LO ·r· •r-
O C CS- L
4-* GJ PC • > IGO U G LO N N O 4-*G • Q Q Q Q C
>- U • • • • '¤ OC N P P P P CU N'f' CD 4-* •r-(U (U LL 'U O
4-* •v·· CU7 U OG LO P
OW G O7 O C'~I— E P L/7 •r-O C
O UIL/I U I./7
4-* C GJ (DP 4-* P P L L3 Q. E •
· I I Q. -I->m QJ (O LO InGJ G V7Of (U 'O
3 P(U
LO P IC 'X C °|*• • Z Z IC GI CL) >3Q O G G G G Ez c.: U Z Z ··— IIC1) UP 4-* I I· I I CU C.Q In M M M M Q. - >„(U (1) (./7 (/7 (/7 (/7 C/7 O.I- I- U <.> c.: <.> -I<
65V
versus time were made which enabled the determination of flow charac-teristics of Bay Mud under isotropic stress. And thus, as a byproduct
of the triaxial tests, it became possible to provide a check on the
value of horizontal coefficient of consolidation obtained from the
oedometer tests. An advanced triaxial equipment is needed as volume
change measurements have to be made under back pressure.
UU Test:
One unconsolidated undrained test was performed which gave a
SU/pé ratio of .33, almost equal to that of unconfined compression
test. A failure strain of 6-7% was noted.
Consolidated Drained Tests:
For an effective stress model, consolidated drained tests offer
the best promise for the computation of model parameters. Four drained
tests were conducted on Bay Mud with confining pressures of l.2 to
2.5 times the in-situ effective overburden pressure. Volume change-
time curves for the consolidation phase of one test are shown in Fig.
4.l2; curves for the other three tests are relegated to Appendix 2.
Coefficient of consolidation for each curve was computed following
Bishop and Henkel (l962). Curves show a considerable range of
secondary compression.During the shearing phase of the tests, an unusually high axial
strain of the order of 35 to 40% is required for Bay Mud before the
peak shear stress is mobilized. This is seen in Fig. 4.l3. what is
perhaps more surprising is the fact that even at 40% axial strain thetest specimens continue to decrease in volume, Fig. 4.l4, and the
66
ft /min.00 6 12 18 24 30 36 42 48 54
1 Test CD-2°é ‘ Api + ^Pé
2
<ä .gg 4Iga
5 Curve for Apéc::>
6Curve for Ap1 1
7
8
Fig. 4.12 Isotropic conso1idation phase of the triaxia1 test(two stage conso1idation).
67
280 CD‘3 · · . _
240 ‘ CD_2· • •
200 2 · °
~ 00-1 2§ 160 . _ ' ·.>£[T"')
O'_ 120 ‘-fl • 0
80 A ° _
40 ,!.
0 0 8 16 24 32 408 %X 0
Fig. 4.13 Stress-strain curves for c0ns01idated drainedtriaxia1 tests.
68
* r—Q FU
°I*
><FU'|“
L+-9
EC\.l r— CQ 1 1 1 -1-E Q Q FUL.) LJ C.) L
'O
ku +-9FU
'O'I*
% ICOU)COULO
- '+—In
ÄÜ S EL3UGJC7
· CE
ko UF" GJE3
|'*
O>IC
°|"
E@ -I-9
U')
·FU -I-9/ ·1· I/7
>< CD<E 4-9
QÖQ
CD Q ® C\| SO
o EISNVHI) HNÜWOA Lg
69
critical state condition, in a true sense, is never attained. This is
due to a highly compressible nature of the Bay Mud.
The stress-strain data for the drained tests is documented inAppendix 2. Test data has been corrected for filter and membrane ef-
fects following the recommendations of Duncan and Seed (l967).
Consolidated Undrained Tests: gDenby (l978) conducted a series of undrained triaxial tests on
Bay Mud following the SHANSEP approach of Ladd and Foot (l974) wherein
the samples were anisotropically consolidated with an assumed value of
KO. Similar tests had been reported by researchers at Berkeley
(Bonaparte and Mitchell, l979), but what distinguished Denby's test
was his use of back pressure of an order of 95 psi to ensure the
complete saturation of samples. while Denby found the same order of
ultimate shear strength as those of previous researchers, a conspicuous
decrease in the failure strain was noted. Denby's samples failed at
approximately 2% axial strain as against 4 to 6% reported previously.
This resulted in an increased value of shear modulus.
Undrained tests in the present investigation were conducted on
isotropically consolidated samples with a back pressure of 90 psi. The
stress state in all samples was brought initially to the virgin con-
solidation line through a consolidation pressure of at least l.5 times
the in-situ overburden stress, and subsequently unloaded to induce
0CR effect if desired. Values of l.0, l.2 and l.5 are used in the
tests. Appendix 2 includes results for the undrained tests. Table 4.6
Lfü RO (*7 (*7 RO RO LO (*7 C7 C IC 1-Q) RO (\I (\| (\I N C RO (*7 IC C\I LO (*7 G IC Q(\I N IC IC G (\I (\I 1*- C7 LO (\I (\l 1- C I"-
I I I I I I I I I I I I I I IQ] IC 1- IC
E
>L)C C C C C C C C C C C
0 <r 1 1 1 1 1 1 1 <1· 1 1 1 <r 1q) 1 CD O O O O O GI O O O1 C)gn C G IC 1- IC 1- FC IC IC G IC 1- IC G IC\ I 1- X X X X X X X IC X X X IC X(\I G X C7 C7 C C (\| LO X (\.| C AI X (\lE 1-; (*7 (*7 (*7 (\I N G C G G7 G N (*7 (*7 (*7U
• • • • • • • • • • • • • •C GI RO C 1-·
LO C C7 C\I (*7 RO N (*7 ON NO:
I I I I I I I I I I I
@-1- RO (*7 RO RO IC N LO LD FC ON (\IC71-EG7 LO N N RO N N C RO N G G C7 C C7-|-7 1- 1- IC (\I IC IC IC
C\IE (\| 1** CO RO G C RO N IC RO N ON LO 1--\ (*7 N C7 C G G 1- (*7 G (*7 N G (\I C C
an (D Z C G C (*7 N RO C FZ LO 1- C C C7 C IZ-|-7 Q') _¥ IZ 1- IC 1-U'1 füGJ LI- G)
><[-1- r- RO C (\.| LO G G LO 1- G RO LO4-) •O ·1- (\I RO IC G C N C (*7 C O7 C7 C C LO¤ Ö I I I I I I I I I I I I I I I
IU Q RO LO N L.O IZ G RO RO N RO RO LO (*7 RO RO1*1- IC 1- IC IC 1- FC1-OL/ICOL)
U U U U U U UU m C C C C C C C
-1- Q): L1.! L.1.| L1J L1.! L1..| L1J L.1JQ O7OO ¢‘ü•1- + + + + + + +L C4-)
4.) 11--1- 1- IC IC 1- 1- IC IC IC 1-O füU fü fü fü fü fü fü fü fü füm L; -1- -1- -1- -1- -1- -1- -1- -1- -1-
1-1 go U U U U U U U U ULJ f‘ü fü fü fü fü fü fü fü fü
N- 1 1 G5 1 1 1 Of. 1 OCO1./'14-)
FC3Ln .CQ) «|-7 LO L(7 LO LO !*' _1- IC 1- FC1 ¤_E . 1 1 1 1 1 1 1 1
Q) RO RO RO RO RO RO RO RO ROCDfü
ROCGJ 1IC 4-* • 1- A1 (*7 C LO G FC (\| (*7.¤ 1.nO 1 1 1 1 1 1 1 1 1fü (DZ G ID D Z7 IJ 3 G G GI- I- L7 L7 L7 L7 L7 (.7 (.7 L7 LJ
'UGJ E+9 I
+9 1- (.7C "" .(D 49 ·E S-E (D +9Q 1- (/7
L7 O.E •(U 1-
(/7 Q.
3(/7 O Q' L(7 LO\ 1 1 1 M N M CD•1- LO C7 LOL|_|
1--
-I<O 0 O 0 0 O(\I -I¢ N ä IZ LO 1-
.. [ | • O Q • • O • • •
6- N (D (*7 <I' 1- 1- CD ON Yi(\I (*7 (*7 (*7 (*7 (*7 (\I (\I (*7
N C7 C7N ä' <' <' LO O3 (*7 (*7 (*7#-1- 1 1 1 1 1 1 1 1 0,.) 1‘ 1
(.1) (D (*7 (*7 (*7 ä' @ (O G] ®(*7 (*7 (*7
LO (O C7 ® <'I1- | G G7 O N (O ON I I [ I<
• • • • • •FZ IZ
(*7 LO §' N <I' (OJI- U M N M M Q- Q- M 1 1 1 1• • • • • • •
"UCU .(JGS (\I LO3(J I ·
•IZ 1*- IZ F- fi r'- I'- ft IZ 1-
C1-1
42 (\.IE G N IC <I' <" 1- C\I ä' G7 (\IV7 ‘ UN E (\.I (\I N (\.I 1- (*7 LO fi LOGJ O Z E d' N C7 <‘ N C\I LOI- X Ii 1- fi IZ IZ 1-
1-(U•'—¤
IX +9 LO LO LO LO LO LO LO IZ IZ fi LO
Q O O O O O O O I l O Q
-1- (1) LO (O (O (O (O (O (O LO (D (O LOLI-
I-I-O D. D. Q. 1-
FU• ••+9
(U +9 +9 +9 OIZ O. (/7 (/7 1/I ·1— 'U 'O 'U 'U3 N C C C -I-7 (U (U GJ (IJ •U') I- O O O C C C C C OQ) L) L) L) GJ 'I- -1- -1- •1-Qt +9 > (U fd (U (U II
U-) 10 ••~ •n C L L L LG) E E E E E O 2 L7
.¤ I- 1 1 1 1 1 U 1 1 1 1(D E (.7 (.7 L7 (.7 (.7 C (.7 (.7 L7 L) C
• 1-1 1-1 1-1 1-1 1-1 1-1 1-1 1--1 1-1 Qä' TJ(D GJ
PC +9 _ YZ (\I (*7 <I' LO (O IZ (\I (*7 Q' m.O (/7 O I I I I I I I I I I tt}f¤ (IJ Z ID D D 3 3 ID E Q Q Q Q LDI'- I- E L7 L) (.7 (.7 L7 L7 LJ (.7 (.7 (.7 -)<
72
summarizes the useful information extracted from all triaxial tests.
Failure strains again appear to have reduced to 3 to 4% as against 6
to l0% reported previously. Such reductions in failure strains are
attributed to the use of back pressure. Again, a critical state for
excess pore pressures is seldom reached.
Determination of initial tangent modulus, Ei, from undrained
stress-strain curves is a tedious task and is often subjective. For
heavily overconsolidated London clay, practice has been to assume the
value of Ei as the secant modulus at l% strain (wroth, l97l). An
unrealistically low value of the modulus is obtained by applying such
technique to soft Bay Mud. Denby (l978) obtained the Ei values of Bay
Mud by fitting initial portion of the undrained stress-strain curve as
hyperbola. Procedure is illustrated in Fig. 4.l5. Values of Ei/SUwere obtained as high as 2500 which appear to be unrealistic for soft
clay like BayMud.In
this work, a new procedure is devised for the determination of
initial tangent modulus. The technique lies in plotting the secant
moduli of the earlier portion of the stress-strain curve against the
corresponding axial strains on a semi-log plot, as shown in Fig. 4 I6.
Extrapolated value of the secant modulus at zero percent strain is the
initial tangent modulus for the test. For normally consolidated Bay
Mud, an average value of 520 for Ei/Su ratio is determined by the methodwhich is in close agreement to a value of 600 suggested by Fig. 4.l7, ·
due to Duncan and Buchignani (l976).
73
.. - - i°L‘.°’2’2*........
E i _ REALP:
Ea •g·3) =.—+E1 (Gi ·¤31„n
E
TRANSFORMED1 4. 4 , L . 4.E, i*91i'
*0
LEaE
Fig. 4.15 Initia1 tangent modu1us, Ei, by hyperb01ic representationof a stress-strain curve (Duncan et a1., 1980).
74 «
40,000
Ei Bay MudTest CU-4
30,000
-E\Z
anLLJU; .; 20,000Q OCJZf;EELJLu”° •
O
10,0000 .1 .2 .3 .4
AXIAL STRAIN, ex %
Fig. 4.16 Determination of the initia1 tangent m0du1us, Ei.
I 75
1600 11$—_IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIUIIInnnununnueeeeeeeeeeuen 11ee eeeeee11e1111111 11111Zä2!1ä§ä2!!111n1111•n111h1• 1„111•11111111I 111*31111
|4OOr4z•u•4•4•nen.~ee eu 1 1111 eeeee 111111111 II n1 1 II—Zh-ZIIIIIQIIICIIIIPIIIIII uns IIIIIIIIIII IIIIIIIIIIIIYAIICVAIIIIIZIKIIGQQI nu uh e Ilullllllllll IIIIInunnuuzuneueeneunes J I1 eu! II Inu 1I uu 1 11200 2111'2!1i1111111111111:Z 1„I1111•1111„«111•1n1111IIIIn1n"·‘· ·
‘ . ”°zF I2ää2!}1111111111111111„f•1111'11•11'1E1I 111111 III2äi‘$2?1F1!1111!11!111111!'i'1IIE111lih' '1'11II111' 1Eé==é··i:€::·=£11é:a:‘::=‘:::‘E;=:·.IäII··:E§I1"·ä4 .1* ' L . 2 2 Z -znugnnennuezece ee¤•1m1•:1g1¤,u äluhm
1K 800 1121}$.'1111111111111111a1!1111111111T hh%2Z11§$111111111!1111§1!111111I111111!1II IIIIII1
uzguemrepneneeneeseg 211 1111:,, neuem II 1 1600 nu 2•z•4g4en•ueue•.11m1•41 queneen 1 u11£1§11'§§&$;¤§1€'”!'11111'111'1I1'1i111111111I*111111111II"II1'1n‘n-“
: ' 2 . I ·::7 ' Ü . ' Ü ' Ü. 1ääääääääää11111«12!1=1·'.1111I1111111111111111n!1IIIIEIIIIIIa1:.•·•e•=1:11¤•e.1e•1o>>·>>·m¤.vg•1 veeneezsezngeaeunäy-.· py
400 sa-Inn 1eu¤.ee.4„4¤«c««m„n ••neue „ee.•1•:• ..111•..·¤r•e see .111 .n111:«I11e1111-•4•.-.-;:—·~¤·o:1m¤:·2mzanssezasussaszaassnrO
1'ZI'II'_F)PlIPlI5II%II•••.'"!Iv)¢0§lX¢0)9))D10C¢¤·„7"|I nm29 zuu1v•e~•.•.¤zeeneuemmeecu, ~··¤o;assssssmass:z1n.:··v.••1•¤¤¤ee•1••11e•4ezeez•e••1em1••Ien•. :—¤2za1¤ss:¢¢1sw:«::-;n------.-„„....-;; ·-·-•1¤211g!g_;=!=ee¤ee„.é::·;·¤:11aggggggggg==lll11"'111'11111I111111···•1 ‘11‘12'!=!1!=*¥*==¤1¤··-•O 1111$$—mIII!Ill IIIIII„IIIgllslllalalllihlllllllllll
I I.5 2 3 4 5 6 7 8 9 IO ·OVERCONSOLIDATION RATIO
•Eu KUNORAINED MODULUS OF CLAY
K • FACTOR FROM QHA RT ABOVES„ • UNDRAINED SHEAR STRENGTHOF CLAY 1
Fig. 4.17 Chart for estimating undrained modu1us of cIay (fromDuncan and Buchignani, 1976).
’
76
The Unconventional Test:‘
A comparison of the values of the coefficient of consolidation
obtained from the isotropic consolidation test, Table 4.6, with those
obtained from the oedometer tests, Table 4.4, shows that the isotropic
consolidation consistently yields lower values of the coefficient of
consolidation. In order to explore if the difference could be
attributed to different stress conditions in two tests, an unconventional
test was conducted in the triaxial cell which approximately simulated
the stress conditions present in the VCON oedometer test.
In this test, by stabilizing the motion of the top cap, tri-
axial sample was consolidated under the condition @1 = 0, Fig. 4.l8a
Radial drainage was permitted. Subsequently, sample was sheared under
undrained condition. Fig. 4.l9 shows the variation of volume change
with time which gives a Cvh value of 6 x l0-4 cm2/sec--a result inclose agreement with those of VCON oedometer tests, Fig. 4.20.
Stress-strain-pore pressure curves for the test are given in Fig. 4.2l.
Finally, it should be mentioned that no laboratory test can
simulate, precisely, the consolidation phenomenon encountered in a
pressuremeter test, Fig. 4.l8b. The closest we can come to is the
unconventional test described above.
77
61 = O/
Soil 62 = 63
Fig. 4.18a The udconventionai test--uniform consolidation.
€·| = 0/ / /
62'
/
Fig. 4.18b The pressuremeter test--nonuniform consoiidation.
78
O<|' U
GJU)R(\I
C E •O U C
KO ·•— Q(\I I ·l-* <' •!—<r IJ cu u 4->
LJ 'U @ fü·n·· r-
‘¤4-* 1- •¤-U7 Ö Ü >< ITGJ U7f„\) G OF" C 1 r— LC7 U)
O 4-* G CKO ^ LJ L= · OO') 'U <' KO U
3 r-·Z eu n u <+-.,. O
>; ‘O .Cfü fü > +-*C.) C
CU(*7 •r-
U••-C
‘+-.,.. q.E (D
OU
Q' r-(\I fü
4-*COl;_—· ~••-S-O
® C .C•— •r—E GJ
.CKO 4-*
- LO 9-KO O
C\.| II C¤—· OCD ·•·C) 4-*n- IU
4-* 3iziitfü
>KO LLJ
O3r-<‘
® C7'I‘
r—· C\.I f*7 LO KO LI-
39 ‘^V EISNVHTJ HNITIO/\
79
10 .\\\
\\ Oedometer (VCON 2)
T• • ~I R d' 'Iriaxia , a ia
ä\ ,.
‘—Drainage (é = 0)
g>>L‘\\_ 0edometer(HCON2)
oLj> ‘\\\\_Triaxial, Radial
O Drainage Under Iso-tropic Pressure
O . . .O 0 Triaxial, Radial & End
Drainage UnderIsotropic Pressure
O
.1 -10 100 1000G10 p' (kN/mz)
Fig. 4.20 A comparison of the values of the coefficient of consolida-tion obtained from different tests.
80
100
80cx:E\E 60 ·Amo°,_ 40
320
0 2 4 6 8 106X %100
80
\E=°’ 40
20
00 2 66 %X
Fig. 4.21 Stress-strain-pore pressure curves forthe unc0nventi0na1 test.
lCHAPTER 5
MODELING OF SOIL BEHAVIOR: EVALUATION AND
, VERIFICATION OF SOIL CONSTANTS
In this dissertation, soil is characterized as a strain-
hardening elasto—plastic material. Strain-hardening is reflected in
bending of a stress-strain curve to the right before failure. why
does it happen? For drained tests, the phenomenon of hardening is
relatively easier to explain (Jain, l979). As the soil decreases in
volume, the effective contact area between the particles increases,
and thus, increasing amount of force is required to cause further
deformations. However, the phenomenon of undrained hardening is not
as easily explained.
Experiments of Taylor and Quinney (l932) on annealed copper have.
shown that metals harden in as much the same way as do clays under
undrained condition. In fact, prestressing a metal is tantamount to
inducing overconsolidation in clays. In what follows, we explain the
mechanism of undrained hardening in the Cam Clay model on the basis
of hardening mechanisms postulated for metals.
5.l MECHANISM OF HARDENING IN METALS
we characterize a macroelement in a continuum as an assemblage
of elastic perfectly plastic elements. For simplicity, consider a
combination of 3 such elements, represented by three bars of identical
dimensions, placed in parallel as shown in Fig. 5.la. we wish to
examine the load-deformation response of the system for the conditions
— 8l
82
I I II I 2vA ---I I ¤I I.5 YA ··— I
v,L ät v,¤. I I _ _C II LI I I
% IL IE E
.
Fig. 5.la A parallel combination of elements with different yieldstrengths.
P
3YA -··— -II
ZYA - I
I Iv,L v,L I
L I II
'· I° II % IP
Fig. 5.lb A parallel combination of elements of different sizes.
83
that the bars have identical elastic properties but possess different
yield strengths. For ease in computations, assume that the second bar
has a yield strength of Y/2, where Y is the yield limit of the other
two bars. By virtue of a parallel arrangement, the displacements are
equal in all three bars. Upon application of load P, the second bar
yields, when
$=§ 2 l.€.,P=§·YA (6.1)at that moment, 6 = igéélk-= ;% . (5.2)
Further, other two bars yield, when
?%= v 1.6., P = 2YA (6.2)
at that moment, 6 = iBlglk—= ik (5.4)AE E
and the system would collapse. Load-deformation response of the system
_ is shown in Fig. 5.la. Thus, when a material is composed, at the
microscopic level, of infinite number of elements with different yield
limits, it is the successive yielding of these microelements which is
manifested in the form of hardening. Perhaps, the reasoning was too
obvious to require any mathematical justification.
But, what if all points in a macroelement have the same yield _
limit (an underlying assumption in the Cam Clay model)? Where does then
hardening come from?
Consider again the parallel assemblage of three bars. Now all
bars have identical properties but the second bar is only half as long
84
as the other bars, as shown in Fig. 5.lb. The load deformation
response of the system demonstrates that a material can exhibit
hardening when its slipping microelements are of unequal dimensions.
Even such a mechanism, which is geometric in nature, does not explain
the hardening phenomenon in the Cam Clay model wherein the material is
assumed isotropic and homogenous.
Finally, we consider a series combination of two elasto-plastic
elements as shown in Fig. 5.2a. As we see from the load-deformation
response, the system exhibits hardening even if both elements have
identical dimensions and material properties. Such a hardening is due
to the build up of internal (elastic) energy in the spring connecting
the plastic elements (Mroz, l973). In other words, a part of the work
done is locked in due to the plastic deformations. It is known that
the stored energy acts as a barrier to further deformations in a system.
This is one of the causes of hardening in metals, where at the
microscopic level, elastic elements are surrounded by a network of
gliding planes. Such a mechanism of hardening helps explain the
hardening phenomenon in clays.
5.2 MECHANISM OF UNDRAINED HARDENING IN CLAYS
Fig. 5.2b demonstrates an edge—to-face arrangement of plate-
shaped clay particles in a triaxial specimen. Application of an axial
force causes particles to bend until the edge—to-face attractive forces
between the particles are overcome. Thus, a part of the work done is
accounted for by elastic deformation of the particles. The amount of
elastic energy stored, as Terzaghi and Peck (l967) indicate, is likely
86
GT
n,gu 1‘ éxcg —» ‘,;§ { <—— cgäx
OT
Fig. 5.2b Triaxiai specimen of ciay with fiocculated structure.
87
to be different for different clay types. In a sand—mica mixture,
for example, much of the deformation is caused by elastic bending or
restitution of the grains. Also, with Fig. 5.2b, it becomes clear thatclays with flocculated structure will show a greater amount of un-drained hardening than the clays with dispersed structure.
A conclusion may be deduced from the foregoing discussion. It is
the unloading index, CS, which primarily governs the extent of un-
drained hardening in clays. 8
In soils, hardening may also be visualized as coming from the
build up of pore pressure due to shear. An increase in pore pressure
causes the mean effective stress in a specimen to decrease which en-
tails an elastic expansion of the soil skeleton. Since no change in
volume is permitted in an undrained test, an elastic expansion must be
compensated for by a plastic compression of equal magnitude. A plastic
compression causes yield surface to expand, which necessarily implies
a hardening phenomenon. Likewise, a decreasing pore pressure during
shear entails a contracting yield surface (if the theory at all per-
mits such a contraction) leading to the phenomenon of strain softening.
5.3 CAMBRIDGE SOIL MODELS
Calladine (l963) extended the classical work of Drucker, Gibson,
and Henkel (l957) to establish that normally consolidated clay was a
strain—hardening plastic material also in the undrained condition and
thus could be modelled within the realm of mathematical theory of
plasticity. In particular, Calladine showed how the concepts of
critical state soil mechanics, then advanced by Roscoe and co—workers
88
(1958,1963) cou1d be combined with the p1asticity theory to derive
stress-strain re1ations for soi1s. within a year of Ca11adine's work,
there came the first Cambridge soi1 mode1 (Roscoe and Schofie1d, 1963)
which was 1ater to be known as the Cam C1ay mode1 (Schofie1d and
wroth, 1968).
By drawing ana1ogy between the experiments of Tay1or and Quinney
(1932) on meta1 tubes under combined axia1 pu11 and torque with the
behavior of norma11y conso1idated c1ay in triaxia1 compression tests
Ca11adine postu1ated that the yie1d surface for the c1ay might be
e11iptica1. The suggestion was 1ater quantified by Roscoe and Bur1and
(1968) in what became 1ater the "modified Cam C1ay mode1." This is the
mode1 used in this dissertation. A brief account of the mode1, with
emphasis on undrained behavior, fo11ows.
Roscoe et a1. (1958) showed that for a norma11y conso1idated
saturated c1ay the void ratio e was a unique function of the mean norma1
stress p' and the deviator stress q. Based on experimenta1 data, they
were ab1e to show that a11 possib1e 1oading paths for a c1ay specimen
1ayed on a unique curved surface in the p',q,e space which they ca11ed
the "yie1d surface." The surface was 1ater renamed as the "state
boundary surface." A typica1 such surface for norma11y or 1ight1y over-
conso1idated c1ays is shown in Fig. 5.3.
Let us consider a triaxia1 specimen norma11y conso1idated to a
point E on the norma1 conso1idation 1ine PER in e-p' p1ane. Now, we
have two choices: Either to shear the specimen, in which case the
specimen wi11 fo11ow the path PS under undrained condition on the state
boundary surface; or, un1oad the specimen under isotropic pressure to
89
Q
P ‘« ~WW O
P NP L
e
Legend:PER CSL - state boundary surface EWV - elastic wall
PER - normal consolidation EV - yield curveCUY‘V€
ES - undrained state pathCSL — critical state line
KUMN — undrained (e = const.)Ew - swelling curve plane
Fig. 5.3 State boundary surface in p'-q—e space.
90
point X on the swelling curve EXW. If the specimen is now sheared
under undrained condition, it first follows an elastic path XX',
perpendicular to the e-p' plane, to yield at X' where it meets the
state boundary surface. Thereafter, the specimen traverses an elasto-
plastic path X's to fail again at point S on the critical state line
CSL.
It is the assertion of critical state theories that on a swelling
line, no matter where the specimen is sheared it deforms elastically on
a vertical path until it touches the state boundary surface. The
phenomenon leads to the formation of an "elastic wall" EWV representing
all possible elastic states of the specimen under undrained condition
corresponding to the swelling curve EXW. To deviate from an elastic
wall, a specimen must undergo plastic deformation. That is to say it
must travel on the state boundary surface, before it can jump over to
another elastic wall. In other words, there corresponds to each point
on the state boundary surface an elastic wall on which a specimen can
unload (elastically).
Indeed the curve EX'w, formed by the intersection of the elastic
wall with the state boundary surface, thus, defining the boundary of
elastic and plastic deformations, represents a yield curve. As evident,
a specimen traversing the path EX'V, i.e. along the yield curve,
experiences no plastic deformations. The yield curve EX'V can beA
taken as a plastic potential since the application of the normality.
rule on the curve does not result in plastic deformation along tangential
direction, thus satisfying the continuity condition.
9l
During an undrained test, the load trajectory ES, Fig. 5.3,
passes through a set of successive yield curves as seen in the projec-
tions of the state boundary surface onto the q-p' and e—log p' planes,
Fig. 5.4. The process of hardening goes as follows. In passing from
state A to state B, Fig. 5.4b, a positive plastic volume change takes
place which is given by AD and consequently the yield surface expands.
However, due to a decrease in p' from A to B, an elastic expansion
occurs which is given by DB, along a swelling line. The elastic expan-
sion DB must be equal to the plastic compression AL) so that shearing
may proceed at a constant volume.
For normally consolidated clays, it has consistently been ob-
served that in p'-q plane, e = constant (or w = constant) curves are
similar for drained as well as undrained tests. Such a curve, which
represents the intersection of the state boundary surface with the
e = constant plane, is indeed the stress path in an undrained test.
Furthermore, corresponding to a given value of void ratio (or water
content), there is only one curve in the p'-q plane, irrespective of
whether such a curve is obtained by a drained or an undrained test.
This fact, Calladine (l963) shows, when combined with the assumptions
that the curves in Fig. 5.4b are straight, and that all elastic swell-
ing curves are parallel leads to the similarity of successive yield
curves on the state boundary surface. Thus, the isotropic hardening
for normally consolidated clay is an acceptable assumption. The
discussion also implies that the same set of yield surfaces can be
used both under drained and undrained conditions.
92
9C.S. C'Line
SuccessiveS _ Yie1d CurvesVi
0 E Y Z p'(a)
E NCLB A
Yz
DCSL
0 I10g p(b)
Fig. 5.4 Projection of the state boundary surface on q—p' ande—10g p' p1anes (Ca11adine, 1963).
93
The essence of this discussion is that for a given combination
of stresses there is a unique state of void ratio regardless of
whether such a state is attained by a conventional drained, undrained,
or a constant-p' test. And, within the framework of critical state
soil mechanics, a single set of stress-strain relations can be used for
predicting drained, undrained or partially drained behavior of
normally consolidated clays.
Model Parameters:
A total of 5 soil constants are needed to completely define the
stress-strain relationships through Cam Clay model. Constant M, the
slope of the critical state line, defines the failure envelope in p'—q
plane. The evaluation of M using triaxial data for San Francisco Bay
Mud is shown in Fig. 5.5. An average value of l.27 is chosen for tri-
axial compression tests. In e-log p' plane, the critical state and —
normal consolidation curves are assumed to be straight parallel lines
in the model. As seen in Fig. 5.6, the assumption is only approximately
valid for the Bay Mud. Normal consolidation curves for Bay Mud show a
concave upward curvature, typical of a sensitive clay. The following
three parameters define the position of state boundary surface in e-log
p' plane. Swelling lines are also assumed straight parallel lines.
C
K 0.062 (6.6)V = 3.75
C
94
280 ( 1 ,G Data, this dissertation /A Data, Denby (1978) jf /
240 ¤ Data, Duncan (1965) / /
200 / /“‘ M = 1.40
///1015E160’F¤
T /·— 120 /
380 __
. A7?40 /’/ Bay Mud
0 O0 40 80 0 •• •• ¤• .•(oi + oé + oé)/3 kN/m2
Fig. 5.5 Determination of the fai1ure enve1ope from the triaxia1 ‘drained and undrained tests on San Francisco Bay Mud.
95
GGGfi
O'!*+-3(U'O
U·r—·CD·r··r—-EOm.!
LOLC4-*O CUOC.) C •m •r— GJ*-" .1 C•— FUSB 5•r¤•(¤
+-3+-* -••—l/) O.L
/ L.) U3/ E|/F/ cu/// .s
_ / CDO' , NE E/ c:» FF
/ D 0Ad./ " - E/ :1 -•->
mO /(U/ Fcf fi
G LCD LO UC' •r·GJ '~+—r—C Of'*'|*%—' 5"’ I3
U3fs(J)C‘O
L.)
LOLO
6-'F"G L!.
FQ
OO Q _ G LO <\! GC\I C\| C\! r--
•*
3 ‘011va uLOA
96
where P, the critical void ratio at unit pressure, is read from Fig.
5.6. .Finally, the elastic shear modulus G is introduced into the
model through the ratio G/SU. Taking an average of four undrained
tests, Table 4.5, G/SU = 250 is adopted in this work. Expressions for
undrained strength SU are derived in the following section.
Undrained Strength in Cam Clay:
Failure occurs when the loading path meets the critical state
line, whereupon shear stress takes on a value,
qf = M p} (5.6)
i.e.,s—'M· (67)u ” 2” pf °
But p} can be computed from the equation of the critical state line in
e-log p' plane, since the void ratio eo remains unchanged during un-
drained deformations; i.e.
eo = P - A tn pé (5.8)
or p% = exp [(P - eo)/A] (5.9)
Therefore,
SU = %—M exp [(P - eo)/A] (5.10)
Eq. (5.l0) can be expressed as,
w + gl tn S = constant (5.ll)S uAnd thus, the Cam Clay model envisions a straight line relationship be-
tween the water content w and the log of shear strength. Now, it also
97
can be shown that Eq. (5.10) a1so imp1ies that SU/p' = constant.
The void ratio e at any stage of the test (drained or undrained),
can be determined from the expressions,pl
e = P - K tn p' - (A — K) tn (5.12)
where pé is the preconso1idation pressure. Substituting eo from Eq.
(5.12) into Eq. (5.10) 1eads toK 2:5.. p' Ä
Su (5.13)
For (a) norma11y conso1idated c1ayspé = pg (5.14)
therefore,ZA-K
1 ASU Z (ä] M pg (6.16)
2Ä-KS AOK .9 Z M (5.16)po nc
for Bay Mud, K = .052 and A = .404, and Eq. (5.16) yie1ds,
SU 1—T = .35 for M = 1.27po
= .33 for M = 1.20
An average of .34 is in accord with the experimenta1 findings.
98
For (b) overconso1idated c1ays
pl—$-= OCRpo
(5.17)i.e. pé = pg · OCR
Substituting in Eq. (5.13), we get
2A-K _Su V]-? ¥7 = — M (0612) 5.18po 2 2 ( )5 S Äii.e. [B?} = [E2} (OCR) Ä (5.19)
O oc O nc
= ,34 (0CR)°87 for Bay Mud.
A PY‘€C1$€ match of = -4O fOY‘ OCR = 1.2 between the Eq. (5.19) and
the experiment, Tab1e 4.3, may be considered fortuituous. However, an
OCR of 1.5 yie1ds SU/pé = .48 which is in excess to the observed va1ue
of .44 for Bay Mud, Tab1e 4.3.
5.4 VERIFICATION OF CONSTITUTIVE MODELConstitutive mode1s represent, to a varying degree, fitting of
the experimenta1 curves. However, the Cam C1ay mode1 requires on1y
five soi1 parameters and then generates behavior beyond that which is
given. In what fo11ows, we wi11 test if the mode1 and the five soi1
constants are ab1e to generate the desired constitutive response of
the materia1. Cam C1ay stress-strain re1ations are given in Chapter 3
and in Appendix 1.
99
(a) Undrained Tests:-
Stress-strain-pore pressure curves under triaxial conditions were
computed for Bay Mud under consolidation pressures equal to those of
tests CU-2 and CU-3. Predicted and experimental curves are shown in
Figs. 5.7 and 5.8. The model shows a slightly stiffer response.
That the Modified Cam Clay model provides predictions of too
stiff a behavior in undrained shear has been a general observation pre-
viously. The problem is in part attributed to the assumed elliptical
shape of the yield surface, which does not produce enough shear strains
at the low values of shear stress. Alternatives lie in either formu-
lating a composite surface of different arc types or resorting to the
original bullet-shaped Cam Clay surface. Both of these proposals were
investigated and it was found that even though an improved undrained
response could be obtained, the modifications created additional limita-
tions. Thus, the slight error in response that exists with the
elliptical surface has been accepted.
A crucial aspect of the undrained modeling is the generation of
excess pore pressures due to shear. In light of the previous discus-
sion, a question can be raised: "Is there a mechanism in the Cam Clay
model which generates pore pressures due to shear? If so, is it
adequate?" Fortunately the model gives reasonable prediction of pore
pressures for both CU tests, as seen in Figs. 5.7 and 5.8, providing
evidence of the model's capability to represent the phenomenon well.
100
Mode150 •
'• _—•
, Experiment40 _
C\lE\E 30 •
¤'_ 20 °QB I
10,
Test CU-2 A00 2 4 6 8ex %
50 _ -1 ·• MOd€·|
40 _ Experiment
NE 30I: O¤°’ 20,
10 1*
00 2 4 6 6Ex
Fig. 5.7 Verification of modified Cam C1ay Mode1 for undrained tri-axia1 test.
4 6 101
00Experimen
60 _ · °° Modelc\| O
E\ •
^ 40 °6* II O
20 60Test CU—3
0 2 4 6 00 ex %
00
60 ,· °
..6 /\ /fc 40 •
CU •Z)
202 0 2 4 6 0
c %X
Fig. 5.8 Verification for the undrained triaxial test.
102
(b) Drained Tests:
Response of the mode1 under drained triaxia1 conditions was
computed for 1aboratory tests CD-1 and CD-3. The stress-strain-vo1ume
change response is shown in Figs. 5.9 and 5.10, where a comparison is
made between the predicted and experimenta1 curves. The match is very
good particu1ar1y considering the fact that the same critica1 state
parameters are used as for the preceeding undrained tests.
5.5 CONCLUDING REMARKS
The chapter began by considering an assemb1age of e1asto-p1astic
e1ements p1aced either in series or para11e1. It is possib1e to
construct e1aborate formu1ations of the constitutive equations by in-
corporating a 1arge number of e1ements in the assemb1age. Such mode1s
are known as the "composite structures mode1s" as against the "inte-
grated hardening variab1e modu1i mode1s" 1ike the one used in this work.
In the 1atter category comes the mu1ti-surface mode1s of Mroz (1967)
and Prevost (1977), and the strain hardening mode1s of the estab1ished
theory of p1asticity such as the Cam C1ay mode1. Appendix 3 presents
a comparison between the two categories demonstrating that integrated
hardening is a superior formu1ation.
It was the purpose of this chapter to show that the Cam C1ay
mode1 represents, to a sufficient accuracy, the drained and the un-
drained behavior of Bay Mud with a sing1e set of materia1 constants.
And therefore, it can be assumed that the same mode1 and the same
materia1 constants wou1d be suitab1e for Bay Mud under the inter-
mediate conditions with partiai drainage.
103
160 •
140
120 ,mE •
. ;;10O Experiment34’?6* 9 80
I IUf"
O
40
20 L Test co-10
0 8 16 24 32 40 48 ex %0
4 •
g 8 ••>Q OQ I12 ,
16
Fig. 5.9 Verification of the modified Cam C1ay mode1 for the drainedtriaxia1 test.
104
280 _, °O//•
240 ./•/
NE 200 _‘
E5. Experiment
„_160 °(Y) I
O O
'_120 ·‘
O •$.4 •’ I80 ,
E 40 _'I Test CD-3
00 8 16 24 32 40 48 ex %
0
463°• Experiment
> 8 °\ •> •< O
12 Mode1 ° .16 ° • _
Fig. 5.10 Verification for the drained triaxia1 test.
UCHAPTER 6
MODELING OF FIELD PRESSUREMETER TESTS:
AXISYMMETRIC-PLANE STRAIN ANALYSIS
The first step in the finite element simulation work was directed
towards establishing that the modeling procedures could reasonably
duplicate the observed field behavior in a pressuremeter test. For
this purpose, finite element analyses were performed for conditions
modeling those of ll of the pressuremeter tests of Denby (l978) and
Benoit (l983) in San Francisco Bay Mud. The tests cover a range of
depths from l6.4 to 49.l ft. (5 to l5 m) and allow a thorough test of
the modeling procedures (see Table 6.l).
A basic assumption of all pressuremeter theories is that the soil
surrounding the probe deforms following the condition of axial sym-
metry and plane strain. Using this assumption the expense of conduct-
ing a finite element analysis can be greatly reduced by modifying an
axisymmetric FE formulation for the additional condition of plane
strain. In such a case, ez = O, and therefore, the stress-strain rela-tion reduces to the form
6; EPl{62 = [pgp]
{ 0 (6.l)0, 3x3 é(00
The finite element formulation is obtained by integrating a one-
dimensional element axisymmetrically around the cavity-—in the hori-A
zontal plane. In this work,-a total of 76 two noded elements, L2P2,
lO5
34-*·4<•-I O O O FZ (\l Ü I- O O (\] Gl(AU • • • • • • • • • • •
I Q I- I-- I- r- IZ IZ IZ I- I- YZ IZC
•--·I
L-C Z·I— G)!-I Ä O C\I Ä I-fü·4—*Z Pi FZ O IZ I-L I'¤\ I I I I I I • • • • •4-*QCä O O O O Ogf) O
C z·sO Z·I- •—• LD LO LO O LD4-* GJX O O O O O O O O Ü O O;¤.p\ I I I I I I I I I I IIi fü!-• I- IZ I- IZ IZ fi O O I" O O*4-ZV)C QI—• O
4-*E 'U - ·LIJ G) 3LI. 4-* Ä Ä Ä O IZ LO I- Ä Ä IZ IZ
fü Ü Ü Ü LO LO LO LO Ü Ü LO O 3Ä E O • • • • • • • • • • n O
” 'O U) .OGJ l.I.I4-* *4-fü UIZ Q
3E *4- Ü·I— Q C7I;) I LO LO LO Ü I\>|\ Ü Ü Ä (*7 O I5
Ch ON OW O G G O O O I- I'- ·I— ·•'~U) 4-) gz I I I I I I I I I I I g4,.) Q3- I- FZ IZ C\I C\l C\I (\| (\| Gi GI I" L -I-*
. In GJ O · _GJ Q *4- 3I- ·I-‘ IZ GJL fü >GJ O O4-* *4·- .QGJ O " füE J.)- 0*7 I- OO LO LO G O OO (*7 Ü 07 QG, I I I I I I I I I I I I 4) 6L .I.> gp- LO LO LO O7 <\I LO LO LO N G IZ I5 II- I3 Q_}.- r'- I- I'- IZ > Q) rsIn GJ ru Q. G7In Q Z NGJ O C7L ^ IZQ__ Q) I- x./
U1-O O (\] Ä O -O fü IZI- • C7 IZ IZ IZ O 3 O " I"GJ 0 I I I I I Z IZ GJ·I- z I\ oo ow cx: <I· nc E Z Z Z Z GJ ·I— .:LL IZ I- I- mn cu cu < <E < < < >I LJ 0 0
4-* I I I I I I I I I I I fü X (./7 4-*Cn In M M M M M M I I I I I cn 0 ·I-C GJ Q Q Q Q Q Q CQ CQ O CQ LI- *4- E'I* I- I I I I I I 'O 'O O 'D 'O O OL ~ U L 'OO UT ·I- • E
CO ·I— <[ +-* füI LJ 3 '
*4- 4-* C C GJIZ I/7 ' fü O 4-* 4-*G) ·I— r'- L 4-* ·I— L(/) IZ LI. IZ C fü
3 fü O O ·I— O Q.-I->-I-I Ix ow s: E co
I- ·I— C C7 I"" fü fü C· L/7 GJ I- -/ L/7 I GJ O
LD I E -/ 4-* QQCu- 4-* fü
Q) IC; L ä•'— •• •• L E
I" GJ .O O IZ CD 3 O.O O. C C •I— 4-* -4-* Lfü X GJ GJ O 'I" fü LI-I—- I.I.1 ca co In cn m —I<
107
with a disp1acement and a pore pressure degree of freedom at each node
are used to mode1 the 11 fie1d pressuremeter tests. The boundary
conditions assumed in the ana1yses are shown in Fig. 6.1.
6.1 MODEL PARAMETERS
On1y two additiona1 constants are required when an e1asto-p1astic
formu1ation is extended to inc1ude the pore-pressure degree of freedom;
the permeabi1ity of soi1 and the unit weight of water. Permeabi1ities
in horizonta1 and vertica1 directions, kh and kv, were determined fromthe resu1ts of oedometer and fie1d pressuremeter tests. Indeed, it is
a difficu1t task to ascertain the horizonta1 permeabi1ity of a stratum.
Benoit (1983) reports permeabi1ity va1ues of 3.5x1O”7 to 7x1O”7 cm/sec
for Bay Mud from his pressuremeter ho1ding tests. Comparing these
va1ues with the Taboratory determined horizonta1 permeabi1ity of
7x1O”8 cm/sec, it was decided to assign a va1ue of 3.5x1O”7 cm/sec to
horizonta1 permeabi1ity of Bay Mud under1ying the Hami1ton Air Force
Base Station.
The soi1 parameters emp1oyed in the finite e1ement simu1ation of
the pressuremeter tests are given in Tab1e 6.2. The Cam C1ay
parameters are the same as those used in the mode1ing of triaxia1 tests
ear1ier, excepting for the va1ue of M which is taken here at 1.2,
instead of 1.27 used for the triaxia1 tests. The va1ue of M for the
ana1yses was backfigured from the fie1d tests by finite e1ement mode1ing.
A va1ue of 1.2 provided the best fit for the se1f-boring pressuremeter
tests in Bay Mud whi1e keeping the other soi1 parameters una1tered.
The reason for the discrepancy in M va1ues wou1d appear to be re1ated
4-*L/7GJ
4-*LI GJ
4-*GJIx EN GJ
L” 3O O an
UIII II GJ
L-3 Q Q
CU•. _;N 4-*N LI-GJ O'OO mC •r—
V74-* >s{ r-
TUCTUC•r-
. O TUaj L
LO 4-* ·CD m~I—>O CIi GJGJ:E
TUGJQ.€UI(.J.C•'—U
LTU+-*0)GJELEO
<'U7E ><C1JE TUU')
·r- 3LO I- (U
· .CGJTO (\| ' I O «I—*L
TUII Il L
OU)(\| W7 •r— LI-4-*
C '·4— CL > c--·•·7 L/IOG 'r· •v—O C E·I—*O•• Cu-
1- Q)·I—*Ea:
GJ GJLO GJGJC 4-*CDC
-4-* -I—*·r—··<E ‘·—
CO.,...3LI-}-
FC
LO
O7'I"LI.
l09
Table 6.2 Model Parameters for Analyses of the Pressuremeter Tests
SoilConstant Value Remarks
l< — .052 CS = .—I2A .404 CC = .93ecs 3.75 at l kN/m2 2.97 at l psi
M l.20 l.27 for triaxial comp. tests
G/Sv 250k 3.5xl0-7 cm/sec kh .D.„ 12kv 3xl0"8 cm/sec kv
llO
to the effect of mode of failure. In the pressuremeter test, the
failure takes place in a vertical plane, as seen in Fig. 2.3 (Chapter
2), and neither a triaxial compression test nor an extension test can
simulate such a pattern of failure.
6.2 COMPARISONS: FIELD VS. MODEL
Figs. 6.2 to 6.lO give comparisons of the analytical pressure-
meter curves with the field curves. Generally speaking the model pre-
dicts the pressuremeter curves very well. This is encouraging in view
of the fact that only one set of soil parameters is used. It also
should be recognized that the analyses suggest that some drainage
occurs in the tests, a subject which will be discussed later.
A comparison of stress paths followed by an element of soil at
the cavity interface is shown in Figs. 6.ll and 6.l2. while the model
is able to generate the basic trend of the total stress path a dif-
ference is noted in the comparisons of effective stress paths. For
instance, consider test HPR-l9, Fig. 6.lla, the effective stress path
predicted by the model compares well with that of the field until point
F, i.e., in the prefailure stage. But, in the postfailure stage, the
predicted path stays at point F while that of field continues right-
ward. No satisfactory explanation could thus far be found for the
discrepancy observed.Fig. 6.l3 demonstrates one of the most significant observations
of the analysis. It shows the variation of vertical stress in a soil
element at the cavity interface as the test proceeds. Contrary to what
would be expected from the theory of elasticity, the effective vertical
111
150
125
NE ' Mode1100o§ 75 „S.ä__ / Test Simu1ationrc ..3; 50 ' HPR—17
¤¤ Depth 5.3 M
25
‘0 2 4 6
Radia1 Strain er %
Fig. 6.2 Mode1ing of fie1d pressuremeter test.
112
125 _ M0de1
***; 100 .$ .Z..>¢. /0 -.„;‘ 75$L •
-1->an
E 60-g Test Simu1ati0n“ HPR-18Depth 6.1 M
25
00 1 2 3 4 5 6
Radia1 Strain, er %
Fig. 6.3 M0de1ing of fie1d pressuremeter test.
113
150
Model
C\|E\ .EL1006.529s.4->m
F6E&’
50l
Test SimulationHPR-19Depth 6.8 M
00 2 4 6
Radial Strain, er %
Fig. 6.4 Modeling of field pressuremeter test.
114
200
Fie1d
150
Exz:.4oJU)0.: ';—5,,100
F?Ero°‘ 0
50 ’
Test Simu1ati0nHPR-22Depth 9.5 M
Q .0 2 4 6
Radia1 Strain, Er%Fig.
6.5 Mode1ing of fie1d pressuremeter test.
. 115
300
250 °
Fie1d
200 °cwE .xz.
ro
$:150 V<us.4->an
FEcc°‘ 100
50 iest s1m01at10nHPR-24Depth 12.5 M
00 2 4 6 _
Radia1 Strain, er %
Fig. 6.6 M0de1ing of fie1d pressuremeter test.
116
350
000 _ Fie1d
250• _
E\ •Z .x
V O*200 VanCDL.-s->“° 150F6-5cu .
100
Test Simu1ation50 HPR-26Depth 15 M
00 2 4 6
Radia1 Strain, er %
Fig. 6.7 Mode1ing of fie1d pressuremeter test.
117
160
4 N 120/
> / •Z /
s.. / °° 80 _
Test Simu1ati0nU JB—HAM-10
400 2 4 6 8 10eY_%
60 ‘
JB-HAM-10
N AQMode1EE
_, ~ F1e1dcu ,
"° 2 ,/ .1//
/0 /0 2 4 6 8 10
Radia1 Strain, er %
Fig. 6.8 M0de1ing of_fie1d pressuremeter test.
118
200
mgl
Fie1dx2 ’¤ 120 /„; , /
‘Q /L
G; 80ÄE Test Simu1ati0n.16-HAM-12
40 0 2 4 6 8 10er %
80mExZAC
Q 60:
253 Mode13 40Lcx.¤J LL ,, «E L ·
’3 E //’ madcu /0 ° />( Z1.1.1
02 4 6 6 10
Radia1 Strain, EP %
Fig. 6.9 M0de1ing of fie1d pressuremeter test.
119
320Test Simulation •JB-HAM-30 · ·280 '6 Field ° ·• _ Model
240N • .EErfg •
L .° 200l'
~ ·
160"
120
80 Model 'mE ° °E 40
•Field
3•
00 2 4 6 8 10’ Radial Strain, Er %
1Fig. 6.10 Modeling of field pressuremeter test.
u
120
40N HPR—19
E\E Fwd TestN F "‘°°°'-_'°‘l•«I—\ ‘Ü••—"‘/_
h
Is.
3 M0de1 ESP0 0 20 40 60 80 100
(6;+ 66.;)/2 kN/m2
(6)
HPR—19 TSPN 40 I
EE1astic-P1astic _
wl SU __ .oc *1 .„ 20CS- Fie1d TestV M0de1
060 80 100 120 140 160
. (dr + d„>/2 kN/m2(b)
Fig.6.11 Effective and t0ta1 stress paths in the h0riz0nta1 p1ane:fie1d vs. m0de1.
121 ·
40JB-HAM-12
cwE\z..>¢fm 20
• “Fie1dÖs.
~· 2 Mode1ESP
00 20 40 60 80 100
(0]; + sé)/2 kN/m2
40
NE JB-HAM-12 O O •
^.„ 20 °¤I • _
.., Fie1dTSP
O 0 20 40 60 80 1000 — 0r 6 2
Fig. 6.12 Stress paths in the h0v·iz0nta1 p1ane: fie1d vs. m0de1.
122
160
140
120
~E\§ 100
N¤
80
60
cx:E\E 40— N¤
20 ·
0 0 2 4 6 6RADIAL STRAIN, er %
. Fig. 6.13 Variation of vertica1 stress.
123
stress 6; sharply decreases at the beginning of the test. This is due
in part to the build up of pore pressure due to shearing of the
soil--a phenomenon not manifested by elasticity formulations. The
failure occurs at approximately 2% radial strain whereupon G; assumes
nearly a constant value. That means after failure an increase in
vertical stress is directly transferred to the pore water and no further
generation of pore pressure occurs.
what is surprising is the observation that total vertical stress
62 also goes down as the test begins. It levels off between 0,5 to 1%
radial strain and increases thereafter. Note that at the moment of
failure, close to 2% radial strain, 62 is approximately the same as it
was prior to the commencing of test.
No field records exist to verify Fig. 6.5. However, laboratory
experiments support the analytical observations. As this research
work was underway, Arno Sinram at Stanford University performed cubical
tests on Bay Mud simulating the elastic-plastic pressuremeter loading
path. The results of one of the tests are given in Appendix 4. It can
be seen that the effective as well as total vertical stress decrease
rapidly in the prefailure stage of the test. Similar observations are
reported by wood and wroth (1977) from cubical tests on Spestone kaolin.
Fig. 6.14 demonstrates how the pressuremeter test alters the in-
situ state of stress in the vicinity of the cavity. while the effective
tangential and vertical stresses, 6é and 6;, reduce from their in-situvalues, the effective radial stress 6; rises. The alteration is maximumat the cavity-probe interface and it diminishes rapidly as we go away
from the cavity. No change in stress state is experienced by soil at a
·0
124
60
50
40NEI>ZZ‘j so · .m
. P-_
uv20
V ¤é· 10
0 1 2 3 4 5 10 20 50RADIAL DISTANCE, r/ao .
Fig.6.14 Variation of vertica1, radia1 and tangentia1 stresses inthe vicinity of the probe at 8% radia1 strain, as shownby the mode1. .
l25
distance of 50 times the cavity—radious from the center of the cavity,
suggesting that a finite element mesh of length 5OaO would have
sufficed for the analysis. The difference between the bell-shaped
curves, o; and og, represents twice the shear strength mobilized.
6.3 SIMULATION OF HOLDING TEST
Fig. 6.l5 shows the finite element simulation of the holding test
JB-HAM-30. The test was controlled by applying the loading increments
(or decrements) every half minute, as indeed was done in the field,
and thus, the radial stress vs. time curve was duplicated as closely as
possible. The resulting variation of the radial strain, er, with time
appears to be in good agreement with that recorded in the field.
The development of excess pore pressures occurs at a more rapid
pace in the model than that observed in the field. However, the
dissipation processes in the field and analytical model are in close
agreement. The slower development of pore pressures in the field may
be related to lag in the measuring systems due to non-saturation, a
common problem in the field and the laboratory. All in all, the
comparison is felt to support the analytical simulation procedures.
126
400l
300 °5 200Os. 100 o Fie1d
—— Model0
0 20 40 60 80 100Time (min)
12 o o o
S 8 O 1wa 4 O <> Field -4}" ModelO o
0 20 40 60 80 100Time (min)
188 O-O Fieldmg A-6 Model
80C\15E 60sm
4020
00 20 40 60 80 100
Time (min) 8
Fig. 6.l5 Simulation of pressuremeter holdingtest, JB—HAM-30. '
CHAPTER 7
EFFECT OF FINITE PRESSUREMETER LENGTH:
THE AXISYMMETRIC ANALYSIS
In the last chapter, a series of plane strain finite element ‘
analyses were performed and shown to yield good predictions for the
field tests. However, it is reasonable to ask if out¥of-plane effects
can influence the test, particularly the excess pore pressures since
it is possible that vertical as well as horizontal drainage can occur.
Thus, the work for this chapter involves performance of a series of
axisymmetric finite element analyses which consider possible drainage
and pore pressure dissipation effects. These are compared to the
plane strain appraoch to test its validity.
A judicious selection of the element type has to be made if
meaningful results are to be obtained from a FEM consolidation scheme.
Here, the eight noded element, Q8P4, with displacement degree of
freedom at 8 nodes and pore pressure degree of freedom at 4 corner nodes
was used. The variation of displacement along each side of the element
is assumed quadratic while the variation of pore pressure is linear.
Thus, the element is isoparametric for displacement and super-parametric
for pore pressure variables. The Q8P4 element, where the pore pressure
variation is of one degree lower to that of displacements, was found
favorable for two dimensional coupled consolidation problems (Johnston
and Clough, l983). Note that in the preceeding chapter, even though
we used identical shape functions to define the variation of
l27
l28
displacement and the pore pressure within each element, no numerical
difficulties were encountered as the problem was one dimensional
(Carter, Randolph and wroth, l979).
However, the use of a O8P8 or Q4P4 element in a coupled consoli-
dation problem results in a sinusoidal fluctuation of pore pressures
about some mean value which is usually the pore pressure variation
evinced by the O8P4 element. A large number of O4P4 elements have to
be used before a comparison between the three types of two dimensional
consolidation elements can be made; this is not considered here a dis-
advantage of the O4P4 element, given the fact that a quadratic element
requires much higher computational effort than of a linear element.
Fig. 7.l shows the finite element mesh used in the analysis of
the pressuremeter test. A total of l84 O8P4 elements carrying 6l3
nodes are used to define a soil medium which extends a minimum of 5OaOfrom the center of the pressuremeter. ao is the radius of the cavity.
Note that the top boundary of the mesh is not constrained against
vertical displacements for, should the medium consolidate, a vertical
constraint on the upper boundary would throw the entire half of the
mesh into a state of extension.
7.l SIMULATION OF FIELD TESTS
The predicted response for the test JB-HAM—l2 is shown in Fig.
7.2 where the stress-strain-pore pressure curves from the axisymmetric
analysis are compared with those of plane strain analysis and the field
tests. The results predicted for the midplane level of the axisym-
metric analyses are used in this comparison. These is no appreciable
129 _
• 584.3 IIIIIIIII6 IIIIIIIII2IIIIIIKKK2 526IIlll—111' 666
I I-1—_——
- ::=====22-IIKililje .. un-¤—111_ 333a ;·-,_un-nziiiV7
___Il11Äl—--•
ät-'___ II]————QIll-——
6IIIlIlI———6 666•¢O3 IIIIIIIIII. 856 IIIIIIIIIIVVao
—>-1 Ht---- 50aO ——-—————>-1
Fig. 7.1 Finite e1ement mesh for the axisymmetric ana1ysis.
130
200P1ane StrainAxisymmetric
160 _ (” Fie1d Test<\I
E I\f§ 120OL
°/80 · _
JB-HAM-12
400 2 4 6 6.
Er,80
P1ane Strain ° °00 Axisymmetric 2
N • •gg 40
Q) .D
, , ,,·’ Fie1d Test'
ZZ/
0 /0\1’ 2 4 6 8RADIAL STRAIN, er %
Fig. 7.2 Mode1ing of pvessuremeter test: Axisymmetric vs.axisymmetric—p1ane strain.
131
difference between the axisymmetric and p1ane strain ana1yses. The
predicted pore pressures in both cases are we11 above those observed
in the fie1d. This may be attributed to two possibi1ities. First, as
mentioned previous1y, the fie1d pore pressure measurements probab1y
exhibit a significant 1ag effect due to 1ack of perfect saturation.
Second, in the case of these ana1yses the 1aboratory determined va1ue
of horizonta1 permeabi1ity, 7x10”8 cm/sec, was used. The data gene—
rated in the preceeding chapter with a horizonta1 permeabi1ity of
3.5x10”7 cm/sec appears to provide a better overa11 fit to the fie1d
pressuremeter tests than that shown in this chapter.
Figs. 7.3 and 7.4 show the stress paths fo11owed by a soi1
e1ement on the wa11s of the cavity. No difference is noted between
the stress paths from the axisymmetric and p1ane strain ana1yses. A1so,
the variation of the tota1 and effective vertica1 stresses shows no
difference. The effective vertica1 stress og in the axisymmetric
ana1ysis dropped by a maximum of 27 kN/mz. Approximate1y the same
reduction is observed for the p1ane strain ana1ysis in the preceeding
chapter.
However, a difference is noted in the magnitude of pore pressures
at a distance of 10 radii from the cavity for the two ana1yses (Fig.
7.5). This may re1ate to the fact that the three-dimensiona1 drainage
in the axisymmetric ana1yses effective1y reduces pore pressures in this
vicinity but not near the probe where the pore pressure distribution is
more uniform.
To ensure the consistency of the resu1ts from the axisymmetric
ana1ysis, one additiona1 test was simu1ated. The test JB-HAM-30 was
132
30
NE 20 °\E . ESP°” 10
0 20 40 60 80 100GI + Ol V
30
Q] Q O O
E 20 ispz¥ •
O- 10
O 0 20 40 60 80 100(p - 00) kN/m2
Fig. 7.3 Stress paths f0110wed by a s0i1 e1ement on the cavity inter-face (unti1 8% strain).
(\Ir-·
O3
C\I (/7
ܤ_ •~
Q /%CO r— QD• O(\I«— +
Q. II N(/7 CI-- O
3 +V7 L
CO C\-Z
IIQ.
U')LQ)
+-7· (O CD
Q) EC (U
•¤- Lrx _] (UC\I I Q.
<\I · O‘ r·—· r- bd (/7I I/7
E n cu<[ L1 Z 4-:
I L-\
Ü If)E"D .-I D. GJ(/7 V7 U7
L) LI.! C Ü.• 3 „,.„
C • V7 LGN+-7\ D. (U
rt: 0 \L.)
1-··3 cx: *4-E \ 0·•—an \
anE4.: LIn \ GJQ) -I-7I., •
CC\Ixs
G (/7 G.C C
KO <I' AI G P_ (Ö I ·: Q Q.¤1 = L
CDL IIP(/7 Ö-
ÜI\
C7-r-
Ä LL
134
„ GlgO
O• z;• E
FUS fs• S; WT6
<: U ° E°|* 'Y"rc: s.. S-is- 4(1 ‘ EQ
OOO
anE >” 4—·2 L 4 J3Ä <>E • ‘
äil · 4:6 ° EE• < 4-*
OC)r¤ 'Or-•[ L E
an</3
ILCLC: Iä;• Q.ZE /Ä g(I)
IÜ Q)
OIä* ·+--„— ‘ 0
4-* I2 ä:: _V*.E 4 EJ) ° g4,; // ä2 • L’
‘ LO
@N
© G GGI- I\
ZW/11*1 {1 ‘3Hf1SS38d 3HOd SSEIOXH LL
l35 .
now selected since the test was located in the overconsolidated zone of
Bay Mud. The comparisons are shown in Fig. 7.6. Again, differences
between the axisymmetric and plane strain analyses are insignificant.
7.2 OBSERVATIONS IN THE VERTICAL PLANE ‘
Thus far we restricted our attention to the horizontal plane
passing through the mid-height of the expanding membrane, where the
feeler strain gages are situated. It is also of interest in the case
of the axisymmetric analyses to examine the deformation patterns and
the distribution of pore pressures along the entire length of the
pressuremeter.
Fig. 7.7 shows the displacement vectors for the test JBQHAM-l2
at 8% radial strain at the cavity wall. A magnified picture is pre-
sented in Fig. 7.8. It is seen that the soil at the mid-height of the
probe does deform in a plane strain mode. Hartman and Schemertman
(l975) reached a similar conclusion from a nonlinear elastic finite
element analysis on sands under fully drained conditions. No consoli-
dation elements were used in their analysis.
Fig. 7.9 shows the contours of percentage strength mobilized.
As evident, only an annulus with thickness ao has reached criticalstate at 8% radial strain on the face of the cavity.
A unique outcome of this analysis is given in Fig. 7.lO where
the contours of excess pore pressure are drawn. Two observations can
be made. First, the flow of water at the mid-height of the probe takes
place radially in the horizontal direction. This is encouraging inas-
much as such a condition is assumed to exist in the plane strain
136
320• Ä :
280 , ,_
NE ° «. P1ane Strain
E 240 _E Axisymmetric
D • [L
ZOO '° 6” JB-HAM-30
1600 2 4 6 8 10
120 //z’ ‘«»cÜ'
GJ P1ane Stra1n,»"5 80 /EE A OE / •
O40 / .
0 Z'•
0 2 4 6 8 10RADIAL STRAIN, Er %
Fig. 7.6 S1mu1at10n of f1e1d pressuremeter test.
137
50a
‘
0
gi. Ä•
° ° ' ' QU
ruO
Ä I Ä Z
. 0 5 101.....:...-I _Sca1e, mm
1(Ü1Sp1äC€1I’1€|'1t V€CtOY‘S) '
Fig. 7.7 Disp1acement vectors at 8% radia1 strain.
136 ·
0 6 IOI._._I..ZI
Node NO' ScaIe, mmg392 „ 393 394 395\ \ I 1
\ I I363 364 365 I 366I I 1I 1 I I334 335 336 337
1 1I 1 1 11..1.1 I I IZ<: I I I Igg 15 305 306 307 308E
I I I Ies I I IE I 1¤ I IZ I
ä 1 I 1 ‘L11 I
276 277 278 279I 1 I 1I I I247 248 249 I 250 1
/ / 1 1216 219 220 221At 1116 At 1.26·· At 2.6·· At 3.76··Interface from the
Interface
Fig. 7.8 DefIect1on patterns in so1I adjacent to cavity at 8% rad1aIstrain.
139
\\
1\\ 1111 19111 1
111 1‘ 1 ..q..¤1·· MW
I1 1,1) 1/1 I/'/
° I/
1 1/
0 5 101
1...lL.-J
Sca1e (inch)
1
1
Fig. 7.9 Contours of percentage shear strength mobi1ized at 8% rad1a1strain.
l40
\‘1I I{ 1
II J5 1,/ \ I I
1 /1 1 \ ·/ / I••§ ll; / [ ‘
\=<·’ 1I _I I I \~ I UE-
‘ s +1,j I?II °f 1 (kN/m2){ 1 I I I1 1 / 1
I
\ ai ,
__1 "’I \I \\ 1\\ /~-../
I IScale J-20. 5 .
Fig. 7.lO Contours of excess pore pressures at 8% radial strain.
l4l
analysis. Second, the negative pore pressures are generated at the two
ends of the probe. Negative pore pressures impart a stiffening effect
to the soil surrounding the probe. Pressuremeter theories neglecting
these effects, will lead to an overestimation of the shear strength of
soil. Fig. 7.ll indicates the precise distribution of the pore
pressures in the soil adjacent to the pressuremeter.
7.3 COST ANALYSIS: AXISYMMETRIC VS. PLANE—STRAIN
_ Table 7.l compares the effort and expense in the two analyses for
the test simulation JB-HAM-l2. Axisymmetric analysis requires a
storage of l5OO K and costs as much as 25 times the plane strain
analysis. A plane strain finite element program, if prepared
specifically for the pressuremeter analysis, could possibly be run on
an IBM PC with 256K memory.
142
’/- Mone 47; N¤d¢ 491 \g.3 .4-7.7 -5.1 -1.2 1.3 1.2
E1ement 145
•1§_3 Q Q Q Q Q- _3-10.3 -4.2 .9 1.3 .8
-25.9 • • • • • ,1-7.6 -.8 1.2 .7 .2"Z(7; Q Q Q Q Q .115.2 2.4 2.0 1.3.6'TQ Q Q Q Q Q Q .1I 29.6 9.1 2.6 1.2 .4 Ig • • • • • -,5
32.7 13.1 4.0 1.0 .2 ILA
1_
E ä1: ä vgg Z g l Q Q Q Q Q -.00‘ gg <__-E 33.3 14.9 5.6 .3 .003
I; E- E I><“" 'TLg Q Q Q _ Q Q Q.003I 31.5 12.5 3.8 1.0 .2 IQ .g; • • • • • -1
_ 27.7 3.4 2.4 1.2 .4°“? Igg • • • • Q -113.2 1.3 1.8 1.3.2:-28.3QQ Q Q Q Q ,3-8.1 -1.0 1.2 1.3 .8
-15.0 • • • • • _g-11.7 -4.5 .9 1.3 .9
E1ement: 46
..7. -5.3 Q.6 -5.3 -1.7 .3 .9 .5Mode 131 Node 143
Fig. 7.11 Excess pore pressures in the vicinity of the probe at 8%radia1 strain, kN/m2.
143
Tab1e 7.1 Cost Ana1ysis
Axisymmetric-Factor Considered Axisymmetric P1ane Strain
Length of storage vector 110,000 5100
Execution time on CPU 2 min 54 sec 3 sec
Execution cost $35.50 $1.35
Cost of print out (resu1tsprinted every 5 increments) $14.50 $2.50
· CHAPTER 8
DRAINAGE DURING SHEAR IN THE PRESSUREMETER TEST
To this point in the thesis the major question of the work,
namely, at what rate of loading does drainage occur in the pressure-
meter test, has not been addressed. The previous two chapters have
shown that: (l) The plane strain model of the pressuremeter problem
reasonably simulates the mid—plane response; and, (2) the pressure-
meter curve can be accurately predicted by the model although the
analytic pore pressures are generally higher than the values observed
in the field. In regard to the latter issue, we may postulate that
either the permeability used in the analyses is too low (thereby not
allowing enough drainage), or that there is a time lag effect in the
measurements. There is no way to know exactly which of these two
possibilities is correct, but some insights can be gained by further 4
analysis.
8.l EFFECT OF PERMEABILITY IN MODELING ACTUAL TESTS
One technique to check the effect of permeability is to determine
its influence in light of actual test results. For this purpose, tests
JB-HAM-lO and JB-HAM-l2 were analyzed. These were chosen since both
total pressure results and pore pressures were successfully monitored.
In the analyses, permeability is varied from the value obtained as a
horizontal permeability in the laboratory (7xlO”8 cm/sec) to 5 and 2O
times this value. In his field holding tests, Benoit (l983) found
that it was about 5 to 8 times the laboratory value.
I44
145
Predicted and observed resu1ts are given in Figs. 8.1 and 8.2.
On1y mixed success is obtained in being ab1e to match both the fie1d
pressuremeter curve and the excess pore pressure curves with the same
va1ue of permeabi1ity. In both cases the use of the highest permea-
bi1ity yie1ds the best agreement for the excess pore pressures, but _
this typica11y causes too soft a response in the fie1d pressuremeter
curve. A11 things considered, it seems that a permeabi1ity simi1ar to
that measured by Benoit in the fie1d in his ho1ding tests yie1d the
best overa11 resu1ts. Interesting1y, the ho1ding test interpretation
is based on the dissipation rate measured in the fie1d, and the finite
e1ement ana1yses of this test showed good agreement with this aspect
of the ho1ding tests. 7
8.2 GENERAL EFFECT OF VARIATION IN PERMEABILITY
For a systematic study hypothetica1 pressuremeter test simu1a—
tions were conducted for tenfo1d variations in permeabi1ity. Condi-
tions were assumed simi1ar to those for the fie1d tests at a depth of
24 ft (7.3 m). The resu1ts are shown in Figs. 8.3, 8.4 and 8.5, where
a pressuremeter curve, the excess pore pressure vs. radia1 strain and
the excess pore pressure vs. radia1 pressure are shown respective1y.
As seen in Fig. 8.5, near1y undrained conditions are attained
corresponding to a permeabi1ity of 1x1O-8 cm/sec, since the excess
pore pressure deve1opment curve is unchanged by a ten-fo1d decrease inE
permeabi1ity. For 1x1O”8 cm/sec permeabi1ity, the soi1 on the cavity
wa11 is found to fo11ow a stress path, Fig. 8.6, which is typica1 of
an undrained test. For higher permeabi1ities, the pressuremeter tests
146
200kh .
PredictedNE
§§ 160 20khVLg§ Faeid iesz
E!ä120E //ETest Simulation
80 JB—HAM—l2600 2 4 6 8 l0
RADIAL STRAIN, er X. _
8Q
_
"NE Predicted,khE60 -¢ ,„
O3
.51
m40 [ ' 'E2L/7 .g
{
O ‘„,’ Z 4 6 8 l0„ RADIAL STRAIN, Gr 3
Fig. 8.l Effect of varying permeability on pressuremetercurves for test JB-HAM-l2.
147
1604
.--2
cx: 1•
’ § Fie1d Test/_//• U I kh
• •·U/
U; /~ • • • 20khPredicted
*1 OO)uu ’ //_, 80 V •Q .<E°‘ •
40 0 2 4 6 8 „ 10RADIAL STRAIN, er %
Test Simu1ation _60 JB-HAM—10 ° Predicted, kh
NE
Ex„ cu:
Ew -,«·’an aLLJMcs.LuM
20m Fie1d TestEtiu.:
0 6 2 4 6 6 10RADIAL STRAIN, er %
Fig. 8.2 Effect of Variation in permeabi1ity for Test JB-HAM—10.
148
20O” kh, 1x10-8 cm/sec
1x10”7160 1x10-6
NEE 1x10°5x
s.8 120V?an1.1.1ocI-</1..1<Z; 80<EQt
40 Depth 7.3 MLoading Rate 1.45 psi/min
0 0 2 4 6 8RADIAL STRAIN, er %
Fig. 8.3 Effect of tenfo1d variation in permeabi1ity on the predicted|DY‘€SSUY‘€TT1€t€Y‘ CUTVGS .
149
100
Depth 7.3 MLoading Rate 1.45 psi/min _
80 8cvE kh = 1x10- cm/se’
. •
G"Undra1ned"Z)„ 60 4
LL!CZDU)U)LuäM · 1x10‘62% 40CL •L/7U')LLJLJ><LIJ
•20 •
•
•
0 0 2 4 6 6RADIAL STRAIN, er %
Fig. 8.4 Effect of variation in permeabi1ity on excess pore ~pY‘€SSUY‘€S .
150
100 1: 1x10“éDepth 7.3MLoadingRate X
801.45 psi/min
GIE\ZgcECU 1
„; 60 1 • mo'6MD .(/7(AL1.1MoeLJ.1ää 40 _5¤- ° 1x10<./>E k cm/secQ h '>«Lz.1
20
80 120 160 200 240RADIAL STRESS, or kN/m2
Fig. 8.5 Diagram i11ustrating the effect of variation in permeabi1ityOU Q€T1€Y‘öt€d DOY‘€··pY‘€SSUY‘€S.
G<I'r-
C·r—E ZI./'I¤ rcs
GZ Q va c\1 >»-
r—· .@PO(\I|\ CU ,2 E Q)
E Z )¤am ‘¥ Ä:G)G C7 Q- LEC 0'O .:fü '—.;.>O rc——)Q.tf)U7G)L4-)<./7
(% ”UG)4-)U••—'OG)LQ.
E .Ggg 4-*
CO_ O
L) U • 3)(1) G) m .,-m In _ „... .\ • .E E ä EZ
G)G LO•
• E3I I ·„ 5__>gGGQ
fiQ.-X X - >'— '—
Cl Ü7füIl II.>„Q)J: -C ÄO L.:—¥ CC Q fü-I->—} cx: >
J) oJ 4-)4-)CUG)(UE¢!—G)
LO-r—LUG)
GRO0 0 0 G C .CO LO <!' (V cg
LU L) •Z /N)| Ü·v—' LL
152
experience a partia1 drainage for the imposed 1oading rate of 1.45
psi/min.An
interesting outcome for these ana1yses is shown in Fig. 8.7
where the subtangent method is app1ied to obtain a stress—strain
response pretending that no drainage occurred in any of the tests.
There is a significant reduction in the interpreted shear strength as
the permeabi1ity increases and drainage occurs in the test.
A simi1ar trend was obtained from the simu1ation of 5 of Denby's
tests for varying permeabi1ity, Fig. 8.8. The tests cover depths of
17.3 to 49.1 ft (5.3 to 15 m). The decrease in pressuremeter shear
strength for a tenfo1d increase in permeabi1ity is again 5 kPa for the
tests in norma11y conso1idated c1ay but is higher in the over-
conso1idated region. Note that the resu1ts shown in Fig. 8.8 are for
a fixed 1oading rate of 1 psi/min.
· 8.3 EFFECT OF RATE OF LOADING ·Fig. 8.9 shows the shear strength profi1e of Bay Mud as obtained
by Denby (1978) and Benoit (1983) from their se1f—boring pressuremeter
tests for the 1oading rate of 1 psi/min. Now the question is, how this
profi1e wou1d have changed had the tests been carried out at a dif-
ferent rate of 1oading? If the reported fie1d tests are indeed un-
drained, then increasing the rate of 1oading shou1d not affect the
computed shear strength since the mode1 does not account for rheo1ogic
or viscous effects. However, if the 1oading rate of 1 psi/min does
permit some drainage during the test, then an increase in the 1oading
rate wi11 1ead to a reduction in drainage, and consequent1y, the
153
II- M'cIC
.C4-*O)CQ)L4-*U)
LfüQ).Cm „LQ)
4-*Q)gg E• ' UQ) 3
U) U)
E QJU L
Q..C
X (D.C•~ 4-*
>-|-— C•—• O..I•*"• >sQ 4-*< *4*L1.] 1--*4-Z .¤ä EN
O•G
P1 Q,
Q.U)
LO C<r ·¤··•C >s1—·r·· S-
E FU
z +°Ü•\ >'|*rum '·4—
00 Z Q. ON U) 4-*
C L)_:·1··· (U
-4-> ·¤ *4-¤. cc *4-Q) O |-4JQ ..I
CO1 NI
‘ICU)
'|"L1.
CD G CD00 t\1 4—·
} ¢Zw/N*1 l H.L5NE1H.|.S HVEIHS
154 _
60. Test Simu1ated
gg HPR-26
aw§ 40 2432
9-[-• ‘ „
ä gg +_ 22ZE;
19 ·I-‘/7 +°‘ 17V1 .
10
0 10-8 10-7 10-6 10-5 . _PERMEABILITY, kh cm/sec.
Fig. 8.8 Effect of varying permeabi1ity on the pressuremeter shearstrength. ·
155
SHE.-XR STTEDIGTH ‘K.tI„·Si·" A _« __3 J I jo gg }O ll, Z; ZU
¥-TQZDZGALE ..1275}Field Jane/
11Z l .
„ä1
JUNCÄN' ::1€1d‘]äf1€ —•\
1:*E
TO 1
TWITCHELL -1NO ;.1NNE (1977)Field ‘/ane \-•
JENBY @2732 anc SENOIT ;Ü233ESé1F··3oring 2r·e,S3ur=£¤€§2r6--/
Fig. 8,9 Comparison of undrained shear strength obtained from these1f—boring pressuremeter tests and field vane tests(Benoit, 1983).
l56
pressuremeter will encounter a stiffer soil system. Hence, with the
increasing loading rate the strength profile should be expected to
move rightward until after at some loading rate no further shifting of
the profile would occur whereupon the undrained conditions would be
attained. And, this is the loading rate we wish to determine.
To do so analytically, each of five of Denby's tests, covering
depths of 5 to l5 meters, was first modelled at the field testing rate
of l psi/min. A constant value of horizontal permeability, 3.5xl0°7
cm/sec is used in modeling all five tests. The resulting strength
profile is shown in Fig. 8.lO where it is compared against the strength
profile obtained by the field tests. Note that the experimental
strength profile, as shown in Fig. 8.l0, is a best fit of over 30(
pressuremeter tests (and not just an outcome of 6 pressuremeter curves,
shown in Chapter 6, against which the validity of the model is
checked).
The analytical simulations were then run for different loading
rates. The resulting pressuremeter curves for the test simulation
HPR-l9 are shown in Fig. 8.ll. The generated pore pressures are shown
for the test HPR-22 in Figs. 8.l2 and 8.l3. The pore pressures are
seen to stabilize at the loading rate of 6 psi/min. (A slight increase
in the pore pressures continues due to the numerical approximations
involved.)
Fig. 8.l0 shows the shear strength profiles corresponding to dif-
ferent loading rates. The optimum loading rate for Bay Mud is deter-
mined as 6 psi/min. It may be noted that the shear strength at this
loading rate is approximately l5% higher than what has been reported
l57
Tf = kN/m20 TO 20 30 40 50
_ -7 cmkh — 3.5
XFIELD TEST(l psi/min)
LOADING RATE(psi/min)
5 · 1--HPR—l7 —-——- • ···· 24---HPR-l9 ——--· • · 6 8 8
2.é+>
·-—— —• „• A Q¤l0
--HPR-24 ————· * ^ ·
15 -HPR-26 ————· • ~ ·
Fig. 8.lO Effect of loading rate on the shear strength profile.
158
200
6 & 8160 4 1
1/2 psi/min€\lE\Z.>¢
,120O
MI·—V)
Q<ZM
40Depth 6.8 M
kh = 3 x10_7
cm/sec
0 0 2 4 6 6RADIAL STRAIN, er %
Fig. 8.11 Effect of 1oading hate on the pressuremeter curves. ·
159
80
Depth 9.6 M••
60 8 °6
O
N
•• 2 .E
E? 40D 0
•20 •
••
l •
0 *4:0 2 4 6
er %
Fig. 8.12 Effect of 10ading rate on generated pore pressures.
160
U)GJLI3U)U)CULQ.GJL
G OG Q.<\|
'UGkOC\I (U0) Q C 1;
•'1 L lE cuX C.,.. Q)U) C7Q. G
G C·-lm «— ALE o(Ü •
·
\ CDZ 4-*A6 fü
LLO U)
C•«—‘ G U
LO fü— (GQ Q •— O\_ r—·\ *4-
O_ 4-*
UGJZIil ÖIll C, Q)
e- CD.C
C 4-*O
4-* CFU Z ·•-·r- • 4-*3 RO 1 FUE · V C) L•v· O) C\1 4-*
C/) C\I •— U)(\1 .C 3
4-* I 4-) r-U) Df. CL •—Q) Q. Q) ·¤··I- I G O— -.: EI o ru
LO')G füO '•—r-- G
G G G GLO <' AI
U,l/ S 2N*1 {1 -Z ooO)•r-
LL.
l6l
for l psi/min. The finding, which is by no means a trivial one,
brings the Stanford pressuremeter tests within the general consensus
of the geotechnical engineering profession that the pressuremeter
yields a higher shear strength than that given by the vane shear test.
As evident from Fig. 8.ll, the rate of loading does not seem to
affect the initial slope, and hence the shear modulus, of the pressure-
meter curves. To precisely verify the fact, the pressuremeter curves
at different loading rates were fitted by cubic splines and the initial
slopes taken as twice the shear modulii. The results are shown in
Table 8.l. The difference in the values of G in the range of loading
rates, l to 6 psi/min, is insignificant. _(
_
8.4 EFFECT OF PERMEABILITY AND LOADING RATE
what is useful for the engineering practice is the Fig. 8.l4
where the test HPR—l9 is analyzed not only for varying permeability but
for different rate of loading as well. Corresponding to a chosen value
of permeability pressuremeter test simulations were made for different
loading rates. The dotted line corresponds to the undrained conditions.
It is seen in Fig. 8.l4 that for a permeability of lxlO-8 cm/sec, a
very slow loading rate shall be needed to cause any drainage and thus
the test may be considered undrained within the normal range of loading
rates in use today (i.e., l to 4 psi/min). On the other hand, for a
permeability of lxlO”5 cm/sec, it is not possible to run an undrained
test within any workable range of loading rates.
The conclusion must therefore be that under permissible range of
loading rates (l to 8 psi/min) undrained conditions can be achieved
162
Tab1e 8.1 Effect of rate of 1oadingon the shear modu1us derivedfrom the FEM pressuremetercurves
Rate of GLoading psi .
.01 psi/min 437
.2 psi/min 4471 psi/min 4692 psi/min 4834 psi/min 504
A
16 psi/min 518
U'!IG FU0:: ,,
-1-maCi .I.-1--1-
I_
E8“‘ '·*
Q O .40 IG -2.J I., Q)
E
O.
_ C7C
-1->, „S.U‘!FGO)
I >>
'
S.I. . s.:
·
OUI. ‘·!-S.
I gwU 4-*4-*GJ CNG)
[ SO U} CEI \ GJCU
I G E s.s.•— U 4-*3I L mm
U!I x s.q;
¢‘US—I.C' 3 "EII'
ä ä"I Ct <u·¤
!.1.! S-C1)I O. Zw-
I(f)1—
= äßäE ·7 ;·¤F G ·¤; r— QJOL .C.C.¤ 4-*4-*g cg-
O
I -CDC
I OOpg-r¤ I
·U .6
‘·äI 3 G4->I
c.¤· 3
I >¤ Q- Em[ FU CU YU
CQ Q Qq)I
1-:4-*
O3I .Q
C> O O@2 Ü:
Q LZw/N>! L HiE)!.!E!E!.LS HVHHS '—·-*<^<I'1-
O
®
O7*|*L1.. I _
l64’
only in soils with permeabilities less than lxlO”8 cm/sec, and that
some drainage will necessarily occur if the permeability is higher
thanlxlO”8
cm/sec.
8.5 RECOMMENDED PROCEDURE FOR CONDUCTING PRESSUREMETER TESTS _
Based on the observations made during this investigation, a
guideline for conducting self-boring pressuremeter tests is set up in
consultation with Dr. Jean Benoit of the University of New Hampshire.
The major question in the test is the estimation of minimum loading
rate for which the test shall be nearly undrained. Therefore, it is
recommended that before embarking on a comprehensive testing program,
a set of two holding tests at different depths should be carried out
and the permeability of the stratum be assessed.
The finite element analysis may then be used to derive a plot
similar to Fig. 8.l4 whereby a suitable loading rate can be determined.
It may be mentioned that only a rough estimate of Cam Clay parameters
is needed to reach at acceptable results, and such an estimate can be
made on the basis of index properties of the soil alone. Wood and
wroth (l978) present useful correlations for this purpose.
In addition, it is suggested that the field testing program p
(awfully expensive, as it normally is) be supported on a daily basis _
by a finite element analysis. At a cost of less than l% of the field
expense, such an analysis may provide a useful insight into the test
data obtained.
·CHAPTER 9
SUMMARY AND CONCLUSIONS
9.l SUMMARY
This dissertation presents a study of the self-boring pressure-
meter test. This instrument has been recognized by the geotechnical
engineering profession for having a high potential for use in in-situ
testing of soils. However, the appeal of the test has been limited, in
part, because of several unanswered questions attached with the test.
Primarily these questions concern the unknown drainage conditions en-
countered in the test, and the effects of disturbance in the soil during
the insertion of the probe. This study investigates the question of
drainage during the pressuremeter test. The investigation involves the
following:
l. Implementation of a numerical finite element model, in-
corporating nonlinear soil behavior with capability of
couple consolidation, to simulate the phenomenon en-
countered in the pressuremeter test.
2. A laboratory test program to evaluate the soil constants
needed in the model._4
3. Calibration of the finite element model using a series of
field pressuremeter tests performed in San Francisco Bay
Mud.
4. Use of the finite element model for parameteric studies
concerning drainage and pore pressure dissipations in the
T65
T l66
pressuremeter test.~
. For the laboratory test program, five tubes of undisturbed soil
samples of San Francisco Bay Mud were imported from the Hamilton Air
Force Base Station in Navato, California, where the self-boring
pressuremeter tests were conducted. The tube samples were tested in
the laboratory by the author in order to determine the stress-strain
and consolidation characteristics of the Bay Mud. A special effort was
made to evaluate the horizontal flow characteristics of Bay Mud in the
laboratory by isotropic and oedometric consolidation tests. It is
found that isotropic consolidation leads in general to lower estimates
of the coefficient of consolidation than estimates produced by the
oedometer test on vertical specimen. The coefficient of consolidation
given by the oedometer tests on vertical specimen is found to be 2 to
3 times higher than the values given by oedometer tests on horizontal
specimen. ·
A finite element program, PEPCO (Johnston and Clough, l980) was _
chosen for the investigation. The program was modified to utilize it
for modeling the pressuremeter test.
9.2 CONCLUSIONS
The investigation asserts that the finite element method incorpo-
rating Biot's theory of consolidation and the concepts of critical
state soil mechanics is capable of modeling, to an acceptable accuracy,
the phenomenon encountered in the pressuremeter test in soft clays. It
presents for the first time a comparison between the finite element
results and the field pressuremeter test in soft clays.
l67
It is shown that through finite element simulation it is possible
to backfigure from the pressuremeter curves a segment of soil constants
if the remaining soil constants are known. The finite element modeling
can also help assess the quality of the test data obtained.
It is proven through axisymmetric analyses of pressuremeter tests
that the soil at the mid-height of the probe indeed deforms in a mode
of vertical plane strain, and that drainage, if any, takes places pre-
dominantly in the horizontal direction. Therefore, it is valid to
investigate the pressuremeter test utilizing a simple axisymmetric-
plane strain element. The finding is supported by comparisons between
the results of the axisymmetric and the axisymmetric—plane strain
analyses.
The study provides an understanding of drainage phenomena in the
pressuremeter test. It is found that the computed shear strength may
significantly deviate from its undrained value if partial drainage
occurs during the test but the pressuremeter data is analyzed assuming
the test to have been undrained. It is discovered that only when the
soil has a permeability of lxl0'8 cm/sec, or lower, it is possible to
run a perfectly undrained test. In soils with permeabilities higher
than lxl0-8 cm/sec, some drainage will necessarily occur during the
test for a feasible range of loading rates (l to 8 psi/min). For ’
permeabilities of the order of lxl0”5 cm/sec, undrained conditions
cannot be achieved for any practable range of loading rates in the
pressuremeter test.U
A comparison of the laboratory—determined values of the hori-
zontal coefficient of consolidation with those of the pressuremeter
l68
holding test provides a limited evidence for accepting the latter as a
promising partner in field investigation of soils. It is found that a
value of horizontal permeability, 5 times that of vertical permeability,
reported by Benoit (l983) from his holding tests, best simulates the
field pressuremeter tests.
Finally, it is shown that, by simulating the pressuremeter test
by a numerical model, it is possible to determine the required loading
rate at which the test should be conducted so the soil deforms under
nearly undrained condition. The proposed test procedure for conducting
the pressuremeter tests in the field is believed to be a step forward
in the process of standardizing the self-boring pressuremeter test.
Added understanding of the mechanics of the pressuremeter test and re-
finements provided by the proposed testing procedure will lead to
consistancy in the results, and therefore, will give the profession an
increased confidence in this novel testing device.
REFERENCES
Amar, S., Bague1in, F., Jezeque1, J. F. and Le Mahaute, A. (1975)."In Situ Shear Resistance of C1ays." Proc. ASCE Spec. Conf. onIn Situ Measurement of Soi1 Properties, Ra1eigh, Vo1. I, pp. 22-45. ·
Atkinson, J. H. and Bransby, P. L. (1978). The Mechanics of Soi1s:An Introduction to Critica1 State Soi1 Mechanics, McGraw-Hi11,London.
Bachus, R. C., C1ough, G. N., Sitar, N., Shafii—Rad, N., Crosby, J.,and Kabo1i, P. (1981). "Behavior of weak1y Cemented Soi1S1opes under Static and Seismic Loading Conditions, Vo1. II,"Report No. 52, The John A. B1ume Earthquake Engineering Center,Stanford University.
Bague1in, F., Jezeque1, J-F., LeMee, E., and Le Mehaute, A. (1972)."Expansion of Cy1indrica1 Probes in Cohesive Soi1s." Journa1 ofthe Soi1 Mechanics and Foundations Division, ASCE, Vo1. 98, No.SM11„ PD. 1129-1142.
Bague1in, F., Jezeque1, J. F., and Le Mehaute, A. (1974). "Se1f-Boring P1acement Method of Soi1 Characteristics Measurements."Proc. ASCE Spec. Conf. on Subsurface Exp1oration for UndergroundExcavation and Heavy Construction, Henniker, New Hampshire,pp. 312-332.
Bague1in, F., Jezeque1, J-F., Shie1ds, D. H. (1978). The Pressure-meter and Foundation Engineering. Trans Tech Pub1ications,C1austha1, Germany.
Banerjee, P. K. and Stipho, A. S. (1978). "Associated and Non-associated Constitutive Re1ations for Undrained Behavior ofIsotropic Soft C1ays." Internationa1 Journa1 for Numerica1 andAna1ytica1 Methods in Geomechanics, Vo1. 2, pp. 35-56.
Benoit, J. (1983). "Ana1ysis of Se1f—boring Pressuremeter Tests inSoft C1ay." Ph.D. Thesis, Stanford University.
Besse1ing, J. F. (1953). "A Theory of P1astic F1ow for AnisotropicHardening in P1astic Deformation of an Initia11y IsotropicMateria1." Nat. Aero. Res. Inst., Amsterdam, Report $-410.
Biot, M. A. (1941). "Genera1 Theory of Three-Dimensiona1 Conso1ida-tion." Journa1 of App1ied Physics, Vo1. 12, pp. 135-164.
169
170
Bishop, A. N. and Henke1, D. J. (1962). The Measurement of Soi1Properties in the Triaxia1 Test, 2nd Ed. Adward Arno1d, London.
Bishop, R. F., Hi11, R., and Mott, N. F. (1945). "The Theory ofIndentation and Hardness Tests." Proceedings of the Physica1Society, Vo1. 57, p. 47.
Bonaparte, R. and Mitche11, J. K. (1979). "The Properties of SanFrancisco Bay Mud at Hami1ton Air Force Base, Ca1ifornia."Department of Civi1 Engineering, University of Ca1ifornia,Berke1ey.
Booker, J. R. and Sma11, J. C. (1975). "An Investigation of theStabi1ity of Numerica1 So1utions of Biot's Equations ofConso1idation." Internationa1 Journa1 of So1ids and Structures,Vo1. 2, pp. 907-917.
Bur1and, J. B. (1971). "A Method of Estimating the Pore Pressuresand Disp1acements Beneath Embankments on Soft, Natura1 C1ayDeposits." Stress-Strain Behavior of Soi1s, Proc. RoscoeMemoria1 Symposium, Cambridge University.
Ca11adine, C. R. (1963). "The Yie1ding of C1ay." Geotechnique, Vo1.13, pp. 250-255.
Carter, J. P., Booker, J. R. and Sma11, J. C. (1979). "The Ana1ysisof Finite E1asto-P1astic Conso1idation." Internationa1 Journa1for Numerica1 and Ana1ytica1 Methods in Geomechanics, Vo1. 3,pp. 107-129.
Carter, J. P., Rando1ph, M. F. and wroth, C. P. (1979). "Stress andPore Pressure Changes in C1ay During and After the Expansion of aCy1indrica1 Cavity." Internationa1 Journa1 for Numerica1 andAna1ytica1 Methods in Geomechanics, Vo1. 3, pp. 305-322.
C1arke, B. G., Carter, J. P. and wroth, C. P. (1979). "In-SituDetermination of the Conso1idation Characteristics of SaturatedC1ays." Proceedings of the Seventh European Conference on Soi1Mechanics and Foundation Engineering, Brighton, Vo1, 2, pp. 207-213.
C1ough, G. W. and Benoit, J. (In Press). "Use of the Se1f-BoringPressuremeter in Soft C1ays." Chapter 5 in Deve1opments inGeotechnica1 Engineering, E1sevier Pub. Co.
C1ough, G. N. and Denby, G. M. (1980). "Se1f-Boring PressuremeterStudy on San Francisco Bay Mud." Journa1 of the Geotechnica1Engineering Division, ASCE, Vo1. 106, No. GT1, pp. 45-63.
171
C1ough, G. N. and Si1ver, M. L. (1983). "A Report of the workshopon Research Needs in Experimenta1 Soi1 Engineering." VirginiaPo1ytechnic Institute and State University, B1acksburg, Virginia.
Denby, G. M. (1978). "Se1f—Boring Pressuremeter Study of the SanFrancisco Bay Mud." Ph.D. Thesis, Stanford University,Ca1ifornia.
Denby, G. M. and C1ough, G. W. (1980). "Se1f—Boring PressuremeterTests in C1ay." Journa1 of the Geotechnica1 Engineering Division,ASCE, Vo1. 106, No. GT12, pp. 1369-1387.
Drucker, D. C., Gibson, R. E. and Henke1, D. J. (1957). "Soi1Mechanics and work-Hardening Theories of P1asticity." Trans.ASCE, Vo1. 122, pp. 338-346.
Drucker, D. C. and Pa1gen, L. (1981). "0n Stress—Strain Re1ationsSuitab1e for Cyc1ic and 0ther Loading." Journa1 of App1iedMechanics, ASME, Vo1. 48, pp. 479-485.
Duncan, J. M. (1965). "Effect of Anisotropy and Reorientation ofPrincipa1 Stresses on the Shear Strength of Saturated C1ays."Ph.D. Thesis, University of Ca1ifornia, Berke1ey.
Duncan, J. M. and Buchignani, A. L. (1976). "An Engineering Manua1for Sett1ement Studies." Department of Civi1 Engineering,University of Ca1ifornia, Berke1ey.
Duncan, J. M., Byrne, P., Wong, K. S. and Mabry, P. (1980). "Strength,Stress-Strain and Bu1k Modu1us Parameters for Finite E1ementAna1ysis of Stresses and Movements in Soi1 Masses." Report No.UCB/GT/80-01, University of Ca1ifornia, Berke1ey, 77 pages.
Duncan, J. M. and Chang, C. Y. (1970). "Non1inear Ana1ysis of StressStrain in Soi1s." Journa1 of the Soi1 Mechanics and FoundationEngineering Division, ASCE, Vo1. 96, No. SM5, pp. 1625-1653.
Duncan, J. M. and Seed, H. B. (1967). "Corrections for Strength TestData." Journa1 of the Soi1 Mechanics and Foundations Division,ASCE, Vo1. 93, pp. 121-137.
Gibson, R. E. and Anderson, W. F. (1961). "In—Situ Measurement ofSoi1 Properties with the Pressuremeter." Civi1 Engineering andPub1ic works Review, Vo1. 56, pp. 615-618.
Hartman, J. P. and Schmertmann, J. H. (1975). "Finite E1ement Studyof the E1astic Phase of Pressuremeter Tests." Proceedings ofthe Specia1ty Conference on the In-Situ Measurement of Soi1Properties, Ra1eigh, North Caro1ina, ASCE, Vo1. I, pp. 190-207.
172
Iwan, N. D. (1967). "0n a C1ass of Mode1s for the Yie1ding Behaviorof Continuous and Composite Systems." J. App1. Mech., Vo1. 34,No. 3, PP. 612-617.
Jain, S. K. (1979). "Some Fundamenta1 Aspects of the Norma1ity Ru1eand Their Ro1e in Deriving Constitutive Laws of Soi1s." M.S.Thesis, Virginia Po1ytechnic Institute and State University,B1acksburg, Virginia.
Johnston, P. R. and C1ough, G. W. (1983). "Deve1opment of a DesignTechno1ogy for Ground Support for Tunne1s in Soi1 —— Vo1. I:Time Dependent Response Due to Conso1idation in C1ays."Stanford University Report to the U.S. Department of Transporta-tion, Rep. No. DOT-TSC—UMTA-82-54, 1.
Ladanyi, B. (1972). "In-Situ Determination of Undrained Stress StrainBehavior of Sensitive C1ays with the Pressuremeter." CanadianGeotechnica1 Journa1, Vo1. 9, pp. 313-319. _
Ladd, C. C., Foott, R., Ishihara, K., Sch1osser, F. and Pou1os, H. G.(1977). "Stress - Deformation and Strength Characteristics."Proceedings of the Ninth Internationa1 Conference on Soi1Mechanics and Foundation Design, Tokyo, Vo1. 2, pp. 421-494.
Ladd, C. C. and Foott, R. (1974). "New Design Procedure forStabi1ity of Soft C1ays." Journa1 of the Geotechnica1 EngineeringDivision, ASCE, Vo1. 100, GT7, pp. 763-786.
Ladd, C, C., Germaine, J., Ba1igh, M. and Lacasse, S. (1980)."Eva1uation of Se1f-Boring Pressuremeter Tests in Boston B1ueC1ay." Federai Highway Administration Interim Report, No.FHNA/RD—80/052.
Lade, P. V. and Duncan, J. M. (1973). "Cubica1 Triaxia1 Tests onCohesion1ess Soi1." Journa1 of the Soi1 Mechanics and Founda-tions Division, ASCE, Vo1. 99, pp. 793-812.
Menard, L. F. (1956). "An Apparatus for Measuring the Strength ofSoi1s in P1ace." M.S. Thesis, University of I11inois.
Mitche11, J. K. and Lunne, T. (1977). Unpub1ished fie1d vane shear _tests reported in Bonaparte and Mitche11 (1979).
Mroz, Z. (1967). "0n the Description of Anisotropic work-Hardening."J. Mech. Phys. So1ids, Vo1. 15, pp. 163-175.
Mr¤Z»Z-(1973,)-University of water1oo Press.
173
Pa1mer, A. C. (1972). “Undrained P1ane-Strain Expansion of aCy1indrica1 Cavity in C1ay: A Simp1e Interpretation of thePressuremeter Test." Geotechnique, Vo1. 22, No. 3, pp. 451-457.
Prevost, J. H. and Hoeg, K. (1975). "Ana1ysis of Pressuremeter inStrain-Softening Soi1." Journa1 of the Geotechnica1 EngineeringDivision, ASCE, Vo1. 101, No. GT 8, pp. 717-731.
Prevost, J. H. (1977). "Mathematica1 Mode11ing of Monotonic andCyc1ic Undrained C1ay Behavior." Int. J. Num. Ana1. Meth.Geomech., Vo1. 1, pp. 195-216.
Rando1ph, M. F. and wroth, C. P. (1979). "An Ana1ytica1 So1ution forthe Conso1idation Around a Driven Pi1e." Internationa1 Journa1for Numerica1 and Ana1ytica1 Methods in Geomechanics, Vo1. 3,pp. 217-229.
Roscoe, K. H. and Bur1and, J. B. (1968). "Dn the Genera1ized Stress-Strain Behavior of 'wet' C1ay." Engineering P1asticity,Cambridge University Press, pp. 535-609.
Roscoe, K. H. and Poorooshash, H. B. (1963). "A Theoretica1 andExperimenta1 Study of Strains in Triaxia1 Compression Tests onNorma11y Conso1idated C1ays." Geotechnique, Vo1. 13, pp. 12-38.
Roscoe, K. H. and Schofie1d, A. N. (1963). "Mechanica1 Behavior ofan Idea1ized wet C1ay." Proc. European Conference in Soi1Mechanics, wiesbaden, Vo1. 1, pp. 47-54.
Roscoe, K. H., Schofie1d, A. N. and Thurairajah. (1963). "Yie1dingof Soi1s in States wetter than Critica1." Geotechnique, Vo1. 13,pp. 211-240.
Roscoe, K. H., Schofie1d, A. N. and Nroth, C. P. (1958). "Dn theYie1ding of Soi1s." Geotechnique, Vo1. 8, pp. 22-53.
Sandhu, R. S. and wi1son, E. L. (1969). "Finite-E1ement Ana1ysis ofSeepage in E1astic Media." Journa1 of the Soi1 Mechanics andFoundations Division, ASCE, Vo1. 95, pp. 285-312.
Schiffmann, R. L., Chen, A. T. F. and Jordan, J. C. (1969). "AnAna1ysis of Conso1idation Theories." Journa1 of the Soi1Mechanics and Foundations Division, ASCE, Vo1. 95, pp. 285-312.
Schofie1d, A. N. and Nroth, C. P. (1968). Critica1 State Soi1Mechanics, McGraw-Hi11, London.
Sinram, A. (1984). "Cubica1 Shear Box Tests Designed to Simu1atePressuremeter Loadings." Thesis in Progress for the Partia1Fu1fi11ment of the Engineer Degree, Stanford University.
l74
Small, J. C., Booker, J. R. and Davis, E. H. (l976). "Elasto-Plastic Consolidation of Soil." International Journal of Solidsand Structures, Vol. l2, pp. 43l-448. I
Szavits-Nossan, A. (l982). "A Kinetmatic Model for Sands." Proc.Int. Conf. Constitutive Laws for Engineering Materials. Tucson,Arizona, pp. 285-288.
Tavenas, F. A., Blanchette, G., Leroveil, S., Roy, M. and LaRochelle, P.(l975). "Difficulties in the In-Situ Determination of KO in SoftDensitive Clays." Proceedings of the Specialty Conference on theIn-Situ Measurement of Soil Properties, Raleigh, North Carolina,ASCE, Vol. l, pp. 450-476.
Taylor, G. I. and Ouinney, H. (l932). "The Plastic Distortion ofMetals." Phil. Trans. Royal Society, Series A, Vol. 230, pp. 323-363.
Terzaghi, K. and Peck, R. B. (l967). Soil Mechanics in EngineeringPractice, Wiley, N.Y.
Tringale, P. T. (l978). Unpublished field vane shear tests, reportedin Bonaparte and Mitchell (1979).
White, G. N. (l95l). "Application of the Theory of Perfectly PlasticSolids to Stress Analysis of Stress Hardening Solids." Div. ofAppl. Maths., Brown University, T.R. 5l.
Wood, D. M. and Wroth, C. P. (l977). "Some Laboratory ExperimentsRelated to the Results of Pressuremeter Tests." Geotechnique,Vol. 27, No. 2, pp. l8l-20l.
Wroth, C. P. (l97l). "Some Aspects of the Elastic Behavior of Over-consolidated Clay." Stress-Strain Behavior of Soils, Proc.Roscoe Memorial Symposium, Cambridge University.
Wroth, C. P. (l975). "In—Situ Measurement of Initial Stresses andDeformation Characteristics." Proceedings of the SpecialtyConference on the In-Situ Measurement of Soil Properties,Raleigh, North Carolina, ASCE, Vol. 2, pp. l8l—230.
Wroth, C. P. and Hughes, J. M. O. (l973). "Undrained Plane StrainExpansion of a Cylindrical Cavity in Clay -- A Simple Interpre-tation of the Pressuremeter Test." Discussion, Geotechnique,Vol. 23, No. 2, pp. 284-287.
Wroth, C. P. and Hughes, J. M. O. (l972). "An Instrument for theIn-Situ Measurement of the Properties of Soft Clays." ReportCUED/C-SOILS TR l3, University of Cambridge, England.
175
wroth, C. P. and Hughes, J. M. O. (1973). "An Instrument for theIn-Situ Measurement of Properties of Soft C1ays." Proceedingsof the Eighth Internationa1 Conference on Soi1 Mechanics andFoundation Engineering, Moscow, U.S.S.R., Vo1. 1.2, pp. 487-494.
wroth, C. P. and Hughes, J. M. O. (1974). "Deve1opment of a Specia1Instrument for the In-Situ Measurement of the Strength andStiffness of Soi1s." Proc. ASCE Spec. Conf. on SubsurfaceExp1oration for Underground Excavation and Heavy Construction,Henniker, New Hampshire, pp. 295-311.
wroth, C. P. and wood, D. M. (1978). "The Corre1ation of IndexProperties with Some Basic Engineering Properties of Soi1s."Canadian Geotechnica1 Journa1, Vo1. 15, No. 2, pp. 137-145.
Zienkiewicz, O. C. (1977). The Finite E1ement Method, McGraw-Hi11,3rd Edition.
Zienkiewicz, O. C., Nayak, G. C. and Owen, D. R. J. (1972). "Compositeand Over1ay Mode1s in Numerica1 Ana1ysis of E1asto—P1asticContinua." Foundations of P1asticity. Ed. A. Sawczuk.Noordhoff Press, pp. 107-122.
Zienkiewicz, O. C., Norris, V. and Nay1or, D. J. (1978). "P1asticityand Viscop1asticity in Soi1-Mechanics with Specia1 Reference toCyc1ic Loading Prob1ems." Proc. Int. Conf. Finite E1ements inNon1inear So1ids and Structura1 Mechanics, Gei1o, Norway, pp.455-485.
TAPPENDIX 1
STRESS-STRAIN RELATIONS
It is instructive to derive the stress-strain re1ations in terms
of triaxia1 stress conditions wherein the effective stress state of
the soi1 specimen is defined by the invariants p' and q as|_.l
p — §—(6] + 263) (A1.1)
q = Gi - Gé (A1.2)
The state of strain is defined by the vo1umetric strain 65 and a
measure of octahedra1 shear strain 6, expressed as
6p=6 +26 (A1.3)v 1 3
=?-(6 -6) (1-11.4)E 2 1 2To formu1ate the p1astic stress-strain matrix assuming isotropic
hardening the essentia1 components needed are as fo11ows.
(i) The yie1d function:
F<p', <1„ pg) = 0 (A1-5)where pg is the materia1 hardening parameter.
(ii) The consistency condition:
zi · Qi E. ·z2
dF Sp, dp + gq dq + gpc dpo 0 (A1.6)
(iii) The norma1ity ru1e:
P z .äE. A1 7däv176
l77
p -E5de— w öq (A1.6)
where w, a scalar, governs the magnitude of the plastic deformations.
(iv) The work-hardening rule:
The rule is obtained by assuming that the strain hardening
properties of soil are governed by the permanent changes in volume as
indicated by the consolidation tests. The equation of the virgin
consolidation line gives the total change in void ratio,
de = — A ägl (Al.9)
whereas the equation of swelling line yields the elastic change in
void ratio,
dee = - K gg; (Al.l0)
Therefore, the plastic change in void ratio is,
dep de — dee
= - (A - K)
(A — ·<)QTLLHence,dg?] — ··
p,whereeo is the initial void ratio. This equation also holds when
p' = pg, I- dp'i.e.v O PO ·
which gives the change in hardening parameter as
dp, _ _ PQ, (‘* eo) dgp (Al.l4)
0_
(A - •<) V
l78
Eq. (Al.l4) expresses the hardening rule for the material. In order
to establish the stress-strain relation, substitute Eqs. (Al.l4) and
(Al.7) into Eq. (Al.6) to have
' (l + e )0F . aF aF po 0 aF _
Eq. (Al.l5) gives the value of 0 as,
-i Qi:. · Qi:.0 —A [Sp, dp + öq dq} (Al.]6)
' (l + e )_ pr po 0 g_where A — 3p« (A:|·:l7)
Substituting 0 in the normality rule, Eqs. (Al.7) and (Al.8), we have
p : i. QE.. • QE. äE..dev A [ap, dp + Sq dq] Sp, (Al.l8)
p = i. ÄE.. • EE. EE.de A [ap, dp + öq dq] Bq (Al.l9)
Rearranging, we obtain
P Qi. . i I QL . E ·ddv 1 pp' pp' :G¤¤' SG dp2Z Z. ——__——————l———————¢—
dp ^Qi,p;E..ps.s»i d(Apo)
B SG pp' : GG pq d
To these equations are added the elastic strain increments,
B a B 1/k 0 dp·pv = (Al.2l)aeB 0 1/26 aqwhere the bulk modulus K is obtained from Eq. (Al.l0)
„ (l + G) ',2.-9);.. ...Q.p. (Al.22)dee K
V
l79
Al.l UNDRAINED DEFORMATIONSThe total volume is conserved during the undrained deformation.
l.€. deV=de5+de$=O (ALZ3)
lL · L L Qi;-or A [Ap, dp + AQ dgl Ap, + K— 0 (Al.24)
L.3LQp;.-..e@'...¤s1.or dq A-+ AE- AE- (Al.25)K ap' ap'
which is the differential equation of effective stress paths followed by
a specimen during undrained deformations. Denoting the right hand
side by Aw, we have
dp' = - AW dq = - due (Al.26)
which indicates a generation of excess pore pressure due due to an
increase in shear stress dq. The scalar Aw, may be interpreted as a
built-in pore pressure parameter of the elasto—plastic model.
APPENDIX 2
TRIAXIAL TEST DATA FOR SAN FRANCISC0 BAY MUD
Table A2-]. Drained Triaxial Test
Effective EffectiveTest Mader Mine? .
Number Axiai Prgncipal Principal Vqlumetricand Initiai Strain ress Stress Strain
veiues e% ci (kN/mz) og (kN/mz)
zCD-]0 75.3 75.3 0.07 77.96 75.3 .08
Depth = 6.] m .]7 83.36 75.3 .]52 .34 89.73 75.3 .28
on = 75.3 kN/m .68 ]0].89 75.3 .53].22 ]03.66 75.3 .93
ei = 2.3] ].69 ]]2.29 75.3 ].282.54 ]]4.93 75.3 ].893.93 ]23.]7 75.3 2.845.08 ]29.25 75.3 3.646.33 ]35.43 75.3 4.457.6] ]4].7] 75.3 5.249.3] ]49.84 75.3 6.20
]].00 ]58.77 75.3 7.]3]2.69 ]65.93 75.3 7.98]6.58 ]82.]] . 75.3 9.74]9.63 ]93.78 75.3 ]0.9222.84 205.45 75.3 ]2.0626.84 2]9.57 75.3 ]3.5]30.]5 227.8] 75.3 ]4.4336.5] 238.]0 75.3 ]5.844].5] 232.7] 75.3 ]6.5]
]80
181
Tab1e A2—2. Drained Triaxia1 Test
Effective EffectiveTest . Pr?ää?ga1 Pr?62?ga1 V010metric
Qälggn Stress Stressz StäainVa1ues 6% ci (kN/mz) oä (kN/m ) Z T} %
CD-2 0 121.00 121.0 0Depth = 6.1 m .35 145.04 121.0 .18
O; = 12] kN/m2 .87 157.98 121.0 .501.75 170.14 121.0 1.06
ei = 2°02 2.62 179.36 121.0 1.583.50 187.70 121.0 2.095.25 202.60 121.0 3.097.00 216.53 121.0 4.038.75 230.46 121.0 4.94
15.75 277.13 121.0 7.9519.28 296.16 121.0 9.1922.74 311.56 121.0 10.2526.24 325.68 121.0 11.1630.06 337.94 121.0 12.0135.90 349.80 121.0 13.0439.10 349.02 121.0 13.5041.53 . 348.62 121.0 13.77
l82Table A2-3. Drained Triaxial TestEffective Effective
h
Näääär Axial PrTää?;al PrTlE$;al Volumetricand Initial Strain Stressz Stressz Stggin
Values 6% ci (kN/m ) dä (kN/m ) Z j7—%
CD-3 0 l48.57 l48.57 0
Depth = 6.l m .355 l80.93 l48.57 .l6
oé = l48.57 kN/mz .887 l96.92 l48.57 .48
ei = l.9l l.77 2l2.2l l48.57 .987 3.55 234.39 l48.57 l.97
7.l0 270.66 l48.57 3.84
l4.37 333.9l l48.57 6.99
l8.99 365.88 l48.57 8.54
23.60 389.62 l48.57 9.84
28.39 408.54 l48.57 l0.95
36.45 426.29 l48.57 l2.4l
38.69 427.08 l48.57 l2.70
40.93 425.74 l48.57 l2.9942.39 422.86 l48.57 l3.l4
l83
Table A2-4. Drained Triaxial Test
Effective EffectiveTest Maier met
and Initial Strain- 2 2 dvValues 6% 6i (kN/m ) ¤ä (kN/m ) 2*TT %
CD—4 0 55.20 55.2 0
Depth = l8.5 m .l66 62.86 55.2 .l39
eé = 55.2 kN/m2 .83l 76.88 55.2 .6l3
ei = 2.36 l.66 86.20 55.2 l.2l3.49 96.99 55.2 2.37
6.64 l09.l5 55.2 4.57
l0.00 l20.62 55.2 6.4
l7.28 l43.86 55.2 9.8
2l.93 l54.94 55.2 ll.42
26.8l l65.44 55.2 l3.03
3l.53 l72.0l 55.2 l4.23
39.02 l72.99 55.2 l5.5l
43.00 l67.20 55.2 l5.95
]84
Table A2-5. Undrained Triaxial Test
Effective EffectiveTest Major Minor Excess
. Principal Principal PoreNumber Axialand Initial Stnain Stressz Stressz Pressur;Values 6% oi (kN/m ) oé (kN/m ) u (kN/m )
CU-2 0 72.67 72.67 0
Depth = 6.55 m 0.5 79.22 56.33 ]6.34
og = 72.67 kN/m2 0.9 8].9] 49.]6 23.5]
].7 82.8] 38.6] 34.06
2.6 8].56 34.33 38.34
2.9 8].64 33.37 39.30
4.0 8].05 3].7] 40.96
5.5 79.43 29.58 43.09
6.4 79.36 29.58 43.09
7.2 78.05 28.54 44.]3
8.2 75.49 26.40 46.27
185
/t Vmin0 6 12 18 24 30 36 42 48 54
0
14 1
Bay MudTest CD-T
· pg = 20 kN/m22 .- 2Ap - 55.2 kN/m _
3
uu>¢1<¤ T
5
2 6
. 7 1 •It'100
8
Fig. A2.1 Isotropic c0ns01idati0n phase of the triaxia1 test. ‘
186
00 6 12 18 t 24 30 "“" 36 42 ·: 4
Ä” Bay Mud1 Test cp-6
_2
3
Stage II4
ou=_ 6<1 1 „
6
7 .
8Stage I
9
10
Fig. A2.2 Isotropic c0nso1idation phase of the triaxia1 test. p
187
80• • • _.•°
•
60 •O
(XIE •\z •A6
0
' !LOJ3 I -
20I
Bay Mud* Test CU-4OCR = 1.2
Oo 2 4 6 6AXIAL STRAIN, ex %
Fig. A2,3a Stress-strain curve from conso1idatedundrained triaxia1 test.
LO‘ 5
(\|'O r- C')3 •Z u r\>5 DSfü O
C\1P- E• U5 \U)
.¥
OOC')O
C7 + C\|
° Q. RO r- ·tf) O <i'IT. \—O—../ I
OU L)GJU 5 4-J
Q 3 ° U)• 'O CO GJO
·
|—|
·••4-*
LLO O• Q-LO
U).C4-*CU
· O.U')U')CUL
4-*V)
O .O• O")[7 O
NEU •
O')U) •¤—
..¥ LL
' ri-5·C\lO+ Gl—•—
OLo OO.;
OLO <!' C') C\.l r— O
lo)
. 189
100
80 .°
ax1E •\ ÖÜ •E’?h¤ I
' 405Bay Mud
ZOI Test CU-54 OCR = 1.5
0Ü 2 4 6 0 86 AX
.8 ·
< O5;.6 • l1-ä!< Igg • • Af = 0.68¤.U; .4gf OmEä
.o¤.
00 2 4 6 8
AXIAL STRAIN, 6X %
Fig. A2.4a Resu1ts of cons01idated-undrained triaxia1 test.
·GG••
LO¥•
II{IG3
. Gt\INE*2( Q-L/> GC\|}* SN
INU 00GJ ¤ ·
( U LnS ¤+ éq•-—• NO 4-)
<¤ $N CD‘ .C49
LS EN‘ anEG ¢'¤ü Q.
- cnU)CUJ L49U)
D• q-•O '__ •
t\l• <[• •• G U')
ON ·•··r-N U-/'\
-00O
¢ G+ON
'|*'
° o\-/
• .
EG G G G G G GLO LO ¢' (Y) C\.| r—
sn ‘Zw/Mx 2/( ¤ — ¤)
T9]
C\|
Z ILGC\I
IJ! I g/11
äI! I F E ::21/ ÜI1J 1 1;
A. 1 f-§’I / I 001
I $3I I SE1 I ·1 (I E
oo 0 <1- 0RQA;w A; A A —6 ‘011va 010A
192
©C\|
ZLO“° /'U LO an
3 GJE .C / v—4-* Q
>, Q. Efü CU fü
CQ 2 chIZ
füU'|*
· 4-*LGJ>
. 'UC
C\I fü/ EL) r-
VÖC7 44r—· X C
OZ -> NO O ·¤··;_> LI O
£CO
/ ·¤44/ "’C1)44
IZ
LG)/ +—>CU/ EO
'Ur-· CDO
Z/ G “—LJ O> C
OU7°!*LfüQEOU
<E
RO<\I<K
C7'|*
LL
fi
GO <r Q ao N GOQ1 C\] C\| r— V"
69 OLLVH (110/\
193
®C\|
Q- ®Z IT@
'U L.)I3 >Z
4-*>, Lnfü GJ
CO I-
LOGJ•—QEfüU7
v-. _r¤
U·«—4-*C\| L
E GJU >\U'! füat
C—> OO
4-*L/7GJ4-*
r- LGJ
4-*GJEO'UGJCD
qu.
O I\-> °C\1
>.4 / _/ U7/ ·•··L1.
I «——
oo <1- 0 no N GQQ; (\| C\1 r—
r····
G ‘011vu 0101
194
ON
CDLO |"
E'U LJ3 >E 4-*A anr¤ GJ
CQ 1-
¤3E-EFUU7
C\] IlE ¢¤U UA ·v··U3 4-*.>c LGJ-> >O
CU4-*anGJ4-*L
_ GJEO‘UGJO
/c~-2/ N
¤ / <G .G'3hy K
oo c Q- U ¤Q <\{ O?Ni ‘“ N N - -9 °OI.LVH CIIO/\
195
Ccx:ul •
Uf"
2* / 3¤3 •0 0
f'*
Z / u:CJ LS / S/ 1 2
/ • 0 0 asI3
N_Z _ mE NO / U EU \ OI / ¤¤ .:• 0 • 0 ac
. -> g0 3/ .=~[ S
/ 2.-0 00 0 ¤— +—*
S9I ES
f cuO
/ ¤3[ -1->#1-
!• •
Oc• 0 -mE’ rc9-
I 6• •• • U
I <1:_ cw
AI1 <E .
| 1 S' E
OO <r CJ no cu coC\J C\.I (\| •·— •—
GC\l
O")
2°UC.)
3 IE -I-J>5 (/7fü CD
CQ }—
4-*L/7GJ4-}
LÄI GJE -•->
L) CUÄ EU') OAd 'U
GJ·> OO •—
füCO·r—
+-7Cr— GJ>COL)G-ä_ ""I
O cx.:·,,.._..—„L„.L• {
·•—L1.
e-co <r 0 no Q Q ·C\_| (\] (M r- r—·
C\.l
<f Or-
ZO
°ULJ3]:Z -4-)>~,L/7füGJ@l—
4-*L/7GJ4-*C\|E L
U GJO7CUx E
O‘ ·>UO G)
Or-füCO
C•— GJ>COL)IT
I""
AI<E
OWLI.
OO <I'_ O ROC\1C\1C\I (\] P- •··—
OC\|
r-ZOL.) O> IT
I•
V) G.)'U1 •—SO C}.Z E
4-* fü>,m anfüü)ml- ·—
fü'
'|*
4-*LG)>füCO4-*
CXI U)E CUU4-*EO')LA6 GJ
4-*->CU0 E
O'OCUOC
1- -1-füL
4-*U)
*4-OCU4-*füL
4-*Cfü4-*U)COLJC\|IO
Cl<E
U)'f*L;
IT
OO <‘ G LO (XI OOC\l (XI (XI IO IO
c9 OILVHGIOA
APPENDIX 3
A NOTE ON MULTI—SURFACE HARDENING MODELS*
Recent years have seen an increasing activity in the application
of multi-surface hardening models. while elegant formulations have
been made, there appears to be a blurring of the distinctions between
two basic approaches to formulating such models. It is the objective
of this note to bring out differences and similarities between the two
approaches. An attempt is made here to show that, while these
approaches employ the common feature of incorporating a set of nested
yield surfaces to extend the classical theory of plasticity, they dif-
fer in methametical formulations and originate from different physical
conceptions.
The first approach is that of composite structure formulations.
In this category are the white (l95l) and Besseling (l953) parallel
model, and Iwan's (l967) series model. The second approach is that of
integrated hardening variables model (Mroz, l967). In what follows,
we first give a brief description of each of these models and then out-
line essential differences between them.
*This note was prepared in association with Professor Z. Mroz whilehe served as a visiting professor in the Department of Civil Engineer-ing, during winter, l984. His assistance is gratefully acknowledged.
l99E
200
Parallel Model_
In this type of model, a number of elasto-plastic elements, each
representing a distinct yield surface, are placed in parallel to form
a composite structure model, Fig. A3.l. with the increase in stress, 6,
an increasing number of elements pass through their respective yield
limits, either to flow plastically under no increase in stress, or to
harden kinematically depending on what type of plasticity model has been
incorporated into the mathematical formulation. In either case a
piecewise linear strain hardening curve is obtained which is very close
to a real stress-strain curve,provided a sufficiently large number of
elements are used. Cyclic stress-strain curves are modelled stipulating
that the curves of reverse loading are enlarged by a factor of two——a
hypothesis suggested by Massing for modelling the Bauschinger
effect.
By virtue of a parallel arrangement, equal strains are induced in
each element but the resulting stresses depend on the material
properties of the element. when elements of equal thickness are
used, the average stress in the system is given as,
(Aw)where N is the total number of elements, and n represents the nth
element. Strain in nth element can be written as
$33) = Eij (A2.2)
201
II IOo(2)
E(n)Ö. ----- ( )
I I = OO(n) · OO
II
oO—(3)M _ E00
<¤> F(“) (M(C)
Fig. A3.l (a) and (b) — Composite parallel model; M(c) — Composite series model;(d) — Stress-strain curves for individual elements in
both models. ‘
202
If 6$g)p is the plastic strain, then the elastic strain in the nth
element is expressed as,
(nlp = (nlpgij eij - eij (A3-3land, therefore, the elastic stress-strain law for the element can be
written as,
(nl - (nlp613 ‘ Dijksa (6ka ‘ 6ka ) (AM)
Defining a yield condition,
(nl (nl (nlF .. - .. - = A3.5l (ow oak] ) op 0 ( )
with Prager's translation rule,
= (VÜP A3 6oaij C eij ( · )
along with a flow rule,
·(nlp - l (nl (nl ·(nl613 ° [GY"13 (“ka 6ka ) (AM)
completes the formulation of basic equations. oij define the position
of the yield surface and op its expansion. K is the hardening modulus
and nij are the direction cosines.An elegant example of a parallel concept is the overlay model due
to Zienkiewicz et al. (l972,l978).
203
Series ModelA
Here, the elasto-plastic elements are assembled in series, Fig.
A3.l, and therefore, the basic formulation goes as follows.
(0) =1 61j 61j (A3.8)
(VI) _ (Ü) _ (Tl) = A3 gr (6111 611) ap 0 (.)•(0) - •(¤)Paij — C 6111 Q (A3-IO)•(¤)1> , l (0) (0) _ •(¤)(nkßE
p = l §E(")" (A3 12)ij N 1 ij ‘
and ;<<·=> = li ;<··>@ 016 16)* ij N 1 ij °
Integrated Hardening Moduli Model
As evident from Fig. A3.2, here an integrated yield condition is
employed which leads to an integrated hardening modulus K(n) with
respect to nth nesting surface.
i e for one block motion f = 6 = 6(A) = 0• •Q Q 1 O O
for two block motion f = 6 0 (A3 l4)’ 2 0 0 0 ‘
for n block motion, fn = 60 = (6é]) + 6é2) + ... + 6én))= O
For the case of mixed hardening, general yield conditions are expressed
204
E(n)JIT
· EU) E(o)
'l> °”< > °— ‘”n 3 (2) (l)Go Oo Go Oo
~
L" ' ° ' ' " ”_-3°° KOO 3
+ ‘"° "' "" 2Q l Q K2Ä I ^+ 'l\-/iiE
..-2 ...2 3
Fig. A3.2 Integrated hardening model.
205
·_ (0) _ (0)2 -fo (cij aij ) op — 0(l) (l)2 -f] ' " Op '0_
(0) (ri)2 -_
fn (oij oij ) - op — O
Flow rule is given as,
EP. = 1 n.. (5.. n )1J 1J
1Jwhere,again, K(n) is the hardening modulus corresponding to nth nest-
ing surface.
A translation rule is employed which gives Ziegler's rule as a
special case.
Discussion5
Now, the difference in the three formulations becomes obvious. In
the first two models, the response of the system at any stress level is
the summation of the responses of the individual elements. with each
element, is associated a hardening modulus, K(n). 0n the other hand,
in the third model, hardening modulus at any stage K(n) is the inte-
grated behavior of all elements in yield state. In other words, in
the first two models, each nesting surface represents one element
representing one slip plane, whereas in the third model each nesting
surface corresponds to an integrated fraction of slipping elements with
respect to the total number of potential slip mechanisms which may
become active, should the collapse of the continuum occur. WG infer
that the first two models are microscopic in nature while the third
206
is macroscopic. And, here lies the fundamental difference between the
first two types of models, which are composite structure models, and
the third, which is an integrated hardening model.
Imagine a particulate system of l000 particles with each particle
making three contacts with adjoining particles. To form a series or
parallel model, ideally we shall need a combination of 3000 elements,
each maintaining a bookkeeping of what is happening at a particular
contact plane. To obtain the response of the system at any state, we
would extract information of its loading history from each element
individually and seek a combination in accordance with a parallel or
series arrangement.
In the integrated hardening model, we will seek to model the
behavior of all contact planes which have crossed the yield limit as_awholg_through a work hardening modulus. This modulus is constantly up-
dated as more and more contact points travel past their yield limit.
Thus, the work hardening modulus in the third model represents an
average property of the yielded portion of the system.
Qualitatively, first model is a stronger system while the second
model a weaker system. The third lies in between. It can be shown
that the first two models attain a steady state under cyclic loading
with no possibility of cyclic creep (Drucker, l98l).
These distinctions having been difficult to discern, several reportsi
demonstrate a confusion about the formulation of these models. For
example, it becomes clear that the foundations and the framework of the
paper by Prevost (l977) are those of the integrated hardening model and
not a derivative of all three models combined, as has been erroneously
207
mentioned by several researchers.
Again recently, Szavits-Nossan (l982) indicated that his
kinematic hardening model is based on Iwan's series model, whereas,
after examination it can be seen that it is a parallel model.
Examples are numerous, but it is not the intent of this note to
bring to readers' attention individual errors. If this brief exposi-
tion has clarified a subtle difference between two basic approaches
of formulating multisurface hardening models, the author would think
he has succeeded in his attempt.
APPENDIX 4
CUBICAL TESTS DESIGNED TO SIMULATE PRESSUREMETER
LOADING*
The tests were conducted in the cubical device, designed by Lade
and Duncan (l973), following an elastic-perfectly plastic total stress
path. Such a stress path is an idealization of the actual stress path
encountered in the pressuremeter test.
Fig. A4.l shows the results of one of the tests. In this par-
ticular test, a cubical specimen of undisturbed Bay Mud was aniso—
tropically consolidated. The specimen was then sheared by increasing
one of the horizontal principal stress while the other horizontal
principal stress was lowered by the same amount to achieve a vertical
total stress path. The strain in the vertical direction is kept
constant to simulate the condition of vertical plane-strain.
*After Sinram (l984).
Ah 208
' 209
...... ...... ..—-•--—
—-•-—- -——— -·--— -———2i -;.:..,1 T"“l2Zil -2-'ZZLLT --2Z1 ·"$'é·?¢:-..:1-=———..-.-‘———„-;$-:1::--:f—=-———’—~-————=—·=·=·——·——f***?E?’iZ§E” =.-EEif.40
——.i*'-—-"°-—’Qf:—-——.._.._._.................-———_'—ä··-————·————~————·—.;.;—.....-.—-—+————_·;;·—f?;¢;·;.;——-i;;:—____—+:··—···—_.. .....·.._..··——;:¤f.-.2-..—..-.=-ff*ff“? T-————“°'%—;Z°———j··§.“
“_f?::f-;;;;‘ff€Qiiiiiiäffgee
I N g —-——:—'*"::—'f·;i;j;j;5;f;‘;;,;— EQ
*1*-—=’—’-T t- *· *--3-::;::.--...;Tit:.-fT:...ff:. .... ‘:’.T”.’jjfJL§II...I-.i.... .1..LI7;.I.L;11.ZI.Z°ITTZIZZL:ZZL?ZI.‘..-.l‘.1.I“'T;i1..:-;-;——--·:·—-—--:j;::r;;·j;:"ei;j;i;;·-L;.:1-:;
E -**:51-—-—-—·..1.I'..l..,....."“"'- Tl
V”:·*;‘*—·l·LIÄLL.:.*éE*i%E·LEE:‘.+.§iE§f?§%·E%é::—**‘LE‘^ EEä#‘===””‘—T~"”’='·=’ fiff:T;tiffit;tf‘‘‘.‘‘ffii? i
ar ;=::UI ....-...--.-..-- .----l·~-—--—;;—;;—_g;;;-5;;:;;; .„;:.::::.;::::::x::r;:;::::;:5:;ggg_{}§g;;;;_;;_{Z_¢Q;;3§}Zi{E?§iZ;g :¥%€E—·————::‘:?*-*'°“————”"'£‘*" ""f—'T”':.”j*:?EiE:”EE;?ifT:i:;i::1:iififi ii;fi‘.fii;:fff?G 499
'i;_fi‘=gj-igfil4 · ‘ ::2.;..i3*i·*5?€?äE5Eä?r?i€f’-:·:r;.;;„ _„· ’° 6 -
300 ‘Q 5 10 15 6, %
Fig. A4.1 Resu1ts of a cubica1 test on Bay Mud as reported by Sinram(1984). The tota1 stress path imposed in the testisshownin the top right corner.
2 1 0
I: ..."=*-......22.-——-———°““ Ä;;.._.-T--2 II I “._“*-Ä;.-...IZ':-Ä*-;2?._.#———-—’ #--1.Q.IE' ’
ää-..;—„_._·—————-**·-*.._.-2——-*-‘-*4‘-..—.—::.=;.:;-:—;——._.-;Ä*-‘-.?.2;Ä‘*·—..—:;:.——E2=$1 :-2;: §
äi¤·————""""Ä'“"—°'-....2._...._..........."—;.'T"''—...—_——..2...""*—*·*-*-**-........'————-——'°....”"“°T:’i§:’:‘--**-—-—-——-——————-*-*-*-‘**·.__..-T-..=_..——-—-——2——-—-.-_ -*-_..-T-. -,.1.*T ·- ·=—¢;_=-*-;:6* · TT
1....-‘—';._...—‘-"“I......_...*"""—————————""*‘——.„.—.*—=—‘—*—"..~——;?;—=___·————".?;;;‘;“f—+—f_;
“'“"-f"Ä._.... ... .'.....“"".. -.-°.f-.:Ä**-*-*‘-‘*‘...'“.'L.°°Z—ß..-°°'—“-‘.;.-T-_.--T—‘._..._....Ä--2-*—-‘-Ä:-T-*"""'"'—=-———-·—-..........2_...-‘————·-—T--'“‘:„>—-2;*-*‘
————·-—---—-'—'=""’—---—"——"°'Ä*-·—:'"—’-Ä"I_"’°°-—*-—-°'f‘—·'-—Ö’Ä?•—Ä*——-—.."“"Ä‘?—-——.;-}.'—”_.—'"; "'”——-·-—-_.'ÖÜ.....“—J°——*‘*‘.;"l**-‘:¥—”“”-TT-°"·Ä__'Z.Z-—.°_°7.l—;’——-Ä.“’...“'”:.‘“-*-‘———-—.I.'“..I“'I.’_.lä$§i+-liizä
'I.....”' ”„———”—'Ü’...°“'._“'I ”......“*-Ä“_......'”"°“-‘Ä_.........-Ä"”°'“°°°.-._.'°-.I ”°..’.'°'*-."._..."“__..-*-""“Ä;——‘...:"‘-I2.i;E-Ä;IIf"I’—°"'“;:·-*-*-Ä‘-*"'°““—..""..———“""". ;—-—.._—-—*·*-Ä-•--—-————-——-—“""'“_ZÜ';_T*·T·'°“Ä
-*:2_-——T°fÄ———*’g———I-l”'”———-——·—““———f—j—————':Ä————————w—:-""""'_-—--i""""'"--’"°°-_:—*-f—j„E_..·;_:‘i.3E:“;:_i..:‘_..;_"‘.;.""..”::....—f.—-.Z....:»_\ 5QQ T .-.... .z .;.··—.:··—...-.:Ä—--Ä·-——-Ä-* -———-""ÄÄ—-—-!
S g
gg 1 -—i’1i1=—.”—f"”:-.Q —";——..-Ö'. Ü;Ä————————...äZ}
450 eiT¤¢·<
'“' """°"""'-"""f""f'f"'"‘ff"' '°"‘f"ff"fff'“"'”"[°'°‘”"f""°'f;;f;;T'_;§f:_;;;;f:"“h• IfT T ' . IT'ITL1TTffZ‘.L'TfTT‘f"'.ff..ZT;' fZLff.f.1';TfI'J„°1l'.T'.f.'.Zfff.ffti-T-: :;;:;-74-Trtggc *1;;:;; *;-1:; ;.:;*:;:;;::::;::.*‘ ;t::1:;-.:;i::‘;;:i::;T;:t:;i::::ii "Z'L.-. . . -...-.4 ..--1*
I eéereäazej
400 :-:::::::2;-.:;:j
'Ä'” .’ :111Zfläié0
I S 10 I · 15- Raum. s‘r=z:m Ey 21,*
Fi g . A4 .1 (contd .) App1ied hori zonta1 pressure.
~:T‘:C‘ :2.. 8
Q *2-
———-2.-.-..._____,_
;.—-;i·*———·—-i:-_;;———=—"é::**‘
-TQ
-i—;;-T,.'-:1?1——-—-*-22-··-2-2:
.5**-:_-—_l·i—i-__i—·l"'*
4-,
22 22
‘ ;_;;3;;;;;j=¢..-;-:::;:„?;.,;*€+-:=*;
$
__ =;?.i5_,;;;·:—;-——-—Q__*:_Q______...-..-—.*****—————-:
V
---iU_EL;_—°——”°°"T'
U}
—··—-i--L
‘: ^
[5:;.:;: —-——..-...-—-.....,__
’"*—-—-—-·"-=‘?=-**;:.-:—..1'T;·,_§‘.- an••- •
"’“f2;=-*—————.._j;_’;_*‘;E S- <v
__*é;:;.——_-;;—__:—é
3 L——— " cn
;:I'T.—·—·—·---—-
4-*"¤ GJQC
"*===—:..*·=E*==‘*
42W"’ CGJ ·•·-
·~~?';;.‘;*—=—:-*=-i}-——-——”
""”¤
.*;2:-:2-2-*2;'—25*:'2m Q)
gg-;
>
—E}..Zl¥.ZL§_ 8•'¤
°"‘ ‘^
*=*=;:;:;:"'-A-;i1?iEt2*=’—:=¢?E?—CT-; '° E S-CU
-;*;;*.::.75}.;;-;.éä7;::i;£_r_;;;T___g..:g._j;:_**;___;***¤1'¤¤=-:1*:**·?EE?-T;.§}Q?g;<r:;:.—:;;_—l____z
‘¤ ¢¤gg
;j:;;~1;;;:;::._,__________ '°?iI‘,_-.:F¥{;§§- ggg:.¤§;2?:::;;1;;_:::.-_:'ffj:*_:g·:3¢*¥1'f?:‘-::::·„:;;:;;éj§5_’j__
• (U Q¤- 4-* '
··.”"?ZEé.-.””C?*¥ééi€E;5:>.:;ZÄl:??Il?;Ii;f€’$E§fi**°=**¤=— Q W
~- li? ‘***···"':r—————·————........;E>*; · .;,_ "‘°"‘*·*··--·*¢===*==:§?';T—§;=E**=¤*r=:::—.éé&3;**·
GJ
:-Ii;:7i;f*=*E€.§—•i;;;;*:1::;.:i§i;llIiiiZlTT;;i**
2 GJ S-.C -4->
, QQji'“"·‘=°*>=’—¥iCi5E*EQ$:§?6;::::;:§}ÄÄ}ÄTLJ'“’TTEf_* Zäéii-ä??:éé;éLé;;;:::;:
JT W^
· :::2;.Ii::€E?:§§?§§}1:::-;-;_•;;_-:.T”,;fffff‘¢¢
*=:=g;.-.;;;.„._; _ __ Z ; _ " 4;
** - ·---..:::::;-;:1;;--gg_*;— ·;;;;E- ·- _; _T·-.g g __;,‘:__:;;:E....L:j·i::;gm *};*:;::::::1 ** *”*E::*‘:::*‘::¥::;:*O
'· :· .. . „_, 8” O(Q sz
I"
<r
—<UvZZ
2l 2
iéäéäräii "*——:_1_—;:-;;+1:*--*‘‘‘‘*.;>-—-—:;i;;*——————————."1;:;;;*..1-+..1**.;;;+*--*1‘——·;;;#le7zääewée
...T.l—’;-;.·——T“'"::.v·-=——:'“*—***.._ «— ..T’...+;’."?é» ’i<£·ä;é?ää* °“?&—**__.———*—··‘.___-...·—”*—_‘"“*—¤—-———————°—’**""—.:.._.'·—··“'—'IT;";;éfäFäé_
‘”'°' ' "”"_-——-———"’_'°""'""""——————“""”‘"'——————"°""_’..°’°;——T;j_;—--jj_Z¢;>———g;g;;;;§;I_;_-*:*1;:1;--—·-—-———;**—T*‘?:—1_—=é;11:1.—*=—-—;—........**_1;i?4501T:_—;11·—_;:.—;:';::_”"°°‘-T1·-—··-——·——·—°—'—_—°1.-..i*——————*_1jr_-ij_i""‘—j;l·-_?.r—
--_......„:._=;;*:i::.:—.._*;;*.T..;?_:*;.;_E&ä;€;ä%;ii;,
Äl".———-—·———.‘*‘*T..‘l.T.'—':*·*—'*——*‘—‘......__'"_""’”'—°"*.—;‘;**ii?-—-°-i——-—-——.....r—********1—;—.:“...Ü—.,f°E
* _________________Z +===-———:_**;—'*‘.._._—-—;—·‘=,.;...=*-¥§fI€i'·?Z?E..'*’;*:'.—Ü..f*'****——_....-....._.**—.·.....**—*‘:_.,_ _} ""'—"°"Z-»-Z G’-1"'-—i—'""'°'""-T'f;§Q_:E—___;-_;:-;:-;L7;;_—---•*'
11111 1*1111-1-1-—1—11 ”“'———————————·—*‘*‘—--———‘“'“‘__„'IQé*
Zi 1;-s-*1-:,;;?**‘—-.:::;:.1-w 1*-Lnl ·r;;;;;;;,—::;:t:;..._...***··•‘•·*:‘**.i."'*' ”‘*‘*"‘""'°°-'-——-——'l’-—;"-L——~'——————'—":_-7:EE:;:.r:rt:::::;::t::?.—_ _·_;:·:·+· ,l;...m_.l"""*‘X..IT .17-T'-°i'l————-'iZ'i··—·— ·—·—-—-·-·-·—-··:·_··;;:Minn;-:";;·_F- ; _1 ·:;;:;::::*·—...._...*"""*7 -I-:_Z-'1‘i‘'°—-+”—i—;}_···;::;:,1;:;:::*;:1:."’a'
400; *;*:1; —;;.;+——*‘*.._-——_-.*.g-——-.—*‘—..„...9}·— · _.;?5it.¢:::;::r:t:;:*;::::rit.;: ‘;‘ ‘ Z "ZYTI•— , 11* :*;*—::.-i*—§:—:———'**—':T:-.?*?;•_:—;·*:_:1.T__l_1·:.:*..:'ia;.?2;r<*1;;=<=Z1 ...—11·..11i1:.1i1T1.:1.:1.;.—.-•-1---1„~-—-- --·-—-··—~—··—--—-—-·—v;r··· —··;;;;,;;·::;;;;::.;..-;:;:.2% .-.1f"1-J-ll
·· ::~;;l:--=-——*t_.“'ZT”—'_— ;v==>«#=2=*222;=—*;·
·‘E;;;;;;:·;;;;;
;2;;;;;;:*e&:a—:a· *:-:2;%.**;*;..=:.=;?:¥s;:+ä:;;;§;iEj;;eäiäggjgüggylé
YZ r:;ä;i@2;-*2132 ÜEE1?1"1T P ‘ ·ä?¥¥17Y¥%Z%ä*—%**?¥—**¥?§?%ias.
‘ 7** 2*;
lS0 5 10 n.¤0•:·_ sv=z····« Gr';
Fig. A4.l (contd.) The resulting variation of total vertical stressduring pressuremeter type loading (éz = O).
213
—'=·—‘°°°° N- _Z—-—————-——'°"°‘ ...°i’“T'i·;%é.;äé%é;ä4FilI-—·;_—·-——·—·-—i·l····—·———···—·——·——·—·····—·—·—i···——·—·——··—··———···· ——· 3 :2;.;;.;.::.;;;il_ E $$111TIT;·
‘ ä>=_——_—————;—-————‘*T_°——”—‘——°°_——————°_°i1”-°—°——..;..——”?£“ é N:
EäääääääääEääääääEääääääääääääääääää*'°—..··——·————·T·-T*—·T·..._._’T’°"“’.._...._"“““'“°'“—“"—“_.._T··:;..._.._""'“"”'“'“°“““"'_.“'°——’"T·T*T‘T'I_.."'”'
"—:'jj:—T;—.'l——-ET—. ”_"-—"""‘——-—·*’*"‘“·-—·—-—‘N’~—-——-——""'_...."""......——-—-—————""-"“"“__...;....‘‘‘‘-’===¤——._......; "”“
...—?§ —.._..*‘—‘-——°—..._.—"-‘——"·”;;—Ü--—-————........—";” ”L_—_„..°_‘”—”“‘;...._—_....."‘—"‘”"”°-””_”.......”-——'”——‘"""‘N——°—..._..”‘_—————.=.·””Ä;_::Ä;._’_"————°:Ä‘—‘¥___. _“......°'“'—.;;.'""‘_""°"'“"'°°°""'.__...;.L-————-—·"'_lI'_:—;"..."'““
; °° ETE-—————TT-———-———·———·—·—·———··;‘TTT‘T——-———N-————TT—————————·‘Ti‘;‘NT"‘·—··N:—·:iiL:;iiT=== L;..Ü
‘h"
(0 1;:*':;*.trr;t"‘*...*;N".....;;.°.:•;·tt.*‘57..-——-l-~—..‘_"‘:.—.."’.........."‘”_*‘—"“.......;..‘*"’....""‘“i"....."‘N..;......i"_."‘ ‘ " jf __
' ..T'ZI:'.flZ'.;.
E—.>
ääeiezääzg "§;-—TI‘L—‘"·-l-"""=—·—————‘......."‘‘“ ”"‘....———-——— ;...=‘——’?:Ee2 ‘ ‘p:3¤.
TEN¤0
A5 10 15
~ R1""3' S"'=""€1* ‘/•
Fig. A4.1 (contd.) The resu1ting variation of effective vertica1stress.