Geology; Rocks Soil Formation Soil Deposits Soil Mineralogy Soil Structure Engineering
DNA-TR-87-4211Q4»to1ii1lliliaiAltt^^ iyW^M&miMMMIIfliliäQi^^JSIXMnü&UaU&UDüttäklK^ parameters...
Transcript of DNA-TR-87-4211Q4»to1ii1lliliaiAltt^^ iyW^M&miMMMIIfliliäQi^^JSIXMnü&UaU&UDüttäklK^ parameters...
^ anc FILE cie DNA-TR-87-42
UNDRAINED COMPRESSIBILITY OF SATURATED SOIL
o Q S. E. Blouln «^j- J. K. Kwang
Applied Research Associates, Inc. 00 New England Division 09 Box 120A r" Waterman Road <J South Royalton, VT 05068
I Q ^ 13 February 1984
Technical Report
CONTRACT No. DNA 001-82-C0042
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UNDRAINED COMPRESSIBILITY OF SATURATED SOIL
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18 SUBJECT TERMS (Continue on reverse if necessary, and identify by block number) • - ''- "" - "' Saturated ^oils-. •'Liquefaction . Soil .Mechanics; Mixture Tfieory;
(U-r-, pr^s/i/-:; pCOT1*' h s 19. ABSTRACT (Continue on reverse if necessary and identify by block number)
sk series of equations for the bulk and constrained compressibility of undrained saturated granular soils are developed. Four sets of equations are developed, each set using a more complicated and realistic set of assumptions describing the soil behavior. A computational parametric analysis describes the applicability of the various assumptions as a functicn of soil parameters. The parametric analysis can be used to select the simplest suitable soil model applicable to a given problem.
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TABLE OF CONTENTS
SECTION
1
2
3
4
PAGE
List of Illustrations iv
Introduction 1
Material Models 3
Material Model and Parametric Analysis 26
List of References 59
Accession For
TTIS GRA&I DTIC TAB Unannounced Justification-
By Distribution/
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Availability Codes
Avail and/or Special
m
a. A«* A ^ mvim msm m*«. «"«:*•-..■*
LIST OF ILLUSTRATIONS
FIGURE PAGE
1 Saturated soil/water mixture under hydrostatic compression, schematic section view . 23
2 Schematic section view of the decoupled model under hydro- static compression 24
3 Schematic section view of the decoupled model under con- strained (uniaxial strain) loading 25
4 Constrained modulus vs. initial porosity for uncemented sand. . 34
5 Constrained modulus vs. in-tial porosity for coral 35
6a Undrained bulk modulus as a function of porosity, uncemented sand 36
6b Undrained bulk modulus for uncemented sand, deviation from fully coupled model 37
7a Undrained bulk modulus as a function of porosity, coral .... 38
7b Undrained bulk modulus for coral, deviation from fully coupled model 39
8a Undrained constrained modulus as a function of porosity, un- cemented sand 40
8b Undrained constrained modulus for uncemented sand, deviation from fully coupled model 41
9a Undrained constrained modulus as a function of porosity, coral. 42
9b Undrained constrained modulus for coral, deviation from fully coupled model 43
10a Undrained bulk modulus as a function of grain bulk modulus, uncemented sand at 35% porosity 44
ICb Undrained bulk modulus for uncemented sand at 35% porosity, deviation from fully coupled model 45
11a Undrained bulk modulus as a function of grain bulk modulus, coral at 35% porosity 46
TV
i^*gK«K5xa<SJi5lJklMJ?s3MJtSGgaaMkAk<M^
LIST OF ILLUSTRATIONS (concluded)
FIGURE PAGE
lib Undrained bulk modulus for coral at 35% porosity, deviation from fully coupled model 47
12a Undrained constrained modulus as a function of grain bulk modulus, uncemented sand at 35% porosity 48
12b Undrained constrained modulus for uncemented sand at 35% porosity, deviation from fully coupled model 49
13a Undrained constrained modulus as a function of grain bulk mod- ults, coral at 35% porosity 50
13b Undrained constrained modulus for coral at 35% porosity, deviation from fully coupled model 51
14a Undrained bulk modulus as a function of grain bulk modulus, uncemented sand at 50% porosity 52
14b Undrained bulk modulus for uncemented sand at 50% porosity, deviation from fully coupled model 53
15a Undrained bulk modulus as a function of grain bulk modulus, coral at 50% porosity 54
15b Undrained bulk modulus for coral at 50% porosity, deviation from fully coupled model 55
16 Undrained constrained modulus as a function of grain bulk mod- ulus, uncemented sand at 50% porosity 56
17a Undrained constrained modulus as a function of grain bulk mod- ulus, coral at 50% porosity 57
17b Undrained constrained modulus for coral at 50% nTosity, deviation from fully coupled model 58
vmttAA aun Am«Lii feA mj\ tw\9u\ Af • iwui i iiM\£m-irmvm.M^xnjenMmjä.MjaammtmAj^n&i^
SECTION 1
INTRODUCTION
Analysis of high intensity dynamic loadings of saturated in situ soil
deposits requires mere sophisticated effective stress models than are used in
conventional soil mechanics. The low intensity loadings and slow rates of
application of concern in most conventional werk permit the use of simplifying
assumptions. For instance, the compressibility of the solid grains is usually
ignored and even the compressibility of the pore water can often be neglected.
Such simplifying assumptions, however, can lead to large errors when used to
describe soil response to high intensity, rapidly applied loads such as those
from explosive detonations.
Detailed modeling and analysis of the response of saturated soil
deposits to explosive loadings require development of finite difference and/or
finite element computer codes which delineate the stresses and strains in both
the pore water and the soil skeleton. Irreversible volume reductions in the
soil skeleton can result in liquefaction and drastically altered behavior of
the soil mass. Use of effective stress models which describe the action of
the soil skeleton within the pore water are required to properly model and
analyze this type of behavior.
While the simplifying assumptions used in conventional soil mechanics
are no*, relevant to the high intensity dynamic loadings, other simplifying
assumprions can sometimes be employed without significantly degrading the
accuracy of the dynamic models. These might include ignoring volume strain in
the soil particles resulting from effective stress and ignoring skeletal
strain resulting from compression of the individual soil particles by pressure
in the pcre water. Use of such assumptions can significantly simplify effec-
tive stress models and great'y reduce computational time and cost.
Applicability of such assumptions is a function of the in situ soil properties
and the loading function, and should be cetermined on a case by case basis.
The following sections develop a series of equations for the bulk and
constrained compressibility of undrained saturated granular soils in terms of
1
11Q4»to1ii1lliliaiAltt^^ iyW^M&miMMMIIfliliäQi^^JSIXMnü&UaU&UDüttäklK^
parameters describing the pore water, the soil grains and the soil skeleton.
Four sets of equations are developed, each set being dependent on a more
complicated and realistic set of assumptions describing the soil behavior.
Finally, a series of parameter studies is presented which graphically descri-
bes the applicability of the various assumptions as a function of soil parame-
ters. These parametric studies can be used to aid in selection of the
simplest soil model applicable to a given problem.
■iievsmmttiMOTcnfmMVMMfevuMteiifemiMBMMa^^
SECTION 2
MATERIAL MODELS
2.1 MIXTURE MODEL.
2.1.1 Bulk Modulus.
The simplest applicable compressibility model for high stress, rapid
loadings of fully saturated soils is the mixture model in which only the
compressibility of the pore water and the compressibility of the solid grains
by the pore fluid pressure are considered. Strictly speaking, this model is
only applicable to suspensions, where intergranular (effective) stress is
zero. As will be demonstrated in Section 3, however, it is a reasonably
accurate representation of compressibility for uncemented low density
saturated soils.
The bulk modulus of the soil-water mixture. Km, is derived from the
relationships illustrated in Figure 1. A total pressure P acts on an ele-
ment of saturated so.l made up entirely of solid grains and water. The total
volume of the soil element is given by V^ and the total volumes of the pore
water and sol id grains within the element are giver, by Vw and Vg, respec-
tively.
Thus,
V » V + v t w g
(1)
and the porosity n is given by
V n = Jl
V (2)
t»Mi^MM.AmiuKmmmmmMt^mjmM&M.tLM.MM»M%jmmmmRMmjmMKMmjinMmM • inurx MW um wn titttrwrwrnwrt M* ■n-iminini wmm MiifunrMTMMWMP"*"—''''
Also,
V
t (3)
The total strain in the soil element, et» is given by
et -r m (4)
where Km is the bulk modulus of the soil-water mixture. The volume change in
the soil element, AV, equals the sum of the volume change in the pore water
AVW and the total volume change in the solid grains AVg' according to
AV = AV + AV w g (5)
The volume change can also be expressed in terms of total volume as
AV = et Vt (6)
Assuming that there is no flow of pore water into or out of the soil
element during loading, the volume change in the pore water is given by
AV = e V w w w (7)
where £w is volume strain in the pore water. In turn, pore water strain can
be expressed by
e P w = r
w (8)
where Kw is the bulk modulus of the pore water. Substituting expressions for
pore water strain from Equation 8 and the volume of water from Equation 2
into 7 gives
AV = -L nv w Kw nvt (9)
The total volume change in the soil grains is given by
AVg = cgVg (10)
where eg is the volume strain in the solid grains. Assuming that the volume
strain in the grains results solely from the action of the pore water pressure
P on each individual grain, the strain is gfven by
r . p IT B ——i
g Kg (ii)
where Kg is the bulk modulus of the solid grains. Expressions for strain from
Equation 11 and for granular volume from 3 are substituted into Equation 10
giving
AVg " IT (1 " n)Vt 9 (12)
It should be noted that the above assumptions ignore any contribution
of effective stress in the soil skeleton to the compressibility of the soil
element. Thus, only the properties of the soil-water mixture contribute to
the overall compressibility.
Substitution of Equations 6, 9 and 12 into Equation 5 gives
an expression for total strain of
£iiiMiMffiiMUMMwmi9<\nry.^luirMifu wu mXn&smmmam'mtMitxvjtuMyui** -AM mmi^s üxmüiüt üfnuuumfuiAAnjuumaiaBMl
et = p K + n(K - K ) w g w
K K g w
(13)
Further substitution of Equation 13 into 4 gives the bulk modulus of the
soil-water mixture as
K K Q W
Km " K + n(K - K ) w g w (14)
Equation 14 is a form of the Wood equation discussed by Richart et al.,
1970.
2.1.2 Constrained Modulus.
Since the assumptions used to derive the bulk modulus of the mixture
ignore any contribution of intergranular stress, the soil element must behave
as a dense fluid. Under uniaxial strain loading, the lateral pressure will
equal the applied stress, and the constrained modulus must equal the bulk
modulus. Thus, the constrained modulus of the mixture, Mm' is given by
M = K = m m
K K _3_w_ K + n(K - K ) w g w (15)
2.2 DECOUPLED MODEL.
2.2.1 Bulk Modulus.
While the compressibility of the soil-water mixture is a satisfactory
approximation of material compressibility for some loose uncemented soils, the
contribution of the soil skeleton is often of significance. The simplest
method of accounting for the influence of skeleton compressibility is to
assume that the skeleton acts independently of the soil-water mixture. This
model, called the decoupled model, is pictured schematically in Figure 2.
The total pressure, P, applied to the soil element is resisted partially by
pressure in the pore water, u, and partially by effective stress in the soil
skeleton, P. The total pressure is given by
>feAA*BAja-j>%«.1fcKj)tt*,.n...a9LanjB ftKAjrAJtrflJi WJI R_M nj* ^JüSJ \Mr--.MVkMMJi ffLM fUt tUL-.P.Jtm ir^ + MnMKMILMnMK* Rh'RMMJlRJiRMHÜlUfRJtfmjfRMmÄmj^I
u + P (16)
which is the effective stress relationship. Assuming no flow during loading,
the strain in the soil skeleton, es' must equal the strain in the soil-water
mixture, em. Each in turn must equal the total strain; thus,
t m s (17)
The total strain can be expressed as
et = if (18)
where K^ is the decoupled bulk modulus of the soil element. The strain in the
soil-water mixture is given by
em " K u
m (19)
where Km is the bulk modulus of the soil-water mixture derived in the previous
subsection. The strain in the skeleton is given by
S = K (20)
where Ks is the bulk modulus of the skeleton. Substituting expressions for P,
u, and P from Equations 18, 19 and 20 into Equation 16 gives
e.K. = E K + E K t d mm s s (21)
mmmmmmm
Because all strains are equal, as shown in Equation 17, Equation 21 redu-
ces to
K , = K + K a m s (22)
Thus, the bulk modulus for the decoupled model is simply the sum of the bulk
modulus of the mixture and the bulk modulus of the soil skeleton.
2.2.2 Constrained Modulus.
The constrained modulus M^' for the decoupled model is obtained in a
manner similar to that used for the bulk modulus. Figure 3 schematically
depicts the uniaxial loading. The applied total stress, av' is resisted by
compression of the soil-water mixture due to the pore water pressure, u, and
by compression of the soil skeleton by the effective stress öv. Thus, total
stress is given by
% = u + 0v (23)
Though strain is constrained to the direction of the applied stress,
Equation 17 still holds, with
et ^ M, (24)
and
cs ^ M, (25)
where Ms is the constrained modulus of the soil skeleton. Since the soil'
water mixture behaves as a fluid, the strain in the mixture is given by
.mm*mm.*m*rmm.^]imr1t]VT-nTrirvmiint!rniftiimiFairfir ftn-mnwwiryiivifWify rw 'if i'Vfif •tifw'rf-ff> *m% i xvmur luru^MA AX mA MIMü MA «.I JU1
Equation 19. Combining Equations 19, 24, and 25 in the same way as
was done for the bulk modulus, gives a similar equation for the decoupled
constrained modulus.
M, = K + M d m s (26)
Thus, the decoupled constrained modulus is simply the sum of the bulk modulus
of the soil-water mixture and the constrained modulus of the soil skeleton.
2.3 PARTIALLY COUPLED MODEL.
2.3.1 Bulk Modulus.
The decoupled model adequately predicts the compressibility of
saturated soils over a range of porosities. At higher densities, however, it
may prove inadequate. A more realistic model can be derived by realizing that
the pore water pressure also acts to compress the soil skeleton. The pore
water pressure acting on each soil particle results in a volumetric strain,
€gU' of
u egu K
(27)
The strain in the soil skeleton resulting from the summation of all strains in
the individual particles, will also be egU. In other words, the soil skeleton
undergoes a uniform volume reduction proportional to egU caused by the reduc-
tion in volume of the individual particles. Thus, if a volume of saturated
soil, initially submerged at a shallow depth, is lowered to a deeper depth,
the soil skeleton will undergo a volume reduction with no increase in effec-
tive stress.
The compatibility equations for the partially coupled total strain
are:
P Et = IT (28)
P
SKm^WNM^m^jSS^^^
where Kp is the effective bulk modulus for the partially coupled model;
et ~ em K u
(29) m
where €m is the strain in the soil-water mixture due to compression by the
pv/r-e water pressure; and
^ JP_ . _u H = S = Ks Kg (30)
where the total strain in the soil skeleton equals the strain resulting from
the intergranular stress plus the strain resulting from skeleton compression
due to the pore water pressure given in Equation 27.
Substituting the value of P from Equation 16 into Equation 30
and setting it equal to 29 gives the pore pressure as
u =
IK K I \ m g /
(31)
Equating 28 and 29 gives
PK m (32)
Substituting Equation 31 into 32 gives
K = K + K - p m s
K K m s
(33)
or, the partially coupled bulk modulus equals the decoupled modulus modified
iy subtracting the last term in Equation 33;
10
•»fw^jri&.i tmim:WiJtr*Mt*m.njmr&mtmmmmMmmm-mm'mkmm..mmxmM-WLmm*ip» urn »« kf» kniu?m. ICH VS«I^ u£m.\xm\m
p d
K K m s K
(34)
2.3.2 Constrained Modulus.
In order to obtain the solution for the partially coupled constrained
loading, the influence of the pore pressure on the total strain in the skele-
ton must first be determined. Because lateral strains are zero by definition
in the constrained case, this influence is not intuitively obvious. For con-
venience, assume that the applied stress is vertical and that the major prin-
cipal effective stress in the skeleton is äv and the resultant radial stress
is är. Application of the vertical effective stress alone would result in ver-
tical and radial strains in the unrestrained skeleton of
ev E (35)
and
s (36)
where n is Poisson's ratio and Es is Young's modulus of the soil skeleton.
Assuming that radial effective stress is uniform, the minor and intermediate
principal stresses can both be given by är. Application of only the inter-
mediate and minor principal effective stresses result in a vertical strain o,
ev= -2lJ r (37)
and a radial strain of
e„ = o o
E ME s s (38)
11
««1U.IIM-!rSl«'M«M"JIM«M.«MÄÜ «MÄ«J!lW«»JrKJim««JIMJ% mit ^"MJlMJimÄl^MÄlLni^JU* 11^1^
Equation 38 takes into account that the intermediate and minor principal
stress are orthogonal to one another. Finally, the components of strain
resulting from compression of the individual soil particles by the pore water
pressure are given by
v 3K
and
u er 3K
In Equations 39 and 40, the skeletal strain is assumed uniform (i.e.
isotropic soil skeleton) and second order terms are ignored.
Summing Equations 35, 37, and 39 gives the total vertical
strain as
(39)
(40)
v - E; (% - 2"°r) + w t =
(41)
Summing Equations 36, 38, and 40 gives the radial strain as
E = 0 = -i r E ' - u (o + o )
r M^ v *■ I u
3K (42)
Solving Equation 42 for the radial effective stress gives
uE
r 1 - u v 3(l-u)K
12
(43)
. ■-. -,^^,>., ~. ^.^-^^ ^-^ ^.: ^.-..^ p.^W.-p.^^^.^^.^p^.^,^,^^^^^^^^^^-^.
Since radial strains are zero, the total strain equals the vertical strain.
Substituting Equation 43 into 41 gives
et Ev av Es(l - y) + Kg 3(1 - y) (44)
Using the elastic relationships between bulk modulus, constrained modulus and
Young's modulus;
K = 3(1 - 2y)
(45)
M = E(l - y)
(1 - 2y) (1 + y) (46)
K 1 + y M " 3(1 - y) (47)
Equation 44 can be rewritten as
0V
K - - _v s t M
+ inru s g s
Comparing Equation 48 to Equation 25 shows that the total skeleton strain
is modified by the last term of Equation 48 as a result of the partial
coupling assumptions.
The remaining strain compatibility conditions for the partially
coupled constrained loading are given bv
(48)
v t M_
(49)
13
mmmaumMxmjmemmmimea&MmeVM-'jMiiiiTmnr. im nm m ■ mi > i im»ill II 11 ■ n »iMMiiiriTwmmmp»*"*"*«*«'««'^«^'M*«<^»»»i«qimanmm wii^
Where av is the total applied stress and Mp is the partially coupled
constrained modulus, and
u et = K m (50)
Substituting the value for effective stress
a = a - u v uv
(51)
into Equation 48, and equating 48 to 50 gives the pore pressure as
u a K K v m g
KM + K K - K K g s m g s m (52)
Equating 49 and 50 gives the partially coupled constrained modulus as
o K
P u (53)
Substitution of Equation 52 into 53 gives
K K M = K + M 1~ p m s K (54)
Thus, the partially coupled constrained modulus is simply the decoupled
constrained modulus from Equation 26 modified by the same term used to
obtain the partially coupled bulk modulus in Equation 34;
K K M - M m S MP - Md ■ ir- (55)
14
temTutmntmvt i i jnt^k m. n-mM -mJsmMmM vmmmmMmM.uJ£mMmM «Jt *MmMMMmMmMmMM£&
2.4 FULLY COUPLED MODEL.
2.4.1 Bulk Modulus.
In the case of the partially coupled model, the influence of the pore
water pressure on the volume change of the skeleton was incorporated into the
modulus equations. In the fully coupled model, the influence of effective
stress on volume strain in the soil-water mixture is added to the assumptions
included in the partially coupled model. Effective stress in the soil grains
results in compression of the individual grains and an additional volume
decrease in the soil-water mixture not accounted for in the previous models.
From Equation 6, the volume strain in the soil-water mixture is
defined as
e = AY (56)
In this case there arc two components contributing to the net volume change;
AV = AV + AV m g (57)
where AVm is the volume change in the soil-water mixture due to the pore water
pressure given by
AV = — v m K vt m (58)
and AVg is the volume change in the solid grains resulting from the effective
stress P. This intragranular stress, äg( is given by
o = g 1 - n (59)
15
k\m%\m%iÄVViMA\vwvuiajiAJin^^
where 1 - n represents the ratio of the volume of the solid grains to the
total volume from Equation 3. Because the grains take up only a portion of
the total volume, the actual stress in the grains is greater than the effec-
tive stress, P. Strain in the grains, eg, due to the effective stress, is
given by
a j r _ _2 „ F g " K„ " (1 - n)K^ (60)
and the volume change equals the strain in the grains multiplied by the volume
of the grains from Equation 3,
_ p AV = :f- V
5 Kq t (61)
Substitution of Equations 58 and 61 into Equation 57 gives the net
volume change as,
Av = — v + — V a K t K t m 9 (62)
Further substitution of Equation 62 into 56 gives the strain in the soil-
water mixture;
u P et " K + K" m g
(63)
The other strain compatibility equations are;
P
'' = ^ (64)
where Kf is the bulk modulus for the fully coupled model, and
16
i»*i^B5aarwaifi.srtt«M-jraM Pw«*«M«MJi«M«Äifc:Smj!r*JSM.ft «j^^niL^
Gt K K s g
165)
Equation 65 represents the volume strain in the soil skeleton resulting from
the effective stress plus the action of the pore water on the individual
grains, and is identical to the compatibility equation used in the partially
coupled model.
Setting Equation 63 equal to 65 and solving for P gives
P= ßku (66)
where
K (K - K ) a - s g m Pk ~ K (K - K )
m g s (67)
Substituting Equation 16 into 64, equating 64 and 63, and solving for
the bulk modulus, Kf, using Equation 66 gives
(B + DK K Af K + BKm g in
(68)
Using the value for ß^ from Equation 67 and a bit of manipulation gives the
bulk modulus as
K, = K + K f m s
K K m s K
+ K K m s
K K K + K - -£-5- - K m s K g
g m s
This is simply the bulk modulus of the partially coupled model from
Equation 33 modified by the last term in 69. Kf can alternatively be
expressed as
17
(69)
irA«uautrj(iuifuiiun«nj(Ä*!Uk%jcJiiuuouuuo^
K.. = K + K K p m s
K + K m s
K K m s K
- K
K K m s
(70)
Equations 69 and 70 agree with results derived through different
approaches by Gassmann (1951), and Merkte and Dass (1983), and also presented
by Hamilton (1971) and Clay and Medwin (1977).
2.4.2 Constrained Modulus.
The difference between the fully and partially coupled constrained
moduli is in the strain compatibility equation for the volume strain in the
so'l-water mixture. Volume strain in the fully coupled case results from
volumetric compression of the pore water and soil grains by the pore fluid
pressure, u. plus volumetric compression of the solid grains by the principal
effective stresses, assumed to be ay and är as in the partially coupled case.
The overall volume strain is given by
M. (71)
where Mf is the fully coupled constrained modulus. The volume strain in the
soil skeleton is identical to that for the partially coupled case given in
Equation 48 and repeated here.
K
M. K M g s
u (72)
The volume strain in the soil-water mixture is derived in the same manner as
that for the fully coupled bulk modulus except that the mean effective stress
in the skeleton, än, is used in (.lace of the effective pressure, P. Thus, the
compatibility Equation corresponding to 63 is
18
MMDwjaaaaaQflQaMüfl J.ilWXiPfct
o _ u . m
m g (73)
where
am = av + 2ör
(74)
Substitution of Equation 43 into 74 gives the mean effective
stress in terms of the vertical effective stress as
n - « 1 •>• V 2 '"s n % " öv 3(1 - y) " 9(1 - y) K^ U
(75)
Using Equations 45 and 47, Equation 75 can be expressed as
K K / K
% ■ 5v ir + ir hr - J <,3 s g \ s
(76)
Substituting Equation 76 into 73, and equating 73 to 72 gives the
effective vertical stress as
o = ß u v m (77)
where
t = 9 s m s "s m s g m s m K^„(K,, - K )
m g g s (78)
Equations 77 and 51 are combined to give
v in (7S)
19
MvmarviryarujCHM. r ■ r in -i n i nunirrrini 'innrnrimnTninf irnn" irr'rniri'r niiiciminrini~"rrif~Mn>nirrrnini
The constrained modulus from Equation 71 can be expressed as
(ß + 1)K M nf K + B K s m g
(80)
using Equation 72 for et, 79 for av, and 77 for äv. Substitution of
Equation 78 into 80 gives
M,. = MK2 + KK2+KK2-MKK - 2K K K s g m s m g s m s m g s
K - K K g m s (81)
Further manipulation gives the constrained modulus in the form
(82)
which is a modification of the partially coupled modulus, M«' from
Equation 54 given by
M, = M + K K f p m s
K + K m s
K K m s K - K
K K K m s
(83)
The modifier in Equation 83 is exactly the same as the modifier for the bulk
modulus in Equation 70. These results agree with those derived by Merkle
and Dass (1983) using a somewhat different approach.
2.5 SUMMARY.
Equations for the undrained bulk and constrained modulus of saturated
granular soil were developed based on four sets of assumptions governing the
20
material behavior. These assumptions and the resulting equations are sum-
marized in Table 1. The first and simplest set of assumptions is that the
soil behaves as a dense fluid. Applied stress is resisted solely by the
compression of the pore water and compression of the individual soil grains by
the pore water pressure. The resultant mixture model neglects any contribu-
tion to stiffness from the soil skeleton.
The second, or decoupled model, assumes that applied stress is
resisted by the stiffness of the soil skeleton acting in parallel with the
stiffness of the soil-water mixture from model one. The resultant moduli are
simply the sum of the mixture modulus and the bulk or constrained modulus of
the soil skeleton, as appropriate.
In the third model, designated the partially coupled model, the
assumptions for the decoupled model are modified by including compression of
the soil skeleton due to pore water pressure acting on the individual grains.
The added strain due to the skeleton compression causes the partially coupled
moduli to be somewhat smaller than the corresponding decoupled moduli.
The fourth and most complex model, designated the fully coupled
model, incorporates all the assumptions of the partially coupled model and
also includes the reduction in volume of the soil-weter mixture due to the
action of effective stress on the individual particles. This additional
reduction in volume strain results in values of fully coupled moduli somewhat
lower than those of the partially coupled model.
21
lntf^Rf»a(UMiaAaiiaiiaMAa«MAMfi '^«vöucnKMwaKiyauogoUf^^
3 ■o o E
x> ttJ c ia i- T3
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19
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SECTION 3
MATERIAL MODEL AND PARAMETRIC ANALYSIS
3.1 MATERIAL MODEL.
A number of numerical models can be developed based on incremental
expressions of the various equations derived in Section 2. Obviously, the
complexity of these numerical models will depend on the complexity of the
assumptions used in deriving the basic equations. It is desirable to use as
simple a model as is adequate for any given problem, thus keeping calculation
times and costs to a minimum. In this section, the four models for fully
saturated undrained soil, developed in Section 2, are compared as a function
of variations in assumed material properties of the soil skeleton. From these
comparisons the adequacy of a particular model can be assessed for a par-
ticular set of material properties.
In order to study the influence of changes in various soil proper-
ties, simplified material models for the soil skeleton were assumed. Models
for both uncemented and cemented sands were developed which should bound the
behavior of most saturated granular materials. These models express the ske-
leton bulk and constrained moduli, Ks and Ms as functions of the porosity, n.
Figure 4 shows constrained skeleton moduli from static loadings of
5 different sands as a function of initial porosity. The moduli are the
secant moduli measured at a strain of 1.5%. This strain represents the strain
at an approximate total stress loading of 1 kbar. A fit to these
data was made using the equation
M. *'{l-t) m
(84)
where Mg is the constrained modulus of the solid grains, u is the porosity,
n0 is a limiting porosity, and m is an exponent controlling the shape of the
26
MMtiyitiMbSMtitä
data fit. At n equal to zero the constrained modulus will equal the modulus
of the solid grains.
A constrained modulus of the grains of 9000 ksi was computed using
Equation 45, and an assumed bulk modulus, Kg' of 5000 ksi and an assumed
Poisson's ratio, ji, of 0.25. The value is representative of typical mineral
constituents of many sands. As shown in Figure 4, values of m = 8 and n0 of
0.8 gave a good fit to the uncemented sand data. The data fit is
thus expressed as
M (ksi) = 9000 s (-Ä)
8
(85)
From this equation, an equation for the bulk modulus of the skeleton can be
established,
K (ksi) = 5000 I1"-) 8
(86)
Equation 86 assumes that the Poisson's ratio of the skeleton is the same as
that of the solid grains (n = 0.25). In actuality, Poisson's ratio of the
skeleton is somewhat greater than Poisson's ratio of the solid grains.
Typical values for Poisson's ratio of the skeleton are 0.3, while values for
the solid grains range from 0.12 to 0.30. However, in order to simplify the
model, Poisson's ratio of the skeleton was assumed equal to that of the
grains.
Figure 5 shows values of constrained modulus as a function of poro-
sity for cemented Enewetak coral computed from data given by Pratt and Cooper,
1968. This coral data represents an upper bound for cemented sand, as uncon-
fined compressive strengths ranged from 2.7 to 5.7 ksi. Using an assumed Kg
of 5000 ksi, and a representative Poisson's ratio for this material of 0.2, a
fit to these data (shown in Figure 5) is given by
27
inrnvturnvKMurKM «u »^f^j»WKiftAAy»wm\f Ä-iäisasvyutfiftÄK/^^
M (ksi) = 10000 s (>-&) (87)
The corresponding equation for the bulk modulus of the skeleton is given by
K (ksi) = 5000 (-Ä) (88)
For purposes of this parameter study, Equations 85 through 88
express the moduli solely as functions of porosity. While this is a
simplifying assumption, these equations represent reasonable upper and lower
bounds to the behavior of actual materials.
3.2 INFLUENCE OF MATERIAL PARAMEÜERS.
3.2.1 Modulus As A Function of Porosity.
Using the equations for skeleton moduli derived in tha previous sub-
section, the undrained bulk and constrained moduli for the four models sum-
marized in Table 1 are computed as functions of porosity. These results for
bulk modulus are shown for the uncemented and cemented cases in Figures 6a
and b and 7a and b, respectively. Corresponding plots for the constrained
modulus are given in Figures 8a and b and 9a and b. A bulk modulus of water
of 300 ksi was assumed in all calculations.
Variation in undrained bulk modulus in the uncemented sand is plotted
as a function of porosity for the four models in Figure 6a. For porosities
greater than 30%, differences between the four models are insignificant. At
the higher porosities, the influence of the soil skeleton is very small, with
the overall behavior primarily governed by the soil-water mixture. At the
lower porosities, significant differences develop. The lower bound is repre-
sented by the mixture model which ignores the stiffness of the soil skeleton.
The upper bound is represented by the decoupled model. In the decoupled model
the skeleton modulus from Equation 86 is simply superimposed on the mixture
modulus. The influence of pore pressure on the skeleton behavior is ignored.
28
i <\£MwrkinijrHjnac«xiu{M)ixAwnraixi^^
Due to this simple summation, at the limit n = 0, the decoupled modulus
approaches 2Kg, In actuality, the modulus should approach the solid grain
modulus. Kg. Thus, the decoupled model introduces large errors at low porosi-
ties.
In the decoupled model, the pore pressure acts to reduce the volume
of the individual grains and in turn reduces the overall volume of the soil
skeleton. This influence of the pore pressure reduces the modulus from that
of the decoupled model. At the zero porosity limit, the partially coupled
model satisfies compatibility with the solid grain modulus. In the fully
coupled model, the modulus is further reduced by additional volume reduction
due to the effective stress acting on the solid grains.
Figure 6b compares the mixture, decoupled, and partially coupled
models to the fully coupled model. Percent deviation of each of the three
models from the fully coupled bulk modulus is shown as a function of porosity.
Above a porosity of 26%, deviation in bulk modulus for all the models is less
than 10%. This porosity range is representative of nearly all uncemented
sands. The decoupled model appears to be entirely adequate in this range of
porosities. At 30% porosity, the decoupled bulk modulus is within 4% of the
fully coupled modulus.
Bulk modulus as a function of porosity for the cemented material is
shown in Figure 7a. Because the skeleton in the cemented case is much
stiffen than that in the uncemented case, there is a significant influence of
skeleton stiffness over the entire range of porosities. The general trends
are the same as those observed in the uncemented material; i.e. the mixture
model gives a lower bound modulus, the decoupled model an upper bound modulus,
and the oartially coupled model a slightly higher modulus than the fully
coupled modulus. At the limiting zero porosity, all models except the
decoupled are compatible with the solid grain modulus.
Deviation of the mixture, decoupled, and partially coupled moduli
from the fully coupled modulus in the cemented material is shown in Figure 7b.
As noted above, the mixture modulus deviates significantly from the
29
»«»ufciiM«*_^M»i^v.ni»sKriiraM*MÄanM».«««*vjr*>rair\rii^rtf^^ TVUV
fully coupled modulus over the entire range of porosities. At a porosity of
45%, the decoupled modulus is 10% greater than the fully coupled modulus, with
deviation increasing rapidly at lower values of n. The partially coupled
modulus gives a good approximation of the fully coupled modulus with
deviations of less than 10% over the entire range of porosities.
The undrained constrained modulus for the four models in the unce-
mented material is shown as a function of porosity in Figure 8a. As was the
case with the uncemented bulk modulus, there is good agreement between the
four models at higher porosities. The mixture and decoupled moduli represent
the lower and upper bounds at lower porosities. At zero porosity, however,
the mixture modulus no longer converges to the same value as the fully and
partially coupled moduli. As shown in the equations of Table 1, the mixture
modulus converges to the bulk modulus of the solid grains while the other two
moduli converge to the constrained modulus of the solid grains. The decoupled
modulus converges to the summation of the bulk and constrained moduli of the
solid grains.
The deviation from the fully coupled constrained modulus in the unce-
mented soil is shown in Figure 8b. The mixture modulus exceeds the fully
coupled modulus by more than 10% at porosities of 34% and less. As was the
case with the bulk modulus, the decoupled model gives good agreement with the
fully coupled model over porosities normally exhibited by uncemented sands.
At 30% porosity, the decoupled modulus is only about 3.5% higher than the
fully coupled modulus.
In Figure 9a, the constrained moduli in the cemented material are
compared over a range of porosities. Due to the high stiffness of the
cemented skeleton, the mixture model significantly underestimates the overall
modulus at all porosities. Convergence at zero porosity is similar to that in
the uncemented material, with the mixture modulus converging to Kg, the par-
tially and fully coupled moduli converging to Mg' and the decoupled modulus
converging to Kg + Mg. The deviation from the fully coupled model is shown in
Figure 9b. Clearly, the mixture model is a poor representation. Deviation
of the decoupled model exceeds 10% below porosities of 34%. The partially
30
i^mssrj&M^Vü^^MMä^v^^^
coupled model, however, is in good agreement with the fully coupled model over
the entire range of porosities, with a maximum deviation from the fully
coupled model of less than 5.5%.
From the comparisons in Figures 6a and b through 9a and b it appears
that the mixture model is an adequate modulus approximation for many unce-
mented saturated sands. The decoupled model represents a good modulus
approximation for all fully saturated uncemented sands. Suitability of the
decoupled model for cemented sands will depend on the in situ porosities and
the degree of cementation. For the strongly cemented material presented in
this study, the partially coupled model would be a significantly better choice
than the decoupled model.
3.2.2 Modulus As A Function of Grain Properties.
Examination of mineral properties reveals a wide variation in moduli.
The bulk moduli of materials making up common rocks and sands vary from about
4000 to 15000 ksi, or from about 13 to 50 times the bulk modulus of water. In
this subsection, the influence of variation in bulk modulus of the solid
grains on the undrained bulk and constrained moduli for the four models is
examined. Again, both uncemented and cemented materials are represented,
using the skeleton models developed in the first subsection. Two porosities
are studied, n = 35%, representative of a dense sand, and n » 50%, represen-
tative of a loose sand.
Figure 10a shows bulk modulus as a function of normalized solid
grain bulk modulus for the four models at a porosity of 35%. The solid grain
modulus is normalized by dividing by the bulk modulus of water (300 ksi).
Thus, while normalized modulus varied from 1 to 50, the range from about 13 to
50 represents the range of moduli experienced in common materials. With the
exception of the mixture model, the models are in good agreement over the
entire range of normalized modulus. The stiffer the grain modulus, the more
inadequate is the mixture model. As was the case for modulus variation as a
function of porosity, the mixture model represents the lower modulus bound,
the decoupled model the upper bound, with the fully coupled model falling just
below the partially coupled model.
31
(fjtiuiiui^fA.^ii?ji;uinjuuuurj^jraji(J>^^
Percent deviation of the mixture, decoupled and partially coupled
models from the fully coupled model for bulk modulus in uncemented soil is
shown in Figure 10b. For normalized grain bulk moduli above 35 (10500 ksi),
the mixture modulus is more than 10% lower than the fully coupled modulus.
The corresponding bulk modulus plots for cemented sand at 35% poro-
sity are shown in Figures 11a and lib. Because of the greater stiffness of
the skeleton, the modulus variation at low values of normalized bulk modulus
is less pronounced than in the cemented soil. For the same reason, the mix-
ture model is much softer than the other models which take skeleton stiffness
into account. There is somewhat more difference between the coupled
models than was the case in the uncemented material. As shown in Figure lib,
only the partially coupled model is consistently close to the fully coupled
model. Deviation is 8% or less over the entire range of normalized bulk modu-
lus.
The constrained modulus and deviation plots for the uncemented sand
at 35% porosity are shown in Figures 12a and b. The trends are the same
as those of the bulk modulus plots of Figures 10a and b, with cfose agreement
between all the coupled models over the full range of normalized grain modu-
lus. Figures 13a and b are plots of constrained modulus and modulus deviation
for the cemented material at 35% porosity. The trends are similar to those
for bulk modulus in the cemented material. The decoupled modulus shows less
variation from the fully coupled modulus, and the partially coupled modulus is
within 6% of the fully coupled modulus over the entire range of grain moduli.
Plots of modulus variation in the uncemented and cemented materials
at an initial porosity of 50% are shown in Figures 14a and b through 17a and b.
Because the soil skeleton has very little influence on the composite behavior
of the uncemented soil, all four of the models for both the bulk and
constrained modulus, shown in Figures 14a and b and 16a and b are in very close
agreement. The moduli for the cemented material, shown in Figures 15a and b
and 17a and b, show the same trends as those at 35% porosity, but with closer
overall agreement Both the decoupled and partially coupled moduli are within
8% of the fully coupled modulus over the full range of variation in grain modu-
lus.
32
In summary, the conclusions drawn from the parametric analysis as a
function of porosity hold independent of variations in the modulus of the
solid grains. That is, the decoupled model is a satisfactory model for unce-
mented sands, over all variations in grain modulus. The partially coupled
model is a satisfactory model for the cemented material over all variations of
grain modulus; though in some instances, the simple decoupled model is also
satisfactory.
33
fj*)fwV&<VJflttiiiKiÄKMXjvWMArj%^^
<A
O 2
C
c o O
düü i i n
A Enewetak Sand, ARA, 1984
+ Ottawa Sand, Lambe & Whitman, 1969
• Minnesota Sand 1 * Pennsylvania Sand ■ Davissen, 1965 1
D Sanqamon Sand
150 l\ B J ■ |vis =9cx;o [[■ QQ )
\
\ *
100 •
\+
V
50
n
0 *\
\ A
D \*
1 1 1 30 50 40
Porosity,n (%)
Figure 4. Constrained modulus vs. initial porosity for uncemented sand.
34
t KlrnjlAXIUUILULM Rjf AJBRSMAM aMMLM-j^aifRMTuniiAjasjmumsomMaKiunuaQauaiufleiK
10000
8000
- 6000 V)
•o o
"g 4000
c o Ü
2000
! j
• Enewetak Coral (Pratt and Cooper, 1968)
— Ms=IOOOO{|-^g)3
20 40 60
Porosity, n (%)
80 100
Figure 5. Constrained modulus vs. initial porosity for coral
35
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LIST OF REFERENCES
Clay, C.S., and H. Medwin, Acoustical Oceanography; Principles and Applications, John Wiley & Sons, New York, 1977.
Gassmann, F., "Über die Elastizität Poröser Medien," VIERTELJAHRSSCHRIFT DER NATURFORSCHENOEN GESELLSCHAFT IN ZURICH, Jahrgang 96, Heft 1, March 1951, pp. 1 - 23.
Hamilton, E.L., "Elastic Properties of Marine Sediments," J. Geophys. Res., Vol. 76, No. 2, January, 1971, pp. 579 - 604.
Merkle, D.H., and W.C. Dass, "Fundamental Properties of Soils for Complex Dynamic Loadings: Annual Technical Report No. 3," Applied Research Associates, Inc., Albuquerque, NM, December 1983.
Richart, F.E., Woods, R.D., and J.R. Hall, Jr., Vibrations of Soils and Foundations. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970.
59
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