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Unformatted text preview: 652 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks TABLE 13.1 Critical Conditions for the Stability of Saturated Clays Foundation Loading Soft (NC and Slightly OC) Clay Stiff (Heavily 0C) Clay Critical condition Remarks UU (n0 draina e). Probably UU but check CD drama 6
g g with equilibrium pore pressures).
Use 925 = O, c = 7): with appropriate corrections for Stability usually not a major problem, sample disturbance, strain rate, anisotropy, age, etc. Excavation or Natural Slope Soft (NC) Clay Stiff (Highly OC) Clay Critical condition Remarks Could be either UU or CD. If soil is very sensitive, it may change from drained to
undrained conditions. CD (complete drainage). Use effective stress analysis with
equilibrium pore pressures. If clay
is fissured, c' (and perhaps 45’) will
likely decrease with time. \
After Ladd (1971b). 13.7 clays) and heavily overconsolidated clays, and see what the critical design situations are. Some of the
critical conditions for stability are summarized in Table 13.1 (Ladd, 1971b). CRITICAL STATE SOIL MECHANICS We have already seen that simpliﬁed, conceptual frameworks of soil behavior, like the Peacock diagram
for sand shear behavior (Sec. 12.4, Fig. 12.11), can provide a means for understanding how soil under a
certain preshear state will behave when sheared to failure. One of the most important conceptual and
theoretical frameworks in all of soil mechanics is known as critical state soil mechanics (CSSM). Devel
oped at Cambridge University, this framework was originally presented by Schofield and Wroth (1968).
It brought together previous wellknown concepts such as the Mohr—Coulomb failure criterion and
Hvorslev parameters (Sec. 13.1.3.2). However, it also provided new levels of sophistication in terms of
our ability to model soil behavior by including prefailure behavior [albeit using a simple elasticplastic
model, Fig. 11.4(d)], the effect of stress history on generalized soil yielding, and drained versus
undrained behavior in clays. The development of this framework served as the foundation for today’s
more sophisticated soil constitutive models, which in turn are used in numerical analyses for simulating
highly complex geotechnical problems. These constitutive models include Modified Cam clay (Roscoe
and Burland, 1968), cap models (e.g., Drucker et al., 1957), nested models (e.g., Prevost, 1977), and
bounding surface models (e.g., Dafalias, 1986; Whittle, 1987; and Pestana, 1994), among others. Consti—
tutive models are reviewed in Sec. 13.8. It is important to mention that the CSSM framework was originally developed for saturated,
reconstituted Clay—that is, clays completely remolded and then reconsolidated to a normally consol
idated state. This reconsolidation can be followed by mechanical overconsolidation to some over
consolidation ratio (OCR). Stated in its most fundamental form, the CSSM framework links a pair of
well—known, twodimensional (2—D) soil mechanical “spaces,” void ratio—effective stress and Shear
stress—effective stress spaces. It combines these 2D spaces into a three—dimensional (3D) space that
describes how a clay with a particular stress history will behave when sheared to failure, including
volume change if it is a drained test, and shear—induced pore pressure if it is an undrained test. Let’s first reintroduce the 2D spaces with which you are already familiar. Figure 13.27(a) shOWS a typical plot of void ratio (e) versus the logarithm (base 10) of vertical effective stress (04,) such as that 13.7 Critical State Soil Mechanics 653 Void ratio, 9
Void ratio, 9 —>
(b) 140
’5
g 120 l:
t 100 3
u; 9 80 r
§ 60 8
S a
.C 40
(n 20 O —> O 1 2 3 4 5 6 7 8 9 1O Displacement, 5 (mm) Effective stress, a; (C) (d) FIGURE 13.27 Simplified critical state framework for direct shear tests that are onedimensionally consolidated,
then sheared to failure: (a) e—Iog «7; relationships for consolidation and critical states; (b) e—cr; relationships for
consolidation and critical states; (c) shear stress versus displacement for direct shear tests at three a; values; and
(d) shear stress versus 0; for the three direct shear tests (after Mayne, 2006). presented in Chapter 8.1 In this case, the slope in the normally consolidated range is the compression
index, CC [Eq. (8.7)]. Figure 13.27(b) shows this same plot, except that 0'; is plotted on a linear scale, resulting in the curved shape of the consolidation relationship. Next, as shown in Fig. 13.27(c), we perform CD direct shear tests on three clay specimens con—
solidated to three different 07’, values. While CSSM can depict test results from other, more complex
tests such as the triaxial test (discussed later), use of direct shear results provides a simple way to
understand CSSM principles. The preshear e—log 0'; states are shown on the compression curves of
Fig. 13.27(a) and (b), and the peaks are plotted in shear stress 1 versus normal stress a; space in
Fig. 13.27(c). These failure 7—0;, states form the familiar Mohr—Coulomb failure envelope defined by 1Schofield and Wroth (1968) originally used speciﬁc volume (1)) instead of void ratio, where v = 1 + e, i.e., v is the
volume of soil for which there is a unit volume of solids. They also used the natural logarithm (base e) instead of log
baselO, and they used the 3—D definition of the mean effective stress, or p’ = (01 + Zia’3) / 3, we mentioned earlier.
To illustrate CSSM principles in familiar terms, we will use e and log 0; and then later use our familiar deﬁnition of
p’ = 1/2 (0'; + ah) to illustrate the usefulness of the critical state concept. 654 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks the effective friction angle qb’, in Fig. 13.27(d). In the CSSM framework, this envelope, which always
has an intercept c’ = 0, is known as the critical state line (CSL). The CSL represents the state of stress
at failure for all soils, regardless of their stress history. In order to link the e—log 01, and 7—0;, spaces together, we return to Fig. 13.27(a) and plot the
e—log 0;, states at failure for the three direct shear tests that were performed. Since these were drained
tests on normally consolidated specimens under constant 0;, we know from Sec. 12.9 that the speci
mens contract during shear, and that the preshear 04, = 0;, at failure. This leads to failure states that lie
directly under the preshear states, and the formation of a new line in Fig. 13.27(a) that is the CSL, only
now it is in e—log a; space (perhaps you can now start to visualize the 3D CSL, which for these tests
would be the combination of e—r—crg). For each of the drained, direct shear tests at different 0;, values, the paths followed during shear
to failure on the CSL can be drawn first on the e—log a; plot [Fig 13.27(b)]; these are the paths labeled
AB, CD, and EF. The same tests can be depicted on the 7—0;, plot in Fig. 13.27(d) —they are also verti
cal for the same reason, that 0;, is constant during these tests. If the paths AB, CD, and EF were plotted
in 3D space (er—ag), we would refer to them as their slate paths, since they track both the stresses
and physical states (as represented by the void ratio) of the specimen during shear. You can begin to see that, like other soil behavior models that we’ve discussed, once the com
pression curve and CSL are established for a particular set of tests, they can be used for simple predic
tions of failure stresses and void ratio changes for given values of preshear 0;, To expand the CSSM
framework to undrained shear, Fig. 13.28 shows the same compression relationships, e—log a; and
e—oQ” and the 7—0; space shown in Fig. 13.27. This time, the direct shear test is performed undrained
after consolidation to (74,0 at point A, so that the void ratio is unchanged as it moves to the CSL. The Void ratio, e Effective stress, FIGURE 13.28 Simplified critical state
framework for a direct shear test that is
onedimensionally consolidated, then
sheared to failure, normally consolidated
state: (a) e—log (7;, relationships for con—
solidation and critical states; (b) e—a;
relationships for consolidation and critical
states; and (c) shear stress versus (7', for Shear stress, r
l Effective stress, a;
the undrained test (after Mayne 2006). (C) 13.7 Critical State Soil Mechanics 655 result is that the 0; at failure (point B) is lower than the preshear 0;, the result of positive excess pore
pressure, Au, being generated during shear. In fact, Au = a; (mama) — 0'3, (failure). We mentioned that CSSM can also be used to model behavior of mechanically overconsolidated
soils—Le, those that were loaded to some maximum stress (agm), then unloaded to a final stress (05f),
which produces an OCR = agm/agf. In Fig. 13.29, a swelling line with slope CS has been added to the
compression curves, e—log a; and e~a§,, and the CSL remains the same in all three spaces as it was for
the normally consolidated specimens. The slope of this unloading or swelling part of the e—log a; path
is C x, which is defined by the same equation as the compression index, Cc [Eq. (8.7)]. The framework can
once again be used for predicting both drained and undrained test results, as indicated by paths AB and
AC, respectively. The undrained path AC indicates that since a; (preshear) is less than 0; (failure), the speci
men experienced excess pore pressure during shear, Au < 0, resulting in a 7—0;, path that curls up and
to the right to land on the CSL in Fig. 13.29(c). One question to ask is this: in an undrained test, is there
a preshear 0;, value that would lead to a vertical r—a'; path during shear (in other words, Au = 0 during
shear)? This would be the case when (Tl, (preshear) is where the swelling line crosses the CSL in
Fig. 13.29(a) and (b), point D. Thus, the CSL in these two figures can be treated as a dividing line of sorts:
when a; (thear) is to the right of this crossing point, the clay is normally or lightly overconsolidated and
will contract during drained shear, or develop Au > 0 during undrained shear. When 01') (preshear) is to
the left of this crossing point, the clay is heavily overconsolidated and will dilate or produce Au < 0. Our use of direct shear tests in the above discussion allowed us to learn some basics of the critical
state framework, namely the relationship between preshear states on the compression curve and failure
states on the CSL. However, this by itself would not have made it that much more advantageous than
the Mohr—Coulomb failure criterion. The other significant piece of the critical state framework is the 0: “i
,9“ 2
s 90 ‘ a
U
"C ._
.5 g
>
(b)
b
m.
(I)
9
FIGURE 13.29 Simplified critical state E
framework for a direct shear test that is g
onedimensionally consolidated, then (7:) sheared to failure, heavily overconsoli
dated state: (a) e—log 0;, relationships for
consolidation and critical states; (b) ea;
relationships for consolidation and critical _
states; and (c) shear stress versus 04, for EﬁeCt'Ve Stress' ‘7 1,1
the undrained test (after Mayne 2006). (c) I
UDD 656 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks concept of the yield surface. To understand what this surface represents, we will now look at the q versus
p’ stress space of a triaxial soil specimen that is hydrostatically consolidated. As Fig. 13.30 shows, there
is still a compression curve for this situation, except that we now plot it as e—log p’ and e—p', instead of
using 0;, which was for one—dimensional conditions used in the direct shear test. There is still a CSL
plotted in these two spaces as well as in q— p’ space [Fig 13.30(c)]. The yield surface in q—p’ space is the
dividing line between elastic behavior and plastic or inelastic behavior, and its size is determined by the
maximum value of p’ to which the soil is consolidated. Figures 13.30(a) and (b) show three normally
consolidated p’ levels for the soil,A, B and C. As p’ increases, the size of the yield surface in Fig. 1330(0)
also increases, defined by the intersections of the surfaces with the p’aXis at points A, B and C. A very simple case using the yield surface is during hydrostatic consolidation. In Figs. 1330(3)
and (b), the soil is mechanically overconsolidated to point D; this is also shown on the p’axis in
Fig. 13.30(c). When the soil is reconsolidated to point B, the recompression portion DB is elastic, since
it lies inside the yield surface, and the portion beyond this (path BC) is plastic, which also increases the
size of the yield surface due to consolidation. Let’s consider a hydrostatically consolidated, drained triaxial test on a lightly overconsolidated
soil. As shown in Figs. 13.31(a) and (b), the soil has been consolidated from point A to point B and
unloaded to point C. These consolidation points are also shown in the stress path space of Fig. 13.31(c).
During the drained shear portion of the test, as shown by path CD in Fig. 13.31(c), the initial part of
this loading, inside the existing yield surface, will be elastic. The soil then yields and begins deforming
plastically, expanding the yield surface until it fails on the CSL at point D. You can see the subsequent
decrease in void ratio on the compression curve. For a hydrostatically consolidated, drained triaxial
test on a normally consolidated soil, the stress path would start where the yield surface crosses the p’
axis, and the yield surface would progressively expand to failure. a
O
o Void ratio, 9
Void ratio, a log p’ (61) FIGURE 13.30 Simplified critical state
framework for a triaxial test showing
the relationship between hydrostatic
yield surfaces and compression curve,
normally consolidated and overconsoli—
dated states: (a) e—Iog p’ relationships
for consolidation and critical states;
(b) e—p’ relationships for consolidation
and critical states; and (c) q versus p’
for different yield surfaces (after
Mayne, 2006). 13.7 Critical State Soil Mechanics 657 Void ratio, 9 (b) FIGURE 13.31 Simpliﬁed critical state
framework for a hydrostatically consoli—
dated, drained triaxial test showing the
relationship between hydrostatic yield
surfaces and compression curve, lightly
overconsolidated soil: (a) e—log p’ rela
tionships for consolidation and critical
states; (b) e—p’ relationships for consoli— Existing yield
surface dation and critical states; and (c) q versus I I I I ——>
p’ for different yield surfaces (after p0 = “no _ “ho pf ,01
Mayne 2006). (C) Undrained tests will obviously behave very differently, since their yield surfaces are ﬁxed by
their preshear p’. Consider two hydrostatically consolidated, undrained triaxial tests, one normally
consolidated and one heavily overconsolidated, shown in Fig. 13.32. The normally consolidated soil
starts at a preshear state, point A. Since all of its deformation will be plastic, it will follow the yield sur
face up and to the left to point B on the CSL; this is consistent with the idea that normally consolidated
soils contract or produce Au > 0 during shear, and this is also seen in the e—log p’ and e— p’ spaces. In
addition, a normally consolidated clay tends to strain harden, monotonically rising to failure as it
climbs up the yield surface. For the heavily overconsolidated clay starting at point C in Fig. 13.32, the
soil would climb up through the yield surface with resulting elastic strains and then fall to the CSL,
where it would fail at point D. This is consistent with expected Au < 0 during shear, and strain soften
ing behavior after peak shear stress is reached. So, how do we determine the yield surfaces (or yield curves in 2D) for a particular soil? We can
run tests following different stress paths, as shown in Fig. 13.33(a) or, as shown in Fig. 13.33(b), we can
use different stress ratios. Both approaches will result in the same yield curve—~i.e., the yield curve is
independent of the stress path used to establish it (Leroueil et al., 1990). An example of the second approach [Fig 13.33(b)] is illustrated by some data obtained by Tavenas,
Leroueil, and their coworkers at Université Laval using triaxial tests on samples of Laurentian clays
from Saint—Alban, Quebec. Stress—strain and pore pressure—strain data at three different consolidation
pressures on overconsolidated specimens from 3 m depth is shown in Fig. 13.34. Note that the strain at
failure (at the maximum principal stress difference) is only about 1%, suggesting that this soil is highly
structured, which is typical for Laurentian clays from Quebec. 658 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks Void ratio, 9
Void ratio, 9 
Normal stress, a; (b) FlGURE 13.32 Simplified critical state
framework for a hydrostatically
consolidated, undrained triaxial test
showing the relationship between hydro
static yield surfaces and compression
curve, normally consolidated and heavily
overconsolidated soil: (a) e—log p’ rela—
tionships for consolidation and critical
states; (b) e—p’ relationships for consoli‘
dation and critical states; and (c) q versus
p’ for different yield surfaces (after
Mayne 2006). Shear stress, 1 Normal stress, a; (C) Mohr—Coulomb Yield surface Envelope In situ Probing stress (a)
——> Different Kvalues during consolidation
 > Different stress paths during drained tests compression tests FIGURE 13.33 Determining the yield curves by (a) different stress paths, or (b) different stress ratios (Leroueil
et al., 1990). Stress paths for the three triaxial tests in
Fig. 13.34 are shown in Fig. 13.35. The data are
plotted with the MIT definition of p’ (Sec. 13.2).
Tests 1, 2, and 3 were conventional CU triaxial
tests consolidated hydrostatically~that is, with
the confining pressure held constant. From the
shapes of the stress paths, you know that the clay
was overconsolidated. The yield curve in Fig. 13.35 was deter
mined from peaks of the three stressstrain
curves in Fig. 13.34 and from the yield points of
four nonhydrostatic consolidation tests—that
is, with the ratio K = 05/03 held constant dur
ing consolidation, as shown in Fig. 13.36. The
yield surface appears to be centered about the
KMCline, with K0 z 1 — sin 95’ [Eq. (11.8)],
rather than about the CSL line as would be
predicted by the Cam clay model shown in
Fig. 13.32. Note that at point A, the major prin—
cipal effective stress (7’1 x 0}, as determined
from 1D consolidation tests. This suggests that
a good estimate of the yield curve can be made
from ordinary 1D consolidation tests performed
on highquality samples. Yield curves obtained from specimens
from three different depths of the SaintAlban
clay shown in Fig. 13.37 have a similar shape, so it
may be possible to normalize the test results from 13.7 Critical State Soil Mechanics 659 40 SamtAman,am
I—U’c=4kPa 30 N
O ﬂ~%W%) 10' FIGURE 13.34 Stressstrain and pore pressurestrain
curves (after Tavenas and Leroueil, 1977). the same deposit with respect to the preconsolidation pressure a}, In fact, as shown by Leroueil et al.
(1990), samples taken from diﬁerent depths and consolidated under a cell pressure 0;, such that the ratio
(72/02, is constant, will have essentially the same (0’1 — 09/01, and ALL/a}, versus axial strain relationships. 30 ,9 CU after hydrostatic
‘ consolidation 0 og/al = constant § m é «a b  N b m ______________ 
 0‘ (kPa) FIGURE 13.35 Yield curve
for Saint Alban clay at 3 m;
see text for description of
the types of tests (after
Tavenas and Leroueil, 1977). m w 660 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks p’ = (kPa) 00 1O 20 30 40 50 60
' {Yield point 2 4 _ _._ _ __ ____ _. _ I _ _ .._._ _ ___ _ _ AV/Vo (%)
0)

g
l
i
 FIGURE 13.36 Volumetric strains
from nonhydrostatic CU triaxial
tests on St. Alban clay (after
Tavenas and Leroueil, 1977). FIGURE 13.37 Yield curves for
specimens of SaintAlban clay
obtained at different depths
(after Tavenas and Leroueil, 1979). 2 pr = 0'; ‘1' 0'5 DiaZRodrr’guez et al. (1992) presented normalized yield curves from tests on 17 natural soft
clays of very different geologic origins, all normalized with respect to 0;], as shown in Fig. 13.38. The
sources and geotechnical characteristics of these clays are given in Table 13.2. All yield curves had the
same general shape as the yield surfaces of the SaintAlban clay from Canada in Fig. 13.37, and they
appeared to be centered about the [gmline, rather than about the CSL line as predicted by Cam clay When the specimens were hydrostatically consolidated, their yield stress (aﬂhydmstmc depended on the preoconsolidation pressure. The ratio (03,)hydmstatiC/0'}, varied between 0.44 and 0.73 with an average 0.6 Bu 0 Winnipeg
Q .............. .‘. . . . . . w _ . ‘6') l x
l
:9: 0.2 O
A (04 — (IQ/20';J (15' = 25~27° l Saint—Louis
0 Ottawa
0 Osaka 13.7 Critical State Soil Mechanics Otaniemi
Riihimaki 49’ = 28«30° I mChampiain sea clay
0 Backebol 0...... ..... _. A Drammen i
I Pornic E A i (d) 0.6
0.4
~Q
b
Q
1;) 0‘2 ........................... ,.
\l". ¢/ = 320
3 o Favren 5“ Cubzaclesponts A SaintJeanVianney FlGURE 13.38 Normalized yield curves
of 17 natural soft clays with friction
angles varying from 17.5° and 43°
(DiazRodriguez et al., 1992). 661 662 Chapter 13 Advanced Topics in Shear Strength of Soils and Rocks “mm—“W
TABLE 13.2 Geotechnical characteristics for the natural clays in Fig. 13.38 (Diaz—Rodriguez et al., 1992) Site Depth (m) w (%) PI 0'; (kPa)
Atchafalaya, Louisiana 21.3 58 44 150
Backebol, Sweden 3.4 87 42 57
Bogota, Colombia 7—12 90—160 100—170 150—255
Champlain Sea clays, Quebec  58—90 17—45 50290
Cubzac—les—Ponts, France 4.5—5 .5 60.80 40 4675
Drammen, Norway — 52 29 —
Favren, Sweden — 60 — 70
Mexico City, Mexico 1.7 460 493 71
Osaka, Japan 30.0 63 — 330
Otaniemi, Finland 2.0 130 63 20
Ottawa, Ontario  65 36 150
Perno, Finland 4.2 100 39 22
Pornic, France 1.2—2.0 75—88 40 3545
Riihimaki, Finland 4 55 25 90
St. Jean—Vianney, Quebec 3.7 41 9 1150
St. Louis, Quebec —— 67 23 190
Winnepeg, Manitoba 8—12 5463 35—60 190~38O (WC (°) Reference
23 Tavenas and Leroueil (1985)
30 Brousseau (1983),Tavenas and
Leroueil (1985)
35 Maya and Rodriguez (1987)
27—30 Brousseau (1983)
32 Magnan et a1. (1982)
30 Berre (1972), after Larsson (1977)
32 Larsson (1977)
43 Diaz~Rodriguez et al. (1992)
25 Oka et a1. (1988)
25 Lojander (1988)
27 Wong and Mitchell (1975)
23 Korhonen and Lojander (1987)
29 Moulin (1988,1989)
27 Lojander (1988)
32 Brousseau (1983)
25 La Rochelle et a1. (1981)
17.5 Graham et a1. (1983) value of about 0.6. The ratio also tended to decrease as ¢’ increasedThe position of the yield curve above
the (ageline changes with the value of the friction angle and with the structure of the clay. The further the
upper part of the yield curve tends to be above that line, the more highly structured it is. Also the height of
this “hump” above the qbgcline decreases with increasing sample disturbance. You will recall a similar
“hump” in the Mohr failure envelopes for CC clays in both drained and undrained shear (see, e.g.,
Figs. 12.26 and 12.34). An extreme example of disturbance is complete remolding or destmcturing of the
clay. The effect on the shapes of the stressstrain curves is dramatic, as shown in Fig. 13.39, and of course
the change is equally dramatic in the shapes of the yield curves, as shown by Tavenas and Leroueil (1985). 0.8 Intact —— Destructured I SaintAlban AtChafalaya II Béckebol
I (Leroueil et at, 1979) (Paré.1983> (Brousseau, 1983)
0O 2.5 5 O 2.5 5 O 2.5 5
e (%) FIGURE 13.39 Stress~strain curves from hydrostatically consolidated CU tests on intact
and destructured soft clays (Tavenas and Leroueil, 1985). 13.8 13.8.1 13.8 Modulus and Constitutive Models for Soils 663 From this brief introduction, you can see that critical state soil mechanics is a powerful frame
work for describing soil behavior and predicting soil response under a variety of preshear and failure
states. However, a number of textbooks, mostly by British authors, have appeared in recent years that
provide additional information and applications of critical state soil mechanics to geotechnical prob—
lems. Recommended are Atkinson and Bransby (1978), Bolton (1979), Muir Wood (1990),Aziz (2000),
Powrie (2004),Atkinson (2007), and Budhu (2007). MODULUS AND CONSTITUTIVE MODELS FOR SOILS Several times in Chapters 8, 10, 11, and 12 we mentioned that soils and rocks were often simply assumed
to be elastic, and that this assumption was important for settlement analyses (stress distributions and
immediate settlement in Chapter 10) and in our discussion of the stressdeformation characteristics of
geomaterials (Chapters 11 and 12). We also occasionally mentioned the modulus of a soil, either a com
pression modulus or a Young’s modulus. You may recall from your courses in strength of materials that
the modulus is the slope of the stressstrain curve. Sometimes the modulus of a material is called its stiff
ness. We showed several different types of stress—strain curves in Fig. 11.4. If the stress‘strain curve is linear,
obtaining the modulus or stiffness is easy, but how is the modulus determined on a nonlinear curve? We begin with a detailed discussion of soil modulus, its definitions, and how it is measured or esti
mated. Soil modulus is part of what are known as constitutive relations in solid mechanics. Constitutive
modeling of soils and rocks has become increasingly important in recent years, because many geotechni—
cal projects require deformation predictions in addition to conventional analyses of potential failure and
factors of safety. Wellcalibrated soil constitutive models are required to make reliable predictions of
deformations. We end this section with a brief description of the hyperbolic (Duncan—Chang) nonlinear
soil model, because it is commonly used in geotechnical practice. Modulus of Soils Although there are a number of ways to describe the modulus of a material, all are basically the ratio
of stress increment to strain increment (or the slope) over a particular range of the stress‘strain rela
tionship for that material. Figure 13.40 shows some definitions of modulus that include: 0 Tangent modulus: slope of the tangent to the stressstrain curve at any point; an important mod
ulus shown in Fig. 13.40(a) is the initial tangent modulus (E i or E ,). 0 Secant modulus: slope of a straight line drawn from the origin to some predetermined stress
level, such as 50% of the maximum stress; a chord modulus is the slope of a straight line between
any two points on the curve. Figure 13.40(a) shows examples of both tangent and secant moduli.  Cyclic loadingrelated moduli: when there is an unloadreload cycle, the modulus may be defined
by drawing a tangent from the lower bound stress on either the unload or reload portion, or by
connecting the end points of the hysteresis loop, as shown in Fig. 13.40(b), The hysteresis loop
modulus is sometime called the unload—reload modulus E”. Besides the loading condition, other factors that inﬂuence modulus include (1) for granular
materials, particle packing (as measured by the dry density, void ratio, and/or relative density, and may
include the inﬂuence of compaction), and (2) for cohesive soils, water content, plasticity index, stress
history, and cementation (Briaud, 2001). It is sometimes difficult to generalize about the effect each of
these factors has on modulus. For example, while dry density tells us something about the packing of
the particles, it cannot be assumed that soils with the same dry density will have the same modulusThe
two soils may have very different structures or fabrics (e.g., ﬂocculated versus dispersed—Chapter 4)
and thus they will have very different modulus values. Another example is the effect of water content.
While higher water content tends to indicate a lower modulus, this assumption would be invalid for
some compacted soils as well as high water content clays that are cemented or highly structured. ...
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