Tải bản đầy đủ - 0trang
III. Forces Required to Deform Soils
K. P. BARLEY AND E. L. GREACEN
where up is the tensile strength of the soil. Analyses of tensile failure for
more complicated configurations are available in the theory of elasticity
( Timoshenko and Goodier, 1951) .
The tensile rupture of bulky structures can also be described theoretically. Applying a spherical model, the zone of plastic equilibrium around
the base or point of a probe can be treated as a pressure bulb of radius R
(see Section 111, A, 3 ) . The radial pressure at R, u ~will
burst a soil clod
if the cross-sectional area of the structural element is such that tensile
resistance is less than the force developed over the cross section of the
pressure bulb. Whether a clod will fail in tension depends then on the
magnitude of uR,the tensile strength of the soil uT,and on the size of the
clod. If rupture occurs during radial enlargement rather than during
penetration a cylindrical model should be used.
Local radial cracks may develop either around individual roots or between adjacent root channels (Fig. 2 ) . Using either a spherical or cylindrical model, the tangential stress U t , which reaches a maximum at R,
closely approaches the tensile strength of the soil. Where the plastic zones
of adjacent roots overlap v(Tt is increased, and local rupture is likely to
2. Shear Failure without Compression
The conventional description of forces acting on the base of a pile
or probe (Terzaghi, 1943) shows that the bearing capacity qp of a
shallow ( z = d ) foundation, of depth z and width d, failing in general
shear, is given by
qp = cNc
+ P Z N ,+ pdN,
where c = apparent cohesion, p = bulk density, and N,, N,, Np =
bearing capacity factors.
The values of the bearing capacity factors depend only on the angle
of internal friction, When saturated clays are distorted with negligible
drainage, the strength of the clay is not altered by an applied load since
the load is carried by the pore water (see Section 111, C, 1). Shear
strength is then determined solely by c, and the soil is called a frictionless or = 0 soil. For circular shallow footings in saturated undrained
clay qpz 7.5 c. According to Terzaghi’s model qp increases continuously
with x. This relation applies to rough probes entering saturated “undrained” clays, the requirement of the “undrained condition being met
either because the clay is so impermeable that it fails to consolidate, or
because the rate of loading or penetration is so high that there is time
for only a negligible amount of consolidation.
MECHANICAL RESISTANCE OF SOIL
With the exception of Terzaghi’s analysis for shallow foundations
there are few analyses of general shear failure appropriate to biological
problems. The general shear failure that sometimes occurs above upward
acting penetrometers and seedling shoots is described in an analysis
given by Balla (1961) for the anchorage of mushroomed pylons. Sohtions require the strength parameters c and + and the configuration of
3. Shem Failure with Compression
Where the soil does not behave as an ideal brittle or plastic material,
but is compressed or consolidated during deformation, conventional
theory is inadequate. For deep piles, z > 3d, a “plasticity” theory
modified from that of Terzaghi is usually employed (Meyerhof, 1951).
Although Meyerhof‘s theory implicitly describes local shear failure, as
shearing is depicted as occurring in a localized zone around the base of
the pile, compression is not described explicitly. According to Meyerhof,
for homogeneous saturated clay soils failing without drainage ( 4 = 0),
qp attains a steady maximum at depth where qp = 10 c. Strictly, qpcannot
attain a steady maximum in such materials, because the shearing zone
would have to extend to the full depth of the pile. But real clays are
neither truly saturated nor homogeneous, and in practice the volume of
the pile may often be accommodated locally, for example by displacement
of the clay into cracks or fissures. In compressible soils, following Terzaghi (1943, p.130) an arbitrary reduction is made in c and 4. The bearing capacity factors have been elaborated by Meyerhof (1961) to include
the shape and roughness of the pile. His theory is useful for saturated
clays and for soils having 4 < 35” and failing with little compression.
Since the factors become highly sensitive to changes in for values >
35”,and as a large arbitrary reduction in + must be made in compressible
soils, the theory lacks general utility.
An analysis of the resistance offered to probes in compressible soils
has recently been made by Farrell and Greacen (1966). Following
earlier work on the distribution of stress in soil around holes (de Jong
and Geertsma, 1953) , tunnels ( Terzaghi, 1943), and around piles
(Nishida, 196l), they postulate the existence of two main zones of compression around the point of a penetrating probe: a zone of shearing
failure called the plastic zone, and outside this an elastic zone (see Fig.
3 ) . Farrell and Greacen assume that the pressure on the base of a probe
is equal to the pressure required to form a spherical cavity in the soil.
This approach is not new. Previously Bishop et al. (1945) had used the
model of an expanding cavity in a study of indentation tests in copper.
Ladanyi (1963) used a similar model to describe pile penetration into a
K. P. BARLEY AND E. L. GREACEN
saturated undrained clay, and Nishida ( 1961) calculated the pressure
required to expand a cylindrical cavity in the soil.
The new contribution of Farrell and Greacen is their treatment of
the compressibility of the soil. The analyses of Bishop et al. and Ladanyi
concerned incompressible material. Nishida assumed that the volume
change was determined by the mean principal stress, ( u1
where the subscripts refer to the principal stresses. Vanden Berg et al.
(1958) also used the mean principal stress, but Sohne (1958) used the
major principal stress. Farrell and Greacen largely overcome this ambiguity by using an experimental curve for compression accompanying
FIG. 3. Compression curves ( a ) associated with the zones of compression I-IV
( b ) around the point of a penetrometer in compressible soil: I , e = emin,11, failure
zone, I l l , rebound zone, and lV, elastic zone.
shear failure. In the plastic zone there are three distinct subzones of
compression (Fig. 3 ) : I, where the soil is compressed to the minimum
void ratio’ emin;11, where the soil undergoing failure behaves as a
material being compressed for the first time; 111, a rebound zone where
the soil behaves as an “overconsolidated”material (see Section 111, C, 2 ) .
After equating the change in volume of voids in the various zones
with the volume of the probe, Farrell and Greacen find the radius of the
plastic zone, R, and, knowing this, the pressure qp on the base of a smooth
(frictionless) cylindrical probe. The theoretical value of qp for a smooth
’I t is mathematically convenient to express the state of compaction of the soil as
void ratio, e, rather than bulk density, p . e = p./p - 1, where p . = absolute density
of solid phase. Similarly, volumetric water content, 8 , is conveniently replaced by e ,
and air space, a, by e,.
MECHANICAL RESISTANCE OF SOIL
probe can be checked experimentally by rotating a real probe to dissipate
friction in the tangential direction. When this was done Farrell and
Greacen found good agreement between theoretical and measured values
of qpin a range of finely structured soils.
Ordinarily, friction is mobilized both at the base (“point” friction)
and along the curved cylindrical barrel (“skin” friction) of a probe.
Point friction is appreciable for metal probes in soil. For example, it
increases the value of qp for real as opposed to smooth probes by as much
as 40 percent when the angle of soil-metal friction, 8, = 23” (Farrell and
Greacen, 1966). When the additional expression for point friction is incorporated, the theory of Farrell and Greacen may be used to predict qp
for real, nonrotated probes. The agreement obtained with measured values for steel probes in three soils is shown in Table I (see p. 15).
It seems likely that qP for root tips is less than qp for steel probes, as
an estimate of the friction angle, 6, for the interface between root tips
< Ssteel-soil (see Section
and sand (Barley, 1962) suggests that SrOOt-SO,l
111, A, 4). However no data are available for the immediately relevant
interface between root cap and soil. It is possible that the well known
secretion of mucigel by cells of the root cap is a means of reducing 6.
Recently Farrell and Greacen have extended their theoretical analysis
to include cylindrical enlargement. Surprisingly, when 4 is large, say
40”,the pressure required for the radial enlargement of a cylindrical
cavity is only one-fifth of that required for a spherical cavity. The difference between the two pressures decreases with decreasing values of 4.
Clearly, the shape of a penetrating object may have a large influence on
the resistance encountered in high 4 soils. The cylindrical model is likely
to be more appropriate when the tip is acutely tapered.
4 . Skin Friction
In foundations-engineering the total axial pressure, q, that a pile can
withstand, or, in other words, the axial pressure that has to be applied to
penetrate the soil, is termed the bearing capacity and is given by
where qp = point pressure; qf = axial pressure needed to overcome skin
friction on the curved cylindrical wall of the pile.
Usually adhesion and skin friction are lumped together and estimated
empirically. For rough piles in “undrained clay, skin friction per unit
curved wall area may be s e t equal to c, and the bearing load due to skin
friction Qf = %JOzcrdz, where r is the radius of the pile. For drained
conditions Eide et al. (1961) represent the radial load on the shaft as
Kuz, where a, is the effective axial pressure and K is a coefficient of earth
K. P. BARLEY AND E. L. GREACEN
pressure. Then, Qr = 2 ~ / o x K tan
~ Z r6 dx. For rough piles 6 may be set
equal to 4.
Little is known about the skin friction and adhesion at the interface
between plant organs and the soil. One value of 8, reported for a root“soil” interface, pertains to the root tip of maize and a moistened plate
of cemented sand (Barley, 1962). This value of 6 was obtained directly
by the following method: first, root tips with a flattened “face” were
obtained by pressing roots against the plate as they grew. The tip was
then severed and secured to a slider with small barbs. Finally, the flat
face of the root tip was forced against a portion of the plate mounted on
a friction trolley. The measured value of 8 was 17”.
Recently Barley and Stolzy (1966) used as a crude measure of Qf the
force required to pull out a penetrating root tip. For peas (Pisum
sativum L.) in a moist loam Q, was one-fifth of the total resistance to
penetration Q. The pulling method is used in engineering to measure Q,
for piles, and it is usefuI in clays. In sands the radial pressure on the pile
is relieved by the upward pull and friction is underestimated.
In contrast to piles, where the whole buried length is pushed through
the soil and meets with frictional resistance, in the root only the short
length from the cap to the proximal limit of the zone of elongation is
pushed through the soil. Friction occurs behind the zone of elongation,
but it is mobilized as anchorage to assist penetration, For emerging shoots
the location of the zone of elongation relative to the apex differs widely
between species (Leonhardt, 1915). In many plants an appreciable part
of the shoot is pushed upward through the soil, and skin friction cannot
be safely neglected in any analysis of the resistance opposed to emergence.
Estimates of the mechanical resistance opposed to growth must be
based on knowledge of the type of deformation produced by the plant
root or shoot. The type of deformation determines not only the soil
properties to be measured, but also, as we shall see, the methods to be
used in measurement.
1 . Determinatwn. of Strength Parameters
The parameters that describe the strength of a soil failing by shear
with little or no compression are the classical strength parameters c and
4. The relationship between these parameters and certain derived measures of strength is described diagrammaticaIIy in Fig. 4.For any particular normal load, un, acting on a plane of failure, c and 4 give the shear
strength, sn, according to the Coulomb equation
sn = c
MECHANICAL RESISTANCE OF SOIL
The Mohr circle for the unconfined compressive strength, uc, is shown
in Fig. 4;it can be seen that uc depends on c and 4. Farrell et al. (1967)
have shown that, at pore water pressures as high as -0.3 bar, compact
loams behave as brittle materials, for which uc = Sor (Griffith, 1924).
Where the sample is in the form of a core, either natural or remolded,
FIG.4. Mohr diagram for an unsaturated soil with the failure envelope described
by c and @, u1 and u3 are the principal stresses; in a triaxial test these are the axial
and the radial stresses, respectively. The shear stress 7 = ( uI - u3)/2. Mohr circles
for the compressive strength, uc, and the tensile strength, uT, are also shown.
can be measured indirectly by means of the so-called Brazilian test
(Kirkham et al., 1959) or uC can be measured by an unconfined loading
test. Both tests are performed in a compression test machine; in the
Brazilian test the lateral load required to rupture the core in tension is
measured, and, in the second, the axial load required to rupture the core
in shear is measured.
Rogowski (1964) has pointed out that the above methods measure
bulk strength of the soil and that the bulk strength is usually limited by
the inter-aggregate strength. Rogowski suggests that intra-aggregate
strength may be more important in controlling root penetration, because
the root may often penetrate by deforming the adjacent aggregates rather
than an extensive zone. He proposes that aggregate density be measured,
strength then being determined on cores of soil remolded and compacted
to the measured density. However soil strength is known to depend on
the stress history of the soil, and there is no simple relation between
density and strength (Section 111, C, 2 ) . Rogowski also developed a techUT
K. P. BAFLEY AND
E. L GREACEN
nique for measuring the crushing strength of small ( 2 to 3 mm.) aggregates, by rupturing them in an unconfined compression test between two
plates. He postulates that roots encounter a resistance that depends on
the crushing strength of the aggregates. However, even if this is so, his
analysis is unsatisfactory as it stands because it neglects deformations
that precede and accompany failure of the aggregates.
Rogowski's criticism of the measurement of bulk soil properties hardly
applies when the deformation spreads over a zone that is large compared
with the size of the aggregates, that is, in finely structured soil. In soils
where the aggregates are commensurate in width with the plant organ
concerned, Rogowski's approach may be profitable.
The derived measures: modulus of rupture, the Brazilian test, the
compressive strength, and the crushing strength each give a single Mohr
circle on the strength diagram (Fig. 4 ) . Because of this any one of these
measures provides useful comparative data only where 4 is constant or
almost so. As mentioned in Section 111, A, 2, saturated, undrained clays
behave as if they were 4 = 0 materials. In unsaturated soils or in fully
drained clays 4 usually varies between 20" and 45" (Fountaine and
Brown, 1959), not being greatly affected by changes in void ratio or
pore water pressure. It should be noted, however, that occasionally much
lower values have been reported (Payne and Fountaine, 1952).
A satisfactory characterization of strength for failure with little or no
compression is obtained by describing the failure envelope on a Mohr
diagram with one of the recognized techniques. The torsion shear box
(Payne and Fountaine, 1952) or the direct shear box (Terzaghi and
Peck, 1948) are often employed, the former being useful for small (25
cc.) samples or peds. The most versatile method for soil cores is the
triaxial compression test, a comprehensive account of which is given by
Bishop and Henkel (1962).
Where the deformation involves local shear failure with compression,
analytical estimates of mechanical resistance require the strength parameters c and 4 together with a measured compressibility curve. The compressibility characteristics may be expressed as a Young's Modulus and
as the gradients of the failure and rebound curves for compression with
shear (see Section 111, A, 2). The parameters c and 4 and the compressibility characteristics are equally important in determining the resistance
to penetration. As Farrell and Greacen (1966) have shown they can be
measured with sufficient accuracy by means of the triaxial cell,
No general relation is to be expected between void ratio, e, and the
resistance that soils offer to penetration, Q. When e>>e,,i, for a particular soil most of the volume change occurs in the zone of compression
with failure; as e approaches eminthe rebound zone and the zone of
MECHANICAL RESISTANCE OF SOIL
elastic compression become important. This change of process is responsible for the lack of any general relation.
2. Empirical Measures of Mechanical Resistance
Although empirical measures of mechanical resistance, such as
penetrometer data, contribute little to physical understanding and
provide little scope for generalization, they may be useful in diagnostic
work. As illustrated in Fig. 5 the point resistance, Qp,offered to a probe
RELATIVE DEPTH OF PENETRATION (Z/d)
FIG.5. Fractional point resistance, Qp/Qp mnx, as a function of z/d for a shallow
( z > 3 d ) test in a compressible soil.
( z = d ) and a deep
increases with z to a steady maximum when x exceeds several diameters.
The force required to indent the soil is customarily measured by a shallow test or “indentation” test in which x = d. It can be seen from Fig. 5
that Qp is still increasing rapidly where x = d. This introduces a serious
source of variability in the shallow test, as errors of +2O percent can
easily be made in measuring the depth of penetration of say a 5 mm.
An alternative to penetrometer testing that has been fashionable in
foundations engineering is the vane shear test (Carlson, 1948). This
method was developed initially for saturated clays that behave in rapid
tests as if 4 = 0. Evans and Sherratt (1948) have shown that for 4 < 10”
the vane shear strength can be related to c and +, but for higher values
of the frictional component becomes overriding. No adequate analysis
has been made of the mechanics of the vane test in high 4 soils.
K. P. BARLEY AND E. L. GREACEN
In a recent study emergence of shoots has been related to indentation
test data using downward acting probes (Parker and Taylor, 1965) (see
Section VI, A ) ; but upward acting probes would seem to be preferable in
that the boundary conditions for the test are then more appropriate
(Morton and Buchele, 1!360). Arndt (1965) devised an upward acting
probe for use in the field, the apparatus being buried in the soil before
weathering of the seed bed had taken place, As the use of Arndt’s device
in the field is extremely tedious, simpler methods should be examined.
Bennett et al. ( 1964) measured the force required to pull up a line buried
horizontally in the soil, and showed that the pull was negatively correlated with the emergence of cotton seedlings. A simple empirical test
that is mechanically more apt could be conducted by using a buried bead
several millimeters in diameter and measuring the force needed to pull
this from the soil with a fine wire.
Although cylindrical probes provide a relative measure of resistance
to penetration, and are useful in correlative studies (see Section VI, B, 2 ) ,
probe data should not be identified with the absolute resistance encountered by growing organs. Discrepancies arise for many reasons; the chief
reasons are as follows: ( 1) Growing organs are flexible and tend to grow
around obstructions. ( 2 ) The shape of plant organs differs from that of
cylindrical probes; moreover the shape is influenced by the resistance of
the soil. ( 3 ) The stress distribution around a plant organ, unlike a rigid
body, depends not only on its shape and on the soil properties, but also
on the anisotropic properties af the tissue. (4)Friction and adhesion at
the interface between plant and soil may differ from that between probe
and soil. (5) Uptake of water by roots causes local changes in the pore
water pressure and hence in the strength of the soil. ( 6 ) In saturated soils
the root creates additional opportunities for drainage.
The biological aspects will be further explored in Section VI, A. Unless the differences between probes and plant organs are understood we
cannot hope to relate theoretical or measured values of Q to the mechanical resistance experienced by roots or shoots.
C. THEEFFECTOF POREWATERPRESSURE
AND VOID RATIO
The data in Table I provide a clear illustration of the extreme dependence of qp on pore water pressure, uw,and void ratio, e. It is worth noting
that the strength of unsaturated soils can change considerably even when
there is little change in the water content; indeed the change in strength
is most rapid when the water-filled void ratio, e,, is appreciable and the
gradient de,/du, is small. Note, for example, that for the Parafield loam
described in Table I, at e = 0.56, qp increases from 20 to 34 bar when
MECHANICAL RESISTANCE OF SOIL
Comparison of Theoretical with Measured Values of Point Pressure ( q p )
for Steel Probes in Three Soils
Pore water pressure uul = -h, where h is the suction in the soil water, both uwand
h being referred to atmospheric pressure as datum. It is more convenient to employ uw
in mechanical studies, as pressures above and below the datum exist simultaneously in
different parts of the soil-plant system.
urnis decreased from -0.3 bar to -0.7 bar, the decrease in e , being
1. Pore Water Pressure and Effective Stress
In a saturated soil a decrease in u, has the same effect on strength as
an increase of equal magnitude in the externally applied pressure
(Childs, 1955). Skempton (1960) has discussed the effect of amon the
strength of saturated soils from the engineering point of view, and should
be consulted for a more detailed account.
Terzaghi (1923) showed experimentally that for a saturated soil the
degree of unidirectional consolidation depended on the “effective” stress,
d,defined as d = u - urn,where u is the applied normal stress. Similarly
the bulk modulus, p, of a saturated soil experiencing isotropic compression is given by p = dp’/dc, = d ( p - u,) /da,, where p and p’ are the
applied and effective pressures and E, is the cubical dilation. Generally,
if c and 9 had been defined in this review as intrinsic properties of the
soil at datum pressure, effective rather than applied stresses would have
had to be substituted in equations such as (5) that contain c or $. In
practice it is often more convenient to work in terms of applied stresses
and use apparent values of c and 0 obtained under conditions of testing
K. P. BARLEY AND E. L. GREACEN
(drainage, rate of deformation) that pertain to the deformation being
studied. For example, if mechanical properties are to be related to root
penetration, tests should be conducted with full drainage at low rates of
deformation (slow drained tests).
In unsaturated soil, where the pores contain both air and water, the
pore water pressure is regarded as acting over an effective area x per unit
area of the soil. The effective stress is then given as
u’ = (u - xuw)
When the soil is saturated x = 1 and Eq. ( 6 ) may be identified with
Terzaghi’s definition given above. Bishop ( 1960) shows experimentally
that x is a nonlinear function of the degree of saturation. The function
exhibits hysteresis and depends on the stress history of the soil. Bishop’s
relations between uw and effective stress are satisfactory where U, is held
constant during deformation, or alternatively where the volume of soil
being strained is so small relative to the bulk of the sample that uw is
buffered by internal drainage. However, where the bulk of the soil is
deformed, as in most testing procedures, u, may differ markedly from the
initial pressure, particularly if the test is rapid or the moisture conductivity is low. Croney and Coleman (1954) show that in undrained saturated soils uw changes considerably with the degree and rate of straining.
Greacen (1960) and Bishop (1960) extended this result to unsaturated
soils. Again, where the deformation involves compression, the influence
of uw on compressibility must be taken into account by measuring the
compressibility curves at a number of initial water contents (Farrell and
In addition to changes in uw arising from deformation of the soil, we
have to remember that the transpiring plant can transmit large suctions
to the soil water. The probable magnitude of gradients in uwaround roots
arising from transpiration is shown, for example, by Gardner (1960). As
the elongating tip of the root is permeable (Rosene, 1937), the tip
presumably takes up water together with the proximal parts of the root.
Indeed the local decrease in uw due to transpiration may often be more
significant than the change associated with deformation.
2. Void Ratio
Although it is obvious that compact soils are hard to deform, failure
to appreciate the nature of the relation between void ratio and penetrability has hindered progress. Veihmeyer and Hendrickson ( 1948) proposed that the inability of roots to penetrate particular soils below a
certain critical void ratio could be attributed to the lack of pores of sufficient width. It is now recognized that the mechanical resistance of the