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III. Forces Required to Deform Soils

III. Forces Required to Deform Soils

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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.





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

the system.

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




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

+ +


U,. (bar)


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,.



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






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








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




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



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








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.

diameter probe.

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.




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.





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




Comparison of Theoretical with Measured Values of Point Pressure ( q p )

for Steel Probes in Three Soils

Void ratio


l’arafield loam

Pore water





Urrbrae loam


Coleraine clay





Total e




















































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

only 0.04.

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



(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

Greacen, 1966).

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

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III. Forces Required to Deform Soils

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