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Empirical Models for the Strength and Deformation of Granular Media 1 Shear Strength of Granular Media



2.4.8. Empirical Models for the Strength and Deformation of Granular Media 1 Shear Strength of Granular Media

rough surfaced particles when compared with worn railway ballast. In addition they showed that the fine-grained, hard-mineral aggregates were associated with lower plastic deformations, results confirmed later by Greene (1990). Thom and Brown (1989) arrived at a similar conclusion that crushed road base displayed lower plastic deformation than that comprised of naturally occurring sands and gravels.

Later, research by Kolisoja (1997) showed that the resilient modulus was higher for crushed rock in comparison with crushed gravel and natural gravel, provided that the gradation and the tests conditions were the same. This agreed with earlier reports by Yeaman (1975), Thom and Brown (1988, 1989) and Thompson (1989), which showed that the stiffness of specimens increased with increasing angularity of the grains.

Furthermore, Brown and Selig (1991) correlated the increase in surface friction between angular particles with the increase in the stress ratio at failure and higher magnitude for the resilient modulus. Using this observation, Zaman et al. (1994) quantified the effect of the angle of internal friction together with other factors on the resilient modulus. In addition, Zaman et al. (1994) reported that, provided that the gradation and test conditions were the same, a significant increase (20-50%) of resilient modulus was observed as the type of material varied, ie. the lowest values were recorded for sandstone gravel, and the highest for limestone gravel with granite and rhyolite displaying intermediate values. Kolisoja (1997) confirmed that the rock type had a significant effect on the resilient characteristics of the material.

2.4.8. Empirical Models for the Strength and Deformation of Granular Media

Chapter 2: Critical review of granular media with special reference to railway ballast

variation of internal friction angle with the logarithm of normal stress based on data from plane strain and triaxial tests on rockfill. Later, Wong and Duncan (1974) proposed a relationship that accounts for the variation of angle of internal friction of rockfill with the stress level:


⎜⎜ ⎞

⎝ ⎛ ′ φ′

Δ φ′

= φ′

a o n

p log σ

- (2.14)

where σ′n is the normal effective stress, φ′o is the φ′ value corresponding to σ′n equal to the atmospheric pressure and Δφ′ is the reduction in φ′ measured from the log σ′n - φ′


DeMello (1977) and Charles (1980) showed that a power relationship between normal stress and shear stress accounted better for the curvature of shear strength envelope of various geo-materials. Its general form is given by:

( )

n B1

1 f =A σ

τ (2.15)

where σn is the normal stress, and A1 and B1 are coefficients accounting for the material characteristics. It should be noted that A1 depends on the system of units of stress and its dimensions vary according to the value of B1.

Later, Indraratna et al. (1993) proposed a non-dimensional failure criterion to eliminate these drawbacks by normalizing both terms of Eq. (2.15) with the uniaxial compressive strength (σc) as follows:


c n c 2

f A ⎟⎟⎠

⎜⎜ ⎞

⎛ σ

= σ σ

τ (2.16)

where the constants A2 and B2 are dimensionless, A2 encompasses the equivalent friction angle and can be regarded as an intrinsic shear strength index and the magnitude of B2 dictates the non-linearity of the failure envelope, particularly at low confining stresses. B2 therefore represents the deformation response of the rockfill, including to some extent the effect of dilatancy and particle sizes. It was also shown that this criterion could describe the behaviour of various rockfill for a wide range of confining pressure (0.1 - 8 MPa). Considering the earlier reports from tests on railway ballast that the failure envelope exhibits a pronounced curvature, it is possible to describe its behaviour to failure (under monotonic loading) by employing this failure criterion.

Earlier reports by Rowe (1962) and Bishop (1966), correlated the significant increase in the measured angle of internal friction at low stresses to the interlocking effect associated with the tendency to dilate. Lee and Seed (1967), using studies on sands having different grain shape and initial relative density, presented the effect of stress level on shear strength in a schematic form (Fig. 27). The sliding friction, dilatancy, crushing and rearranging of the grains were identified as the contributing factors to the shear strength of fine grain cohesionless materials at low level of confining pressure.

Koerner (1968) reached the same finding for coarse grain particulate media having compact grains. Nevertheless, at higher confining pressures breakage of the grains becomes more significant and causes a decrease in the angle of internal friction (Bishop, 1966; Vesic and Clough, 1968).

Chapter 2: Critical review of granular media with special reference to railway ballast

Figure 27. Schematic illustration of contribution of sliding friction, dilatancy, remoulding and crushing to the measured Mohr envelope for drained tests on sand

(after Lee and Seed, 1967)

Efforts have been made to separate the frictional component from the volume change component (dilatancy) (Newland and Alley, 1957; Ladanyi, 1960; Rowe, 1962, 1971;

Bolton, 1986). Newland and Alley (1957) and Ladanyi (1960) reported that for fine grained soils the rate of dilatancy decreased as the confining pressure increased. Rowe (1962) and Biship (1966) found that for sand there is a linear variation of dilatacy factor at failure (Dp= 1-dεv/dεa) with the peak principal stress ratio (Rp = (σ′1/σ′3)p). Later, . Rowe (1971) showed that Dp tends towards a limit value of 2 at low level of confining pressure (Rowe, 1971). Raymond et al. (1976) reported, from a series of triaxial tests on coarse grain uniform railway ballast, that the rate of dilatancy at failure decreased as the maximum principal stress ratio increased, however, no relationship was presented.

Bolton (1986) proposed a relative dilatancy index which could predict the behaviour of sand at failure. To describe the dilatancy behaviour of dense sand when subjected to

low range of stress, he made use of a saw-tooth analogy as presented in Figure 28. He suggested that the maximum angle of shearing resistance could be estimated from:

( )


10 lnp 1


3 D


max−φ′ = Ι − ′ −

φ′ (2.17)

where φ′max and φ′cs are the maximum and the critical state angles of shearing resistance respectively measured in degrees, ID is initial relative density and p′ is the mean effective stress at failure, measured in kPa. While this model fits well a wide range of data from published research on sand, it does not fully describe the behaviour of coarse-grained particulate media which are affected by degradation even at lower levels of stress. Also, due to the significant difference in the grain size from sand to railway ballast the application of this failure criterion might require extensive modifications.

After reviewing existing failure criteria for granular materials, Maksimovic (1996) pointed out their limitations as follows:

• power type expressions are valid in a limited stress range (40-400 kPa);

• the semi-logarithmic plot assumes the validity of the relationship over a rather wide stress range, and requires a sharp cut-off at an elevated level of normal stress where the dilatancy effect is suppressed.

Therefore, he proposed a hyperbolic type of function to interpret the results from conventional triaxial compression tests:

TX 3 B

s B

1 Pσ′

+ φ′

+ Δ φ′


+ Δ φ′


φ′ (2.18)

Chapter 2: Critical review of granular media with special reference to railway ballast

Figure 28. The sawtooth model of dilatancy (after Bolton, 1986)

where φ′s is the secant angle (from a tangent to a single Mohr circle that passes through the origin), φ′B is the basic angle of friction, Δφ′ is the maximum angle difference, σ′3 is the effective confining pressure and PTX is the normal stress corresponding to the median angle, as illustrated in Figure 29.

While for rockfill and sand there is a general agreement that the crushing process is a significant phenomenon that should be accounted for, there appears to be no agreement in accepting the same concept for railway ballast. For practical purposes the straight Mohr-Coulomb envelope with an associated ‘apparent cohesion’ is extensively used (Raymond and Davies, 1978; Siller, 1980; Chrismer, 1985; Kay, 1998). Though, this is a reasonable approximation of the state of stress at failure for only a limited range of effective stresses, and the term ‘apparent cohesion’ is in contradiction with the definition of cohesionless materials. Nevertheless, Kay (1998) reviewing published data from triaxial tests on different types of railway ballast, emphasized that it is because of this ‘apparent cohesion’ that angular materials with high dilatancy tendencies are used in railway tracks to provide lateral support. Plastic Deformation

Initial investigations on the cyclic behaviour of railway ballast were carried in the

(a) (b) Figure 29. Variation of secant angle of shearing resistance φ’

(after Maksimovic, 1996)

conventional (cylindrical) triaxial equipment at constant confining pressure. It was generally agreed that the permanent strain after N cycles, εPN, is related to the permanent strain after one cycle ε1 by a logarithmic relationship (ORE, 1970; Siller, 1980; Ford, 1995; Kay, 1998):


1 A logN



εPN = N=1 + ′ (2.19)

where A′ is a dimensionless parameter controlling the rate of growth of deformation, with typical values between 0.19 to 0.4 depending on ballast type. Later, ORE (1971, 1974), proposed a relationship that accounts for the initial porosity of specimens and the repeated deviator stress on the permanent strains of railway ballast. From controlled conditions in the laboratory the following relationship was proposed:

N) log 0.2 (1 ) (q 38.2) n


εPN =0.82 − r C − (2.20)

where N is the number of load applications, C is a coefficient depending on the level of

Chapter 2: Critical review of granular media with special reference to railway ballast

applied stress (which ranges from 1 to 2 for low stress levels, but may reach 3 for high levels of stress), and qr = σ13 is the repeated deviator stress where σ1 and σ3 represent the repeated vertical stress and the constant confining pressure, respectively.

Alva-Hurtado and Selig (1979, 1981) discussing the cyclic triaxial tests on granite ballast (Dmax = 30 mm and D50 = 20 mm) claimed that, independent of the state of stress and degree of compaction of the specimen, the following non-linear relationship better predicts the permanent strains (εPN) after N load applications:

2 1 N 1


PN (0.85 0.38 logN)ε (0.05 0.09 logN)ε

ε = + = + + = (2.21)

Alva-Hurtado and Selig (1981) also reported that there is a good agreement between the plastic deformation measured after the first cycle (ε1) and the permanent deformation monitored during monotonic loading tests, confirming earlier observations reported by Barksdale (1972), Monismith et al. (1975) and Knutson (1976). It was suggested that the strain after the first load cycle (ε1) could be approximated from the (axial) plastic strain (εa) measured during monotonic loading tests for the same stress level.

Janardhanam and Desai (1983) used a similar method to describe the behaviour of granite ballast subjected to true triaxial tests (σ2≠σ3).

In early 1960, Japanese railway companies published a relationship that enabled the estimation of the settlement of railway ballast when subjected to cyclic loading.

Originally developed from laboratory results (Okabe, 1961; Sato, 1962), the following equation is currently used to estimate the deformation (settlement) of both heavy haul narrow gauge and high speed standard gauge (Sato, 1995):


1 e



sN =χ − αN + (2.22)

where N is the repeated number of loading or tonnage carried by the track, α is the vertical acceleration required to initiate slip and can be measured using spring loaded plates of the ballast material on a vibrating table, β is a coefficient proportional to the sleeper pressure and peak acceleration experienced by the ballast particles and is affected by the type and conditions of the ballast material and the presence of water, and χ is a constant dependent on the initial packing of the ballast material. The first term

represents the initial rapid settlement and corresponds to the process in which the gaps between ballast particles are closed and the ballast consolidates, whilst the second term expresses linear settlement after this and corresponds to the lateral movement (flow) of ballast particles under the sleepers. Unfortunately, Sato (1995) has provided no information on the values of the constants used in this equation, the units of the variables and the derivation of the formula.

After analysing an extensive database of worldwide field measurements, Shenton (1985) concluded that Eq. (2.19) gives a reasonable approximation over a short period of time but a significant underestimation can occur for large number of axle loadings (above 106 load cycles). Therefore, a relationship was proposed that reasonably estimates the short-term and long-term track settlement, and is given by:


sN = 1 0.2+ 2 (2.23)

where sN is the settlement, and C1 and C2 represent material constants. The C1N0.2 term predominates up to 106 load cycles, whereas C2N term becomes relatively

Chapter 2: Critical review of granular media with special reference to railway ballast

significant only above this value. However, Kay (1998) argued that due to the degree of inconsistency and scatter of the acquired data, Eq. (2.23) could, at best, approximate the behaviour of ballast in any particular case.

Jeffs and Marich (1987) used a semi-confined device to investigate the behaviour of various types of railway ballast under cyclic load. Load of various intensities and frequencies was applied through a rigid plate on the loosely placed ballast. They reported that independent of different test variables, a linear variation of settlement with the number of load cycles was observed beyond 200000 cycles. Although a linear variation conflicts with predictions of Eq. (2.19), it confirms trends reported earlier by other researchers (Okabe, 1961; Sato, 1962; Shugu, 1983; Shenton, 1985).

Selig and Devulapally (1991) analysed the results from box tests on dry dolomite ballast subjected to large numbers of cycles and compared them with settlement trends reported earlier by Selig et al. (1981) from field observations. They agreed with Shenton (1985) and Jeffs and Marich (1987) observation that the semi-log relationship increasingly underestimates the cumulative plastic strain as the number of cycles increased.

However, they showed that a power relationship (as given below) better estimates both laboratory and field measurements:

f 1

N s N

s = (2.24)

where s1 is the settlement recorded after first load application and f is a material parameter. The power relationship was in good agreement with earlier reports by Schultze and Coesfeld (1961) and Morgan and Markland (1981) from plate-loading

tests and by Diyaljee and Raymond (1982) from drained triaxial tests on both sand and railway ballast. In spite of the lack of agreement with regard to the trends reported, which might be due to other test conditions, one point should be made, that the semi-confined devices employed by Jeffs and Marich (1987) and Selig and Devulapally (1991) allowed ballast movement (lateral flow) from under the load applicator providing a closer simulation of the in-track conditions.

Paute et al. (1993) using results reported by Pappin (1979) proposed a relationship between the axial strain rate and the number of cycles to characterize deformation of granular materials subjected to repeated loads. The relationship that considers only the plastic strains is commonly used in France and its expression is given by:

N ln h ε g

ln&Pa = + (2.25)

where ε&Pa is the rate of axial plastic strain per cycle, and g and h are material coefficients. Tests results presented by Pappin (1979) were quite variable for the first 100 cycles and were regarded as a settling-in phase of the tests. Therefore, Paute et al.

(1993) presented an equation to estimate the plastic strain developed after 100 cycles as follows:



⎡ ⎟

⎜ ⎞

= ⎛ -i

PNa 1

100 - N 1

ε c (2.26)

where εPNa it the plastic strain for N > 100 load cycles, i is a material coefficient and the parameter c1 was related to the peak applied stress ratio as follows:

Chapter 2: Critical review of granular media with special reference to railway ballast


= η k

-c1 j (2.27)

in which j and k are material constants related to the stress ratio at failure k


f =

η ,

similar to the hyperbolic relationship proposed by Lentz and Balady (1980) for sands.

Equation (2.27) implies that as η approaches failure, c1, and therefore the accumulated strain, becomes very large. Elastic Deformation

As discussed before, the researchers are in agreement about the fact that granular materials tend to a quasi-elastic state after a number of load repetitions. However, in the literature there are conflicting opinions upon the number of load repetitions required to reach the resilient state. Hicks and Monismith (1971) and Kalcheff and Hicks (1973) reported that well-graded crushed limestone (Dmax = 40 mm) needed only a few hundred cycles to arrive at constant elastic deformation. Profillidis (1995) stated that the magnitude of the modulus of elasticity after 1000 cycle of loading had doubled from that after first cycle, and did not change any further. Based on the results on partially saturated well-graded crushed granite (Dmax = 5 mm) Brown (1974) argued that several thousand load repetitions are necessary to reach constant values of the resilient modulus. Furthermore, work on railway ballast by Shenton (1975) and Alva-Hurtado (1980) suggested that the magnitude of the resilient modulus gradually increased with the number of cycles.

Commonly the interpretation of resilient deformations uses the resilient modulus (Er), which is defined by the equation:


r ε

E = qr (2.28)

where qr = (σ13) is the cyclic axial load and εr is the recoverable part of axial strain.

While the plastic strains decreased with each load cycle, the resilient modulus increased, as presented in Figure 30.

It is well established that, provided that shear failure does not occur, the resilient modulus can be related to the horizontal stress (confining pressure) as reported by Hicks (1973), or to the sum of the principal stresses (Hicks and Monismith, 1971; Kalcheff and Hicks, 1973; Brown and Pell, 1975) as given below:

( )


r k1

E = θ′ (2.29)

where θ′ = σ′1 + σ′2 + σ′3 is the effective bulk stress, and k1 and k2 are material constants. This equation, known as the k-θ model, is the simplest and whilst it is widely used it has proven to have some limitations. Firstly, the equation is dimensionally unsatisfactory and Hicks (1973) attempted to eliminate this by normalizing the equation with reference to atmospheric pressure (pa). Secondly, Shackel (1973) and Brown (1974) indicated that the level of shear stress affected the relationship between Er and θ. Studies by Boyce et al. (1976), Cole et al. (1981) and Johnson et al. (1986) reported that the relationship also changed with the stress ratio.

Brown (1996) suggested that Eq. (2.29) should rather be regarded as a lower bound to experimental data, as proposed earlier by Uzan (1985) and Nataatmadja (1994).

Chapter 2: Critical review of granular media with special reference to railway ballast

Figure 30. Definition of resilient modulus

Selig and Waters (1994), based on research by Stewart (1982), offered an alternative equation for the calculation of the resilient modulus:

( )

re rc

re r εrc

q -q

E = 2 (2.30)

where qrc is the shear stress in compression, qre is the shear stress in extension, εrc is the compression resilient strain and εre is the extension resilient strain given by the following expressions:



( )

3 -1.15



rc 0.00087q σ

ε = ′ (2.31)

( )

3 -1.52

re re 0.00143q σ

ε = ′ (2.32)

where d1=0.924

( )

σ3 0.088, σ′3 is the constant effective confining pressure for cyclic test and the stresses are measured in kPa. Considering that Eq. (2.30) was developed

from results on granite ballast, its application to other types of ballast may require some modifications to fit the results. Also, it should be observed that it does not account for the effect of the degree of unloading. However, if the resilient properties are defined in terms of the parameter Er, then a corresponding value for Poisson’s ratio is required, which should itself be stress-dependent as noted by Sweere (1990).