Scale effects

JCS=6 9#10^{^0 0087. rR Ln]g+0 16. h} (eqn 5.33) where JCS is in MPa units, r is the rock density in g/cm³ units and R_n(L) is the rebound number of the L-type Schmidt hammer. Caution is suggested when using this correlation due to the large dispersion of values commonly found. There are several correlations between the uniaxial compressive strength of rock and the Schmidt hammer rebound number (see Zhang 2005). Alternatively, the ISRM empirical field estimates of s_c shown in Table 2.3 can be used.

5.3.2.6 Scale effects

Although discussions about the effects of scale on the shear strength of defects as defined by the Mohr-Coulomb

failure criterion (c_j and f_j) are limited, the available data indicates that:

■ laboratory tests frequently overestimate the shear strength of discontinuities, especially the cohesion;

■ the results of several back analyses of structurally controlled instabilities indicate that the peak shear strength of clean structures with sound hard rock walls, at scales from 10–30 m and in a low

confine-Table 5.12: Typical values of the basic friction angle, f_b, for some rock types

Rock type f_b dry f_b wet Rock type f_b dry f_b wet

Amphibolite 32° Granite, fine-grained 31–35° 29–31°

Basalt 35–38° 31–36° Granite, coarse-grained 31–35° 31–33°

Chalk 30° Limestone 31–37° 27–35°

Conglomerate 35° Sandstone 26–35° 25–34°

Copper porphyry 31° Schist 27°

Dolomite 31–37° 27–35° Siltstone 31–33° 27–31°

Gneiss, schistose 26–29° 23–26° Slate 25–30° 21°

Source: Data from Barton (1973), Barton & Choubrey (1977)

Table 5.13: Defect roughness profiles and associated JRC values

Source: Modified from Barton & Choubray (1977)

Table 5.14: ISRM-suggested characterisation of defect roughness

Class

Scale

Typical roughness profile JRC20 JRC100

Intermediate Minor

I Stepped Rough 20 11

II Smooth 14 9

III Slickensided 11 8

IV Undulating Rough 14 9

V Smooth 11 8

VI Slickensided 7 6

VII Planar Rough 2.5 2.3

VIII Smooth 1.5 0.9

IX Slickensided 0.5 0.4

Notes

The length of the roughness profiles is intended to be in the range of 1–10 cm The vertical and horizontal scales are identical

JRC₂₀ and JRC100 correspond to joint roughness coefficient when the roughness profiles are ‘scaled’ to a length of 20 cm and 100 cm respectively Source: Modified from Brown (1981) and Barton & Bandis (1990) by Flores & Karzulovic (2003)

Table 5.15: Estimating JRC from the maximum unevenness amplitude and the profile length

0.1 0.2 0.3 0.5 0.8 1 2 3 5 8 10 Profile Length (m)

Uneveness Amplitude (mm)

1 PROFILE LENGTH (m)

1 UNEVENESS AMPLITUDE (mm)

PROFILE LENGTH (m)

400 300 200

100 80 50

30 20

10 8 5

3 2 0.81

0.5

0.3 0.2 0.1

20 16 12 108 65 4 3 2 1 0.5

Joint Roughness Coefficient, JRC

Source: Barton (1982)

Table 5.16: Estimating the uniaxial compressive strength, s_c, of the defect rock wall from Schmidt hardness values

Source: Hoek (2002)

ment condition (the predominant condition in the benches of an open pit mine) is defined by nil to very low values of cohesion and friction angles in the range of 45–60°;

■ at low confinement and scales from 50–200 m, struc-tures with centimetric clayey fillings have typical peak strengths characterised by cohesions ranging from 0–75 kPa and friction angles ranging from 18–25°;

■ at low confinement and scales from 25–50 m, sealed structures with no clayey fillings have typical peak strengths characterised by cohesions ranging from 50–150 kPa and friction angles ranging from 25–35°.

Both JRC and JCS values are influenced by scale effects and decrease as the defect size increases. This is because small-scale roughness becomes less significant compared to the length of a longer defect and eventually large-scale undulations have more significance than small-scale roughness (Figure 5.24).

Bandis et al. (1981) studied these scale effects and found that increasing the size of the discontinuity produces the following effects:

■ the shear displacement required to mobilise the peak shear strength increases;

■ a reduction in the peak friction angle as a consequence of a decrease in peak dilation and an increase in asperity failure;

■ a change from a brittle to a plastic mode of shear failure;

■ a decrease of the residual strength.

To take into account the scale effect Barton and Bandis (1982) suggested reducing the values of JRC and JCS using the following empirical relations:

JRC JRC L

L ^{. /}

F O

O F

JRC 0 02 _O

=

-e o (eqn 5.34)

JCS JCS L

L ^.

F O

O F

JRC 0 03 _O

=

-e o (eqn 5.35)

where JRC_F and JCS_F are the field values, JRC_O and JCS_O are the reference values (usually referred to a scale in the range 10 cm–1 m), L_F is the block size in the field and L_O is the length of reference (usually 10 cm–1 m).

These relationships must be used with caution because for long structures they may produce values that are too low. Ratios of JCS_F /JCS_O < 0.3 or JRC_F /JRC_O < 0.5 must be considered suspicious unless there are very good reasons to accept them.

The Barton-Bandis strength envelopes for

discontinuities with different JRC values are shown in Figure 5.25, which also shows the upper limit for the peak friction angle resulting from this criterion.

From Table 5.14, the following values can be assumed as a first estimate for the joint roughness coefficient:

Figure 5.24: Summary of scale effects in the shear strength components of non-planar defects. f_b is the basic friction angle, d_n is the peak dilation angle, s_a is the strength component from surface asperities, and i is the roughness angle

Source: Bandis et al. (1981)

Figure 5.25: Barton-Bandis shear strength envelopes for defects with different JRC values

Source: Modified from Hoek & Bray (1981)

■ rough undulating discontinuities: JRC ≈ 15–20

■ smooth undulating discontinuities: JRC ≈ 10

■ smooth planar discontinuities: JRC ≈ 2 5.3.2.7 Stress, strain and normal stiffness

Numerical slope stability analyses require, in addition to the strength properties, the stress-strain characteristics of defects. Detailed discussions on the stress-strain behaviour of defects can be found in Goodman (1976), Bandis et al. (1983), Barton (1986), Bandis (1993) and Priest (1993).

The loading of a discontinuity induces normal and shear displacements whose magnitude depends on the stiffness of the structure, defined in terms of a normal stiffness, k_n, and a shear stiffness, k_s. These refer to the rate

of change of normal (s_n) and shear (t) stresses with respect to normal (v_c) and shear (u_s) displacements (Bandis 1993):

d d

k k

dv du 0

n n 0

s c s

st == G

' 1 ) 3 (eqn 5.36)

where:

kⁿ v_c

n u_s

2 2s

=_{f p} (eqn 5.37a)

k^s u_{s v} 2 c

2t

=_{d n} (eqn 5.37b)

Therefore, a discontinuity subjected to normal and shear stresses will suffer normal and shear displacements that depend on the following factors:

■ the initial geometry of the discontinuity’s rock walls;

■ the matching between the rock walls, which defines the variation of the aperture and the effective contact area (Figure 5.26);

■ the strength and deformability of the rock wall material;

■ the thickness and mechanical properties of the filling material (if any);

■ the initial values of the normal and shear stresses acting on the structure.

It is assumed that the defect cannot sustain tensile normal stresses and that there will be a limiting compressive normal stress beyond which the defect is mechanically indistinguishable from the surrounding rock (Figure 5.27).

Figure 5.26: Examples of discontinuities with matching and mismatching rock walls

Source: Flores & Karzulovic (2003)

Figure 5.27: Determination of the normal stiffness of an artificial defect by means of uniaxial compression tests on specimens of granodiorite with and without a discontinuity. (a) Normal stress-total axial displacement curves. (b) Normal stress-discontinuity closure curves

Source: Goodman (1976)

The normal stiffness of a defect can be measured from a compression test with the load perpendicular to the discontinuity (Goodman 1976), or from a direct shear test if normal displacements are measured for different normal stresses (Figure 5.12). The following comments can be made.

1 Normal stiffness depends on the rock wall properties and geometry, the matching between rock walls, the filling thickness and properties (if any), the initial condition (before applying a normal stress increment), the magnitude of the normal stress increment and the number of loading cycles.

2 Generally, normal stiffness is larger if the rock wall and filling material (if any) are stronger and stiffer.

3 For a given set of conditions, normal stiffness is larger for defects with good matching than for mismatching ones.

4 Normal stiffness increases with the number of loading cycles. Apparently, the increment is larger in the case of stronger and stiffer rock walls.

5 The values quoted in the geotechnical literature indicate that normal stiffness ranges from 0.001–

2000 GPa/m. It typically takes the following values:

→ defects with soft infills: k_n < 10 GPa/m;

→ clean defects in moderately strong rock: k_n = 10–50 GPa/m;

→ clean defects in strong rock: k_n = 50–200 GPa/m.

The normal stiffness of a defect increases as the defect closes when s_n increases, but there is a limit that is reached when the defect reaches its maximum closure, v_cmax. Assuming that the relationship between the effective normal stress, s_n, and the defect closure, v_c, is hyperbolic (Goodman et al. 1968) it is possible to define the normal stiffness (Zhang 2005):

where k_ni is the initial normal stiffness, defined as the initial tangent of the normal stress-discontinuity closure curve (Figure 5.29). As the defect’s tensile strength is usually neglected, k_n = 0 if s_n is tensile.

Hence, to determine the normal stiffness of a defect it is necessary to know the initial value of this stiffness and the defect’s maximum closure. From experimental results, Bandis et al. (1983) suggested that k_ni for matching defects can be evaluated as:

. . . coefficients of the Barton-Bandis failure criterion and e_i is the initial aperture of the discontinuity, which can be estimated as:

For the case of mismatching structures, Bandis et al.

(1983) suggested the following relationship:

. . where k_ni,mm is the initial tangent stiffness for mismatching defects. Regarding the scale effect on the normal stiffness, it can be implicitly considered by using ‘scaled’ values for JRC and JCS, and an ‘adequate’ value for e_i. Although these relationships have several limitations there are few

practical tools to estimate k_n. Some reported values for the normal stiffness of discontinuities are listed in Tables 5.17 and 5.18.

Figure 5.28: Definition of k_n and k_ni in an effective normal stress-discontinuity closure curve

Shear displacement, u_s

sn

Shear displacement, u_s nn

Figure 5.29: Determination of secant peak shear stiffness of a defect from a direct shear stress

Source: Goodman (1970)

Table 5.17: Reported values for normal stiffness for some rocks

Fresh to slightly weathered,

good matching of rock walls 1 4–23 s_ni = 1 kPa Bandis et al.

(1983)

2 11–35

3 18–62

Moderately weathered, good matching of rock walls

1 4–26

2 9–27

3 15–45

Weathered,

good matching of rock walls 1 2–5

2 9–14

3 11–20

Shear zone with clay gouge 1.7 Estimated from data in

reference, assuming a 3 cm thickness

Wittke (1990)

Bedding planes, good matching (JRC = 10–16)

13–24 Direct shear tests with s_n ranging from 0.4–0.9 MPa

Rode et al. (1990) Bedding planes, good matching

(JRC = 10–16) 7–12

Fresh fractures, good matching

(JRC = 12–17) 17–25

Fresh fractures, poor matching (JRC = 12–17)

8–12

LIMESTONE

Fresh to slightly weathered,

good matching 1 8–31 s_ni = 1 kPa Bandis et al.

Weathered, good matching 1 4–13

2 40–50

3 42–65

Joints in weathered limestone 0.5–1.0 s_n = 5 MPa Bandis (1993)

Joints in fresh limestone 4–5

QUARTZITE Clean 15–30 s_n = 10–20 MPa Ludvig (1980)

With clay gouge 10–25

DOLERITE

Fresh, good matching 1 21–27 s_ni = 1 kPa Bandis et al.

(1983)

2 59–75

3 103–119

Weathered, good matching 1 8–13

2 24–92

3 37–130

GRANITE

Clean joint (JRC = 1.9) 1 121 Estimated from ref.

Biaxial tests s_n : 25–30 MPa

Makurat et al.

(1990)

Clean joint (JRC = 3.8) 1 74

Clean joint 352–635 Mes. Sist. Pac-ex.

s_n: 8.6–9.3 MPa

Martín et al. (1990) 50–110

Shear zone 2–224 Mes. Sist. Pac-ex.

s_n: 0.5–1.5 MPa 7–266 Mes. Sist. Pac-ex.

s_n : 18–20 MPa k_n = Normal stiffness

s_n = Normal stress k_ni = Initial normal stiffness s_ni = Initial normal stress

Pac-ex: Measured by the system Pac-ex, a special instrumentation system developed in the Underground Research Laboratory by Atomic Energy of Canada Ltd.

Source: Flores & Karzulovic (2003)

Table 5.18: Reported values for normal stiffness for some rocks Rock Discontinuity

Load cycle

k_ni (GPa/m)

k_N

(GPa/m) Comments Reference

SILTSTONE

Fresh, good matching 1 14–26 s_ni = 1 kPa Bandis et al. (1983)

2 22–64

3 22–70

Moderately weathered, good

matching 1 10–11

2 20–22

3 20–26

Weathered, good matching 1 7–14

2 27–29

3 29–41

QUARTZ MONZONITE Clean 15.3 Triaxial testing (?) Goodman &

Dubois (1972)

PLASTER Clean, artificial fractures 2.7–5.4 s_n: 3.5–24 MPa Barton (1972)

Clean, artificial fractures 2.7 Karzulovic (1988)

SLATE

Fresh, good matching 1 24–47 s_ni = 1 kPa Bandis et al. (1983)

2 98–344

3 185–424

Weathered 1 11–14

2 19–40

3 49–78

RHYOLITE Clean 16.4 Triaxial testing (?) Goodman &

Dubois (1972)

WEAK ROCK With clay gouge 5–40 Increases with s_n Barton et al. (1981)

HARD ROCK

Soft clay filling 0.01–0.1 Typical range Itasca (2004)

Clean 37–93 Triaxial testing. Increases with

number of loading cycles Rosso (1976) 8–99 Direct shear tests

Clean fracture 1620 Estimate for numerical

analysis Rutqvist et al.

(1990)

Good match, interlocked > 100 Typical value Itasca (2004)

Fault with clay gouge 0.005 30–150 cm thick Karzulovic (1988)

Rough structure with a fill of

rock powder 0.8 Mismatching

GYPSUM Fresh joints (JRC = 11) 1 3–11 s_ni = 0.2 MPa Rode et al. (1990)

Fresh joints (JRC = 11) > 1 10–13

k_n = Normal stiffness sn = Normal stress k_ni = Initial normal stiffness sni = Initial normal stress

Pac-ex: Measured by the system Pac-ex, a special instrumentation system developed in the Underground Research Laboratory by Atomic Energy of Canada Ltd.

Source: Flores & Karzulovic (2003)

There are simple cases for which it is possible to compute the normal stiffness of the structures. If the Young’s moduli of rock, E, and of rock mass, E_m, in the direction normal to the defects are known, and the rock mass contains only one set of defects with an average spacing s, then the normal stiffness of the structures can be computed as:

k s E E

In the case of the defects with infills, if the defects are smooth or the infill thickness is much larger that the size of the asperities the normal stiffness can be computed as: where E_inf and n_inf are the Young’s modulus and Poisson’s ratio of the infill and t is the infill thickness. This equation assumes that the infill cannot deform laterally, i.e. it is in an oedometric condition.

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