La aceptación incondicional

In this section, which principally concerns dynamic moduli and dynamic Poisson’s ratio, the use of V_pand V_smeasurements to derive the four standard dynamic elastic properties of rock masses will be discussed. The validity or otherwise of these dynamic, small-strain,

elastic moduli to rock engineering design will also be addressed. The four standard equations, two of which were introduced earlier in Chapter 1 are reproduced here together, for ready reference (ISRM, 1998).

Dynamic Young’s modulus E_dyn:

(6.5)

(6.6)

Dynamic Poisson’s ratio _d:

(6.7)

Dynamic shear modulus : (also with symbol G)

(6.8)

Dynamic bulk modulus K_bulk:

(6.9)

Figure 6.8 Effect of stress on V_pvalues beneath plate load tests at the Inguri arch dam in Russia. Savich et al., 1974.

(a)

(b)

Figure 6.9 a) Uniaxial, and b) triaxial test of coal, showing velocity and attenuation changes caused by cleat and microcrack behaviour. Shea and Hansen, 1988.

It will be noted that if the dynamic Poisson’s ratio is estimated (rather than derived from V_pand V_s), the three dynamic moduli can theoretically be estimated from V_p measurement alone. As pointed out later, this can cause significant inaccuracies, and such values given in the literature should be treated with caution.

The manner in which V_sand V_pvalues are distrib-uted in relation to the general quality of the rock mass was illustrated in Chapter 1 (Figure 1.7 from Sjøgren, 1984). Further sets of data, both from laboratory sam-ples and from comparable field data (Ribacchi, 1988) are illustrated in Figure 6.10.

Ribacchi’s data shows a particularly clear demarcation between the lower dynamic Poisson’s ratio in the case of the higher velocity laboratory data, and the opposite trend for the lower velocity field data. High dynamic Poisson’s ratio are a clear sign of the influence of joint-ing. In shear zones and faulted rock, high values of _d are common.

During ‘static’ flatjack biaxial loading tests of rock masses, ‘static Poisson’s ratios’ (or lateral expansion coef-ficients) in excess of 0.5 may even be measured. In special

cases, values in excess of 1.0 have been registered. This occurs as shear failure is approached, in the case of biax-ially loaded model rock masses, having two conjugate fracture sets that are under significant levels of shear stress. (Barton and Hansteen, 1979, Barton, 1993a, and Barton, 2004b).

Measurements of V_pand V_sat an unweathered site in Norway are shown in Figure 6.11, and indicate ratios of V_p/V_s of about 1.8 to 1.9. The corresponding dynamic Poisson’s ratios were found to lie in a narrow range, more than 80% of the values were between 0.26 and 0.32. The mean value of 0.28 in fact lay close to the maximum RQD and minimum joint frequency trend for this par-ticular site, and corresponded to a ratio V_p/V_sbetween 1.8 and 1.85. In the rock masses investigated, the authors found that the full range of dynamic Poisson’s ratios was from 0.15 to 0.39.

Deere et al., 1967, addressed the important differences between field measurements of E_{F dyn} and laboratory measurements of E_{L dyn} of the intact rock, by utilising their observation that (V_Field/V_Lab)² resembled RQD/

100. (This was referred to in Chapter 5, Table 5.1.) This was based on the Onodera, 1963, suggestion of using the field/lab velocity ratio V_F/V_Las a measure of rock quality.

The resulting method for estimating E_{L dyn}/E_{F dyn} is shown in Figure 6.12. As can be noted from the spread of

Figure 6.10 Contrasting laboratory and field data for the V_pand V_s values of limestones, with calculated dynamic Poisson’s ratios. Note the over-riding tendency of higher dynamic Poisson’s ratio in the case of the lower velocity in situ data. Ribacchi, 1988.

Figure 6.11 Ratio of V_p and V_s at a hard igneous rock site in Norway. V_p/V_s
1.8–1.9. The full range of dynamic Poisson’s ratios was from 0.15 to 0.39, with higher values when velocity was lower. Sjøgren et al., 1979.

usual observation concerning the different strain ampli-tudes involved (perhaps 10^3 and 10^6 respectively) but also point out that the different moduli are thereby directly caused by the presence of pores, cracks (and joints). These are deformed in static tests, but hardly deformed by dynamic waves, but when stresses are very high and pores, cracks and joints are (almost) closed, the static and dynamic moduli are likely to be very close.

The possibility of estimating E_dynvalues from V_p meas-urement alone, by just estimating the value of the dynamic Poisson’s ratio (_d) instead of measuring both V_p and V_s was referred to earlier. This would involve using equation 6.6 instead of the correct method utilising equation 6.5. In warning against this short-cut, Stacey, 1977, assembled what he called ‘reliable’ E_dynvalues from the literature. These values are shown in Figure 6.13a.

(For the numbered references see Stacey, 1977).

Stacey also assembled a large number of E_e-E_dyndata (Figure 6.13b) and E_d-E_dyndata (Figure 6.13c). These two figures show the apparently irrelevant nature of E_dyn in comparison to the standard rock engineering methods of testing deformation modulus for design purposes.

The doubt nevertheless remains that most of our large scale methods are testing an excavation disturbed zone, or EDZ, rather than the undisturbed state, with its higher and maybe more isotropic in situ stresses.

There is a contrary factor that most rock masses observed or tested, are actually on a major unloading curve due to erosion or due to rock excavation, and may have correspondingly higher joint stiffnesses, than if loaded up without this prior unloading.

The mismatch of the static and dynamic moduli in jointed rock masses, except where rock qualities are very high and strains are very small, is probably a universal rule, unless extremely small strains are actually involved in the ‘static’ loading.

In the last two decades, it has apparently been recog-nised in soil engineering that strain levels associated with normal foundation designs are rather small, for example 0.01 to 0.1%, and therefore stiffnesses may be success-fully described by the correlations obtained from in situ

(b)

(c)

Figure 6.12 Utilising the velocity ratio (squared) or RQD%/100, to estimate the ratio of the dynamic modulus for intact samples (E_{L dyn}), compared to the dynamic field modu-lus (E_{F dyn}). Deere et al., 1967.

seismic measurements (Matthews et al., 1997). Such measurements also have the great advantage of register-ing the stiffness of the ground at in situ stress levels and in the undisturbed condition.

A corollary to the above is the predicted high deform-ation moduli for jointed rocks at depth, shown in Figure 6.14 (Barton, 1995). A pre-condition here is that the rock mass is undisturbed and strains are small, corres-ponding more closely with the higher seismic velocities seen at greater depths in the same rock masses. When a tunnel or test adit is constructed at considerable depth, the EDZ effect will alter the above conditions in a com-plex way, to a degree that depends on rock quality and the care with which the adit-excavation has been performed.

One must expect a certain seismic velocity gradient, a deformation modulus gradient, a deformation gradi-ent and even a permeability gradigradi-ent and pore pressure gradient, and finally a possible gradient of saturation.

The natural complexity of a site may also tend to increase the range of moduli and moduli ratios in rela-tion to those measured in one particular lithology. The Latiyan Dam site in Iran was founded on granites, peg-matites, migmatites and gneiss, and weathered layers of each of these. Lane, 1964, compares four of the basic moduli commonly obtained at dam sites:

● laboratory E_{L dyn} (from laboratory V_p, V_s and Poisson’s ratio)

● field E_{F dyn}(from field V_p, V_sand Poison’s ratio)

● field deformation modulus (D, E_d or M) (from plate or flatjack loading)

● field elastic modulus (E_e) (from plate or flatjack unloading)

and also gives ratios of each, as obtained in three exploratory tunnels in the dam foundations. The surpris-ingly large ratios of these moduli, for rather poor rock mass qualities (Q mostly  0.1?) are shown in Table 6.3.

Link, 1964, also gave a wide-reaching comparison of dynamic and static moduli from projects (usually dams) in many countries. He found that the ratio of E_{F dyn}/D from field tests (category 6 in Table 6.3), ranged from about 1 to 16, with most values of the ratio lying in the range 3 to 7. A large number of the static loading tests were pressure chamber tests from Central European dam sites, and the author pointed out that the seismic measurements (from V_p, V_sand dynamic Poisson’s ratio) were also of quite large scale. As we see from the above table, ratios of E_{F dyn}/D can be even larger than the above when the rock quality is poor, due to the basic inequality

(a)

(b)

(c)

Figure 6.13 An extensive collection of ‘reliable’ E_dyn.data, and of the ratios E_e/E_dyn. and E_d/E_dyn.. Stacey, 1977. (See individual references in article).

of the two static unloading/loading moduli: E_e/E_d (or E_e/D), shown, for example by Kujundzíc and Grujíc, 1966.

Link, 1964, made special reference to an extremely high value of E_{F dyn}measured (interpreted) under the lower slopes of the Vajon limestones. The value of 140 GPa was considered the result of high overburden, and/or residual stresses.

Graphic presentation of the inequalities between E_{F dyn}and E_ewere given by Kujundzíc and Grujíc, 1966.

Figure 6.15a and b show the significant inequality of E_dynand E_efor the case of the limestones tested in their

Yugoslavian dam site tests. A general trend was noted as follows:

(6.11)

where E_eand E_dynare expressed in GPa.

The lower the dynamic modulus the larger the ratio of the dynamic/static moduli. When the inequality of E_e and E_d is also considered, the very large ratios of E_dyn/E_d of 10 to 20 given by Lane, 1964, and Link, 1964, are more readily understood.

E E

e E

dyn dyn

 ( .5 30 05. )

Figure 6.14 High values of static deformation modulus E_mass(or M) for the rock mass (also referred to as D, or E_d by some authors), are pre-dicted where stresses and rock quality are high. Here E_massapproaches E_{F dyn.}. Barton, 1995.

Table 6.3 Ranges of moduli and ratios of moduli at three complex sites (after Lane, 1964).

Modulus Tunnel 1 Tunnel 3 Tunnel 4

1 Laboratory Dynamic (E_{L dyn}) (GPa) 35.8 45.4 35.8

2 Field seismic (E_{F dyn}) (GPa) 17.7 25.4 16.4

3 Average modulus of deformation field loading (GPa) 1.8 1.3 1.3

(D or E_dor M, depending on author)

4 Average modulus of elasticity field unloading (E_e) (GPa) 4.8 4.6 3.8

5 Ratio laboratory dynamic to field seismic (E_{L dyn}/E_{F dyn}) 2.02 1.78 2.18

6 Ratio field seismic to modulus of deformation (E_{F dyn}/D) 10 20 12

7 Ratio field seismic to unloading modulus of elasticity (E_{F dyn}/E_e) 3.7 5.6 4.4

8 Ratio laboratory dynamic to modulus of deformation (E_{L dyn}/D) 20 36 27

9 Ratio laboratory dynamic to jacking modulus of elasticity (E_{L dyn}/E_e) 7.4 9.9 11.5

6.3 Some examples of the three