Experience has shown that shortly after excavating a gallery in pressurized rock, the surrounding rock shows signs of being strained or even over- strained. Rock bursts occur and deformations are observed. Interpretation of these rock deformations depend on the basic approach to the general state of residual stresses in rock masses.
10.5.1 Talobre's interpretation
Talobre assumes that the rock is in a hydrostatic state of stress (Heim's hypothesis) and that the horizontal residual stress, ah, is equal to the vertical stress, av, which is equal to/?*, i.e. ah = kav, with k = 1. According to the theory developed in section 5.4 for circular galleries the circumferential stress at at the periphery of the rock excavation is then at = 2/?*, at whatever angle <j> (fig. 10.28a). This is correct so long as at = 2p* is smaller than the elastic limit oel of the rock.
Whenever the circumferential stress at a distance r from the centre of the gallery reaches this limit of elasticity, the rock is plastically deformed or even crushed (fig. 10.286), which relieves the stresses as indicated on the figures. At the same time there is a small inward displacement of the rock.
Crushing of the rock will occur if crmax — amln > aCr where aCr is a critical value obtained from the Mohr circle. It is likely that the rock crushing leads progressively to stabilization of the deformations of the rock and the tunnel.
256 Underground excavations
(a)
plastic deformations'' >crushed rock Fig. 10.28 Stresses about cavities in isotropic rock, uniform residual stresses (ov = ar = p')\ (a) 2/?' < ael; (b) 2p' > <rel; (c) crushed rock, cel = elastic limit (after Talobre, 1957).
It should be understood that these three sketches give only qualitative in- formation, as they cannot take into account the lack of homogeneity of natural rock.
10.5.2 Rabcewicz's interpretation (1957,1964, 1965)
Rabcewicz assumes that the gallery is excavated in rock strained by a vertical field of forces av = p* (with k = 0). A new equilibrium is progres- sively obtained.
Stress rearrangement generally occurs in three stages (fig. 10.29), pro- vided the rock in the neighbourhood of the cavity has not been disturbed by earlier tunnelling. At first, wedge-shaped bodies on either side of the cavity shear off along the Mohr surfaces and move towards the cavity, the direction of the movement being perpendicular to the main pressure direction (fig.
10.29a). The increased span thus produced causes the roof and floor to start
Hi II
•P*HIM
11 ft t
= />* (a) "* =
Fig. 10.29 Stresses and strains about a cavity in rock. Initial residual stresses uniaxial (av = p*, ah = 0). Progressive redistribution of stresses and rock displace- ments towards the cavity (after Rabcewicz, 1964,1965).
converging (fig. 10.29&) and lateral pressure develops in a horizontal direc- tion. In the next stage (fig. 10.29c) movement is increased and the rock buckles under continuous lateral pressure and may protrude onto the cavity.
Squeezing pressures - the last stage - though common in mining, are seldom encountered in civil engineering.
This description is consistent with diagrams developed for the case of a vertical potential field of parallel forces. It is also consistent with experience and observations published in much earlier textbooks on this subject.
The theories developed by Schmidt, Fenner and Terzaghi could also be used to analyse the more general case when oh = kav, with k < 1. It leads to conclusions similar to those of Rabcewicz and indicates that dangerous compression stresses develop along the horizontal tunnel diameter. Such stresses have been observed in many tunnels, in particular in the large Mont Blanc road tunnel.
The theories developed here assume a reasonable degree of homogeneity and isotropy of the rock masses. The image of the rock deformations as sketched by Talobre and Rabcewicz are fundamentally altered when the rock is fissured, fractured or stratified. Jaecklin (19656) said that high tensile stresses develop near the tunnel soffit in horizontally stratified rock to a far greater degree than indicated by the theory of stresses about cavities in iso- tropic rock. Similarly Talobre (1957) and Rabcewicz (1964) emphasize the great differences between isotropic and fractured rock masses concerning the behaviour of rock about tunnel excavations.
10.5.3 Fenner's equation and comments
Talobre and Rabcewicz in estimating the stress distribution about a gallery excavated in overstrained rock use Fenner's equation (1938):
( r\2 sin <£/(l-sin <t>)
- j
where
r = radius of the cavity,
R = radius of the protective, overstrained zone,
Pi = arr the required radial 'skin resistance' (fig. 10.30) and/?0 the uniform residual rock stress (av = ah = p0 = /?*).
Talobre assumes that usually p{ = 0 and the equation gives R, when c and
<f> are known. Talobre gives the following example for a 6-m-diameter tunnel at 1500 m depth:
p0 = 400 kg/cm2 = 4000 t/m2, px = 0,
sin <f> = 0-50 and c cot <f> = 50 kg/cm2 = 500 t/m2. Fenner's equation becomes
- 5 0 0 + [500 + 4000 x \] I - = 0 ,(rV
258 Underground excavations or
With r = 3-0m,i? = 3 ^ 5 = 3 X 2-23 = 6-70 m = 3 + 3-70 m.
The width of the crushed protective zone would be 3-70 m. Rabcewicz assumes that/?j can be positive, corresponding to the 'skin resistance' of the shotcrete layer or to the radial stress due to rock bolting.
overstrained zone
Fig. 10.30 Schematic representation of Fenner's equation (after Kastner, 1962).
Fenner's equation has been criticized, because, in some cases, it fails to yield acceptable values. In section 10.10, on the estimate of the required rock support for tunnels, solutions differing from Fenner's equation will be developed, based on Kastner's and Lombardi's suggestions (Jaeger, 1975, 1976).
Fenner's equation and the alternatives proposed by others are important to many basic engineering problems to be discussed in detail in other chapters.
Figures 10.28 and 10.30 could possibly explain the large local deformations of rock vaults measured in situ. The measured total settlement of the soffit of a large excavation in the Swiss Alps has been found locally to total about 20 cm since the beginning of the excavations, well in excess of what had been expected (Buro, 1970). Such a large displacement of a cavern roof in rock with no preferred joint direction could be explained by assuming locally a very low modulus of elasticity, E, of the rock at the limit of the excavation (see Kujundzic, Jovanovic & Radosavljevic, 1970). Very recently Kujundzic (1970) has tested the rock about a similar gallery with seismic waves and shown that, at some distance from the gallery, the rock is compressed and the dynamic modulus of elasticity consequently higher than in the virgin rock mass, whereas some decompression occurs near the gallery, where the modulus of elasticity drops to lower values. The same compression and decompression effects have been checked by a team of French experts about
a cavern excavated in chalk, but because the cavern had a flat roof and so was not circular the pattern of stresses and the compressed and decompressed zones varied with the progress of excavations.
Hayashi and his colleagues (Hayashi & Hibino, 1970; Suzuki & Ishijima, 1970) in Japan have established correlations between the stresses in rock material and the E modulus. The modulus is shown to rise with compression and to drop in relaxed rock, whereas the Poisson ration v follows the opposite trend. These opposite trends of E and v have already been established by Jaeger (1966a). Hayashi introduces into a computer program the variables E and v for progressively deteriorating rock conditions, whereas Miiller &
Baudendistal (1970) have developed a program for finite element analysis where displacements are assumed to occur along weak joints characterized by low values of tan <f>. With both methods, important rock strains and large settlements of the cavern roof are obtained for mediocre rock conditions, which are nearer to values really observed in difficult underground excava- tions.