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Hence
(2.42)
Eq. (2.42) can be rewritten as
(2.43)
Where = reaction pressure caused by support
= in situ stress and , where and are unit weight of rock mass and depth, respectively
= radius of a circular opening
R= radius of inelastic zone
Equation (2.43) is just the famous modified Fenner Equation, which indicates the relationship between the nonelastic zone and the reaction pressure produced by support in a definite rock mass and ascertained depth.
2.4 The “Q” System
On the basis of an evaluation of a large number of case histories of underground excavations, Barton et al (1974) of the Norwegian Geotechnical Institute proposed a “Q” system. The numerical value of the index Q varies on a logarithmic scale from 0.001 to a maximum of 1,000 and is defined by:
(2.44)
Where = Rock Quality Designation
= joint set number
= Joint roughness number
= joint alteration number
= joint water reduction factor
= stress reduction factor
In explaining the meaning of the parameters used to determine the value Q, Barton et al (1974) offer the following comments:
The first quotient ( ), representing the structure of the rock mass, is a crude measure of the block or particle size, with the two extreme values (100/0.5 and 10/20) differing by a factor of 400; The second quotient ( ) represents the roughness and frictional characteristics of the joint walls or filling materials; The third quotient ( ) consists of two stress parameters. SRF is a measure of : 1) loosening load in the case of an excavation through shear zones and clay bearing rock, 2) rock stress in competent rock, and 3) squeezing loads in plastic incompetent rocks. It can be regarded as a total stress parameter; The parameter is a measure of water pressure, which has an adverse effect on the shear strength of joints due to a reduction in effective normal stress; the quotient ( ) is a complicated empirical factor describing the ‘active stress’.
Based upon analyses of case records, Grimstad and Barton (1993) suggest that the relationship between the value of Q and the permanent roof pressure is estimated from:
(2.45)
Where Q = rock quality value
= joint set number
= Joint roughness number
2.5 Pressures of Ground-Structure Interaction
The processes of ground pretreatment, excavation, and ground stabilization alter the preexisting state of stress in the ground, before the lining comes into contact with the ground. An underground structure is not an independent structure acted upon by well-defined loads, and its deformation is not governed by its own internal elastic resistance. As shown in Figure 2.9, if this tunnel is subjected to concentrated loads at crown and invert, it will bulge inward at those points and outward at the spring lines. These deformations will be large compared with those of the uniformly loaded ring. Consider now a partially confined ring, in which passive pressure is developed on at the springline. When a fully confined ring is subjected to local active pressure at the crown and invert, and passive pressure develops around the full balance of its perimeter. Both inward and outward bulging are further reduced from the previous case. When a general case of fully confined ring is subjected to randomly distributed active pressure loading, in which the passive pressure is distributed relatively uniformly, and the deformations and curvature changes of the ring are slight and smoothly distributed.