To design structures like bridges, buildings and the like, first, we must define the loads acting on them, and then choose the type of structure, materials and dimensions. When it comes to tunnel supports, it is very natural to follow the same procedure, and to devise many theoretical and empirical formulae to obtain the loads acting on the supports. However, for tunnel supports, the distinction is not so clear between the load and the supporting members because the load varies according to the stiffness ratio between the ground and the supports. This relationship was known empirically to old miners, and formulated by R. Fenner (1938).
The relationship was put to practical use firstly through variable length steel supports (TH channel), and secondly through double shell arch lining. The latter method made progress through the development of building materials (shotcrete, anchor bolts), rock mechanics, and in-situ measurements aided by many field experiences, and became known as the so-called New Austrian Tunnelling Method (NATM). As mentioned above Fenner demonstrated that the load acting on the supports decreases according to the increase of displacement of the tunnel inner surface (Fig. 1 curve a,b,c,d). But F. Pacher, concerned that too much deformation would induce tunnel failure, modified Fenner's proposal (Fig. 1,a,b,c',d') by taking into consideration the de- crease of c, Φ of the surrounding rock by loosening. A somewhat smaller deformation than pi minimum is recommended for suitable support systems.
However, these relationships have not been worked out numerically, so we cannot apply these principles directly to actual tunnel Work. The author has studied the theoretical formulae for this relationship, but the results do not confirm the phenomena suggested by F. Pacher, so he will propose other principles to dimension the most suitable supports, and will modify the theories of F. Pacher and L.v. Rabcewicz.
(Figure in full paper)
Stresses and deformations around a cylindrical hole in an elastic solid have been worked out already.
(Figure in full paper)
By means of these formulae, we can obtain the relationship between Pi (support reaction) and Ua (displacement). However, these formulae neglect the effect of the gravitational force on the rock, therefore, even if we assume that C and θ decrease nearly to zero, we cannot obtain the ascending curve beyond the pi minimum. More rigorous solutions including gravitational force have been given by Noboru Yamaguchi as follows:
(Equation in full paper)
Where σrradial stress, σθ: tangential stress, Pi: internal pressure (i.e. tunnel support reaction), these stresses and pressures are designated by a (+) when tensile and by a (-) when compressive.
Trθ: shearing stress, Ur: radial displacement, Uθ: tangential displacement, a: radius of hole or tunnel, r, θ: polar coordinate as shown in Fig. 2. h: distance from center of the tunnel to. the ground surface, pg: unit weight of rock, V: poisson's ratio, k: lateral pressure coefficient = V/(l − v)