A classical problem of mining engineering is how to estimate the stability of a self-supporting underground structure in which the stresses are carried on the walls, pillars or other unexcavated parts of the openings, rather than on linings or steel supports. Whereas hydro-power and railway engineers
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tend towards a more or less circular shape for tunnels and galleries, mining engineers frequently have to deal with rectangular openings, with rounded corners, or multiple parallel excavations.
Assuming the square opening to be in a vertical field of stresses p, the sides of the square being vertical and horizontal, the maximum tangential stress at, will depend on the ratio r/B, where r = radius of the rounded corner
(a)
Fig. 10.1 Rectangular opening with rounded corners in a field of forces/? and A/?.
(a) Field of forces parallel to sides of rectangle; (b) field of forces inclined to sides of rectangle.
and B = width of the square (fig. 10.1). The maximum stress at occurs for an angle </>* (</> angle with the vertical). Then at maximum is given by table 10.1
Table 10.1 Values of at for h = B {square) and a vertical stress field = p
r\B = 0
<!>*
90°
i 3
— -1
i 1-8 2-78 -0-8
i 1-78 3-35 -0-8
A
1-68 4-6 -0-8 The case rjB = \ obviously corresponds to the circle r\D = \.
For rectangular openings the maximum relative stress concentration ajp for rjB = 0-10 rises to about 5 for hjB = J, where h = height of rectangle, width B.
Assuming two parallel cylindrical openings (fig. 10.2) such that the distance B between the holes is equal to the diameter D of the circles (the distance between the centres of the circles is therefore 2D). It can be shown that the maximum circumferential stress at rises from at = 3/7 (one opening) to at = 3-4/7. The vertical stress in the middle of the distance B is then about 1-7/7 (/7 = vertical uniform stress field in the rock). Such a result could be expected to be due to the rapid decrease of the circumferential stresses at
with the distance from the circle (see section 5.4).
228 Underground excavations
l
Fig. 10.2 Stress concentration. Two parallel galleries in a vertical field of forces/?
(B= D = 2a) (after Obert & Duvall, 1967).
Elliptical galleries and three-dimensional cavities have been analysed mathematically (Terzaghi and others). The case where the principal axes of the cavity are inclined towards the field of stresses has been solved equally (fig. 10.1).
The use of photoelastic methods yields rapid results which can be compared with field measurements.
Direct calculations of underground cavern systems (like machine-hall and transformer-hall excavations, downstream surge tanks) using the finite element method yield results differing from the simple analytical approach (Benson, Kierans & Sigvaldson, 1970). This divergence of the results can be explained by the special shape of the excavations. For Ruacana on the Cunene River (South West Africa), the upper part of the vertical downstream surge tank had to be reshaped completely to reduce tensile stresses obtained when calculating the case k = 0-40. Power-house excavations or downstream surge tanks may have a height: width ratio > 1; in the case of so-called 'rock pillars' separating caverns calculations disclosed tensile stresses in a direction perpendicular to the high walls (for k < 1). For the analysis of such systems of caverns, the use of the finite element method analysis is essential.
10.2.2 Subsidence and caving
The first manifestation of subsidence may be convergence of the walls and roof of the gallery or a succession of local failures in the rock surrounding the openings. This phase of the process is termed sub-surface subsidence, as opposed to surface subsidence, which causes a depression in the overlying surface (fig. 10.3a). Subsurface subsidence is largely an uncontrolled process.
Caving is a form of subsurface subsidence which is at least partly controlled by the mining method.
When excavation occurs in relatively thin-bedded deposits with overlying weak sedimentary rocks, surface displacements occur. Any point at the
lateral surface
o A displacement j
(a)
excavation I
Fig. 10.3 (a) Idealized representation of trough subsidence (after Rellensman 1957). (b) Circle of Mohr for subsidence conditions.
surface has a vertical downward displacement and a horizontal displacement versus the centre Go of the excavation. The vertical surface displacement is a maximum at G' over the Go of the excavation. The horizontal surface strain is tensile at points outside the limits of the excavation and compressive with- in the limits (fig. 10.3a). D is a point of zero strain and there is an in- flection point B at the surface of the soil, a is the angle of break and point A corresponds to the maximum of tensile strain at the surface. /? is the angle of draw.
Table 10.2
Clay Sand
Moderate shale Hard shale Sandstone Coal
Angle of friction <f>
(measured)
15-20°
35-45 37 45 50-70 45
Angle of break (calculated)
52-5-55°
62-5-67-5 63-5 67-5 70-80 67-5
In fig. 10.36 the angle of break a determines the line along which rupture occurs, r is the shear stress along this line and r = c + a tan <f>. Tracing the circle of Mohr for such conditions yields:
2a = 90° + <f>.
Seldenrath (1951) quoted by Obert & Duvall (1967) published table 10.2.
230 Underground excavations