(a) Completely suspended rock lamina (fig. 10.42). A lamina of length L and width B9 thickness t and unit weight yr is to be bolted. There are n± rows of
v\\l lh IJMA^ zA
(a)
Fig. 10.42 Laminated roof, supported by bolting (after Obert & Duvall, 1967) bolts, containing n2 bolts per row. If the lamina is completely suspended by the bolts, the load per bolt is
(b) The lamina is held at the edges (fig. 10.426). It acts as a clamped beam, The lamina 2 is bolted to lamina 1. Let q1 and q2 be the loads per unit length, Ex and E2 the moduli of elasticity, and I± and 72 the moments of
274 Underground excavations
inertia of the two beams. The deflections yx and y2 of the two beams must be identical so that
and therefore:
or
f Aq)x\L - x2)
(ft + Aq) q
A/7 *
(?2 —
2 — &q
£±q)x\L - x2) 24E2I2
which gives the load Aq per unit length on the bolt. In the case that I± ^ oo Aq = q2.
(2) Bolting an inclined fissure or fracture plane. F is supposed to be the force parallel to the rock surface, c/> the angle of friction along the rock fracture, and a the angle the normal to the joint plane makes with the surface (after Obert & Duvall, 1967). B is the force in the bolt.
\B
f surface ^ ^ " j surface
VB ^ L
Fig. 10.43 Bolting of a rock fault (after Obert & Duvall, 1967).
When the bolt is normal to the joint (fig. 10.43a) then:
Fsin a B + F c o s a or
< tan <f),
n
— > sin a (cot <f> — cot a).
F If a < <f> no bolt is necessary.
When the bolt is normal to the surface (fig. 10.436), then F sin a — B cos a
— — < tan <p.
F cos a + B sin a
In the two examples, bolting is not effective unless F is small.
(3) Bolting the roof of a gallery (after Talobre 1957). Talobre assumes implicitly in his approach to the problem of rock bolting that the gallery is excavated in a rock where horizontal and vertical components of the residual stresses are identical (oh = kav; k = 1) and there is no preferred direction of stratification. When the tunnel roof is circular, it can be assumed that bolts consolidate a circular rock arch within the rock mass. This arch is compressed radially by the mass of rock and circumferential stresses develop in it. Talobre (1957) gives an example (shown in fig. 10.44).
anchor
Fig. 10.44 Rock-bolts reinforcing a circular roof (after Talobre, 1957).
Tunnel internal radius
Thickness of the supporting arch Radius of the supporting arch Load on the arch (assumed) Thrust N in the arch Circumferential stress
R = 2-50 m e = 1-00 m
*m = 3-75m p = 3-0 t/m2
i V = 3 x 3-75 = 11-25 t at = (N/e)= 11-25 t/m2 Assuming the intrinsic curve (Mohr circles are
to be traced) for the rock to require a radial component ar minimum for maintaining rock stability
Area covered by one bolt Required force in one bolt Tension force used for one bolt
ar = 2 t/m2
1-5 X 1-5 = 2-25 m2 T = 2 0 x 2-25 = 4-5 t 2 x 4-5 = 9 t
(4) Designing rock-bolts after L. v. Rabcewicz (fig. 10.45). The rock fault has the direction a-a at an angle a to the horizontal and the rock-bolts are
= -K- COS (a + /0
COS {i
Fig. 10.45 Rock-bolts reinforcing a roof (after Rabcewicz, 1957)
276 Underground excavations
inclined at an angle of 45° to a-a. R is the force in the arch formed by the roof, H the horizontal component of R, /? is the angle of R to the horizontal and R = H/cos /?. The angle of friction on the rock fault is <j>. The thickness of the arch is e, B its width and h its height. The force in the bolts cut by the fault a-a is 2g& and S the shear force along a-a.
From fig. 10.45 we take that
COS/?
and
sin (a + P) tan <j> + -^-b (1 + tan </>),
[cos(a + P) - sin (a + fltenfl
Assuming horizontal fissures and a = 0, Rabcewicz gives the following example:
a = 0, B = 14 m, h = 4-5 m, g = 5 m , tan <£ = 0-70, tan p = 0-70, # = 401
= H{\ - tan/Stan ^) V2
1 + tan cf>
= 40(1 - 0 - 4 9 ) ^ = 17 t.
Neglecting the friction tan (f> along the fissure (tan <f> = 0) yields
(5) Anchored prestressed cables', Veytaux undergroundpower-station {Switzer- land), (Rescher, 1968). The underground power-station at Veytaux has been excavated in average to indifferent quality limestones and marl of the Dogger formation (80% to 90% limestones, 10% to 20% schists). The strata, 0-20 m to 1-50 m thick, are nearly horizontal. They are cut by three different systems of faults and fractures, some of them filled with clay-like material or with mylonites. The rock is characterized in that a gallery wider than 3-50 m would not be stable without support. Many vertical fractures, parallel to the axis of the cavern, and some crushed zones were encountered.
The main cavern is 136-50 m long, 30-50 m wide and 26*65 m high.
The excavation was obviously difficult, and support had to be adequate.
Several alternatives were considered; reinforcement of the roof and sides with prestressed cables was chosen. The design of the bolting was based on a modulus of elasticity for the stable loads E = 100 000 kg/cm2, a friction factor <f> = 31-5° and a cohesion factor c = 3 kg/cm2. The problem was
assumed to be two-dimensional, stresses in the direction of the length axis of the cavity were neglected, and a ratio k = ahjav = 0-33 assumed.
Cables (38 wires of 8 mm, 11 to 13 m long) were placed in 115-mm- diameter boreholes, tensioned to 170, then grouted under a 140 tensile force. Special Danish cement was used which hardened after five days, the cable was then finally tensioned. Other cables of 125 tonnes maximum strength (24 wires of 8 mm) in 102-mm-diameter boreholes were also ten- sioned with a force of 115 tonnes. In all, 652 cables were used. Additionally, 1700 smaller bolts of 3-5 to 4-5 m length, capable of carrying 15 tonnes were used between the larger meshes of the larger cables (the mesh 4-30 x 2-90 m for the roof).
The whole design was analysed as a two-dimensional problem with the mathematical method of finite elements of Zienkiewicz, checked by a series of photoelastic models. The last of these was a fissured model, with the main geological joints observed on site reproduced. The tan <f> on the model was only 0-19 but stability of the arch was nevertheless obtained by annular compression.