Planta primera

The state of stress due to tunnel excavation can be calculated from analytical elastic closed form solutions. Kirsch's elastic closed form solution is one of the commonly used analytical solutions and is presented in Appendix E. The closed form solution is restricted to simple geometries and material models, and therefore often of limited practical value. However, the solution is

considered to be a good tool for a "sanity check" of the results obtained from numerical analyses.

The interaction between rock support and surrounding ground is well described by the ground reaction curve (Figure 6-26), which relates internal support pressure to tunnel wall convergence.

General description of ground reaction curve is well described Hoek (1999).

Figure 6-26 Ground Reaction Curves between Support Pressure and Displacement (Hoek et al., 1995)

As shown in Figure 6-26a, zero displacement occurs when the support pressure equals in-situ stress, i.e., Pi = Po. When the support pressure is greater than critical support pressure and less than in-situ stress, i.e., Po >Pi >Pcr, elastic displacement occurs. When the support pressure is less than the critical support pressure, i.e., Pi < Pcr, plastic displacement occurs. Once the support has been installed and is in full and effective contact with the surrounding rock mass, the support starts to deform elastically. Maximum elastic displacement which can be accommodated by the support system is usm and the maximum support pressure, Psm is defined by the yield strength of the support system. As shown in Figure 6-26b, the tunnel wall displacement has occurred before the support is installed and stiffness and capacity of support system controls the wall

displacement.

Hoek (1999) proposed a critical support pressure required to prevent failure of rock mass surrounding the tunnel as follow:

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Where:

Pcr = Critical support pressure Po = Hydrostatic stresses

σcm = Uniaxial compressive strength of rock mass φ = Aangle of friction of the rock mass

If the internal support pressure, Pi is greater than the critical support pressure Pcr, no failure occurs and the rock mass surrounding the tunnel is elastic and the inward displacement of tunnel is controlled.

A more realistic design, especially for large tunnels and large underground excavations, is based on the true behavior of rock bolts: to act as reinforcement of the rock arch around the opening.

This rock reinforcement increases the thrust capacity of the rock arch. The design objective is to make that increase in thrust capacity equivalent to the internal support that would be calculated to be necessary to stabilize the opening.

The increase in unit thrust capacity (ΔTA) of the reinforced zone (rock arch) shown in Figure 6-27 is given by the equation (see Figure 6-27) developed by Bischoff and Smart (1977):

Figure 6-27 A Reinforced Rock Arch (After Bischoff and Smart, 1977)

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where ΔTA is increase in unit thrust capacity of the rock arch, φ is effective friction angle of the rock mass, Tb is stress at yield of the rock reinforcement steel (fully grouted rock bolts), Ab is cross-sectional area of the reinforcement steel, S is spacing of the reinforcement steel, in both directions, t is effective thickness of the rock arch (= L – S), and L is length of the reinforcement steel.

Analytical solutions to calculate support stiffness and maximum support pressure for concrete/shotcrete, steel sets, and ungrouted mechanically or chemically anchored rock bolts/cables are summarized in Table 6-11.

Table 6-11 Analytical Solutions for Support Stiffness and Maximum Support Pressure for Various Support Systems (Brady & Brown, 1985)

Support System Support stiffness (K) and maximum support pressure (Pmax)

Concrete/Shotcrete lining

Blocked steel sets

Ungrouted mechanically or chemically anchored rock bolts or cables

NOTATION: K = support stiffness; Pmax = maximum support pressure; Ec = Young's modulus of concrete; tc = lining thickness (Figure 6-28a); ri = internal tunnel radius (Figure 6-28a); σcc = uniaxial compressive strength of concrete or shotcrete; W = flange width of steel set and side length of square block; X = depth of section of steel set; As = cross section area of steel set; Is = second moment of area of steel set; Es = Young's modulus of steel; σys = yield strength of steel; S = steel set spacing along the tunnel axis; θ= half angle between blocking points in radians (Figure 6-28b); tB = thickness of block; EB = Young's modulus of block material; l = free bolt or cable length; db = bolt diameter or equivalent cable diameter; Eb = Young's modulus of bolt or cable; Tbf = ultimate failure load in pull-out test; sc = circumferential bolt spacing; sl = longitudinal bolt spacing; Q = load-deformation constant for anchor and head.

Figure 6-28 Support Systems: (a) Concrete / Shotcrete Lining, (b) Blocked Steel Set The size and shape of wedges formed in the rock mass surrounding a tunnel excavation depend upon geometry and orientation of the tunnel and also upon the orientation of the joint sets. The three dimensional geometry problems can be solved by computer programs such as UNWEDGE (Rocscience Inc.). UNWEDGE is a three dimensional stability analysis and visualization program for underground excavations in rock containing intersecting structural discontinuities. UNWEDGE provides enhanced support models for bolts, shotcrete and support pressures, the ability to

optimize tunnel orientation and an option to look at different combinations of three joint sets based on a list of more than three joint sets. In UNWEDGE, safety factors are calculated for potentially unstable wedges and support requirements can be modeled using various types of pattern and spot bolting and shotcrete. Figure 6-28 presents a wedge formed by UNWEDGE on a horse-shoe shape tunnel

Figure 6-29 UNWEDGE Analysis: (a) Wedges Formed Surrounding a Tunnel; (b) Support Installation