LECTURE 0 6. STRESSES AROUND UNDERGROUND OPENING contd. CASE : When σ x = 0 For σ x = 0, the maximum tangential stress is three times the applied stress and occurs at the boundary on the X-axis that is θ = 0 or π. When θ = π/ and 3π/, the tangential stress at the boundary of the opening is equal to the applied stress but is of opposite in sign. Figure 6.7. Tangential and radial stresses around circular openings when σ x =0 165
Figure 6.8. Tangential stresses around the periphery of the circular openings when σ x =0 6.3 CIRCULAR HOLE IN AN ELASTO-PLASTIC INFINITE MEDIUM UNDER HYDROSTATIC LOADING Figure 6.9: Circular hole in an elasto-plastic infinite medium under hydrostatic loading 166
Using Tresca s yield criteria For a r c r s rp = hp ln a s θ p r = hp 1+ ln a For r c ha s r = p 1 e r (1 h) / h s θ ha = p 1 + e r (1 h) / h Here, σ r, σ rp = radial stress in elastic and plastic zone respectively σ θ, σ θp = tangential stress in elastic and plastic zone respectively a = radius of the circular opening θ = central angle with x-axis r = radial distance of the element from the center of the opening c = ae (1-h) / h = radius of boundary between the elastic and plastic zones p = applied hydrostatic pressure, compressive h = k / p. Elasto- Plastic Observations The tangential stresses at the boundary of cylindrical openings are considerably lower for the elasto-plastic rock mass than for a perfectly elastic one. The tangential stresses beyond the plastic zone are larger than the perfectly elastic case at the same radial distance The zone of influence due to opening is larger than that in the case of perfectly elastic rock. 167
Figure 6.10: The stress distribution along the axes of symmetry in elasto-plastic medium Figure 6.11: The stress distribution around the circular excavation in elasto-plastic medium 168
6.4 PLASTIC BEHAVIOUR AROUND TUNNELS When the tangential stress around an opening is greater than about one- half of the unconfined compressive strength, cracks will begin to form. At large depth, such rock failure can cause violent bursts. Weak rocks like shale reach the condition for rock cracking at small depths. In such rocks, new cracking may initiate further loosening as water and air cause accelerated weathering. The zone of broken rock is driven deeper into the walls by the gradual destruction of rock strength. As a result, the load on the tunnel support system will increase and the supports experience a gradual build up in pressure known as squeezing. To gain a better understanding of the mechanics of a squeezing tunnel and to provide an analytical framework to provide appropriate support systems, the theoretical model proposed by Bray (1967) can be considered. The following are the assumptions made in the elastic- plastic model proposed by Bray (1967) so that it can be applied to field problems, The failure of rock is by Mohr-Coulomb theory. State of stress is axis- symmetric i.e., k=1. Within the plastic zone, which extends to a radius R, Bray assumed the fractures were log spirals inclined at δ degrees with the radial direction. 169
Figure 6.13: Conditions assumed in Bray s (1967) elastic-plastic solution Bray s solution of log spirals is acceptable in shales and clays. For minimum strength, the appropriate value of δ is 45+ϕ/, where ϕ is the angle of internal friction of the intact rock. It proves useful to define a quantity Q given by: tanδ Q = 1 tan( δ φ ) j where, ϕ j is the friction angle for a joint. 170
Assuming that the broken rock inside the plastic zone contains log-spiral surfaces with shear strength characteristics τ p = S j + σ tan ϕ j, the radius R of the plastic-elastic zone boundary is given by: φ p qu + [1 + tan (45 + )] S j cotq j R = a( ) q [1 + tan (45 + )]( pi + S j cotq j ) 1 Q Where, p is the initial rock stress, a is the radius of the opening, ϕ is the angle of internal friction of the intact rock, ϕ j is the friction angle for a joint, S j is the cohesion, q u is the unconfined compressive strength of the intact rock, p i is the internal pressure in the tunnel provided by the supports. Bray s solution for radial and tangential stresses in the elastic zone is as follows: σ r = p-(b/r ) σ θ = p+(b/r ) where, φ {([tan (45 + ) -1] p + q b = φ [tan (45 + ) +1] u ) R The radial and tangential stresses in the plastic zone is defined by, σ r = (p i + S j cot ϕ j )(r/a) Q - S j cot ϕ j σ θ = (p i + S j cot ϕ j )[ tan δ/ tan (δ- ϕ j )](r/a) Q - S j cot ϕ j For various values of r, the values of σ r and σ θ are found out, plotted and compared with Kirsch solution shown in Figure 6.14. From the plot, it is evident that for some distance behind wall, the tangential stresses lower than that predicted by elastic theory, thereafter they are higher. 171
Figure 6.14: Stresses around the yielding tunnel given in the example (After Bray, 1967) The plastic behavior of the region near the tunnel has the effect of extending the influence of the tunnel considerably farther into the surrounding rock. In the wholly elastic case, the tangential stresses would have fallen to only 10% above the initial stresses at a radius of 3.5 times the tunnel radius but in the elastic-plastic case, the elastic zone stresses are 70% higher than the initial stresses at this distance and 10 radii are required before the stress falls down to 10% of initial stress. Thus, two tunnels that do not interact with one another in elastic ground might interact in plastic ground. 17
6.5 ZONE OF INFLUENCE Zone of influence of an excavation is very important for underground tunneling and mining applications where multiple excavation/ tunnels are excavated. With considerable simplification of a design problem, idea is to get the domain of significant disturbance of the excavation stress, and get the stresses near field and far field of an opening. Stress distribution around a circular hole in hydrostatic medium, a s r = p 1 r s θ p 1 a = + r τ rθ = 0 at r =5a, σ θ = 1.04p σ r = 0.96p At r = 5a, the state of stress is not significantly different (within 5%) from the field stress σ/p σ θ /p a σ r /p 3 4 5 Figure 6.15: Radial and tangential stresses corresponding to different radial distance in a hydrostatic stress field 173
Table 6.1: Tangential and radial stresses corresponding to different radial distance in hydrostatic stress field Distance from centre (r) Tangential stress (σ θ ) Radial stress (σ r ) a P 0 a 1.5P 0.75P 3a 1.11P 0.88P 4a 1.06P 0.94P 5a 1.04P 0.96P Figure 6.16 shows a closer examination of the deformations induced in the rock mass by the excavation of the underground powerhouse and transformer gallery. It is seen that, the smaller of the two excavations is drawn towards the larger cavern and its profile is distorted in this process. This distortion may be reduced by relocating the transformer gallery and by increasing the spacing between the galleries. 174