The 14-72-km-long Fionnay-Riddes tunnel had to cross difficult rocks
10.10 Estimate of required rock support based on rock
10.10.5 Some results obtained by more detailed calculations
They show that p* depends on /?, and on the angle of friction, <f>, implicitly included in the parameter £. The/?* value is also shown to depend on the ratio s = ploc, the physical interpretation of which is obvious for many tunnelling problems.
There are many other cases where this simplified approach cannot cope with the local geological or geomechanical situation, for example when the ratio k = ahfav is very different from unity, or when a weak rock seam crosses the cavity at an unfavourable angle. In such cases, the finite element method is the tool to be used.
Papers by Zienkiewicz et al (1964/65), Hayashi (1970), Miiller et al. (1970), Daemen (1975) etc., discuss typical examples and analyse important case histories. Lombardi (1970), sometimes critical of the finite element method, has published a series of graphs obtained with a method equating radial deformations and stresses at the boundary of the elastic and the plastic rock zones and at the contact of rock mass and rock support. They confirm that RL depends on <f> and on /?, as indicated by the theoretical analysis, but also on k = an\av and on the direction of the joints and faults (see section 16.3 on Waldeck II). A few typical diagrams of Lombardi (1974/75) are reproduced.
Figures 10.53 and 10.54 show how RL depends on/?* and <f>. Figure 10.55, from the same author, refers to an underground power-station of 30 m width, its arch being secured by means of stressed rock anchors. The forces in these anchors are not constant over the rock surface nor are they always radial. They are adapted to local conditions. The failure zone is mainly at
p + Ca - 10 t/rri
0 10 2030 225 (p + CaY
Fig. 10.53 Extension of the failure zone as a function of the reaction pressure;
(after Lombardi, 1970), Ca = rock cohesion.
1 0 -
70
Fig. 10.54 Extension of the failure zone as a function of the friction angle (after Lombardi, 1970).
i +
-135 + 135°
+ 90°
Fig. 10.55 Subterranean power station with roof secured by rock anchors. Anchor forces, failure zone and distribution of stress (after Lombardi, 1970).
the bottom of the excavation. The distribution of the stresses about the cavity are illustrated on the figure, so are the stress trajectories and the sliding lines.
(b) Advanced research on the progressive expansion of the visco-plastic zone inside the rock mass surrounding a cavity being excavated. Most rock deformations are not instantaneous. A first elastic rock deformation occurs very rapidly after blasting. Further deformations follow, progressing with the excavation. Visco-plastic deformations develop within the rock mass, which, in large excavations, may take several months to achieve equilibrium.
In some cases, engineers are faced with a measurable creep of the rock mass.
A first attempt at solving these difficult problems consists in developing
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mathematical models for plastic rock deformations with time (Hayashi, 1968/70, Daemen, 1970), and solving these most complex equations with the finite element method. Some new physical parameters are introduced in such equations for which numerical values are difficult to guess. Hayashi decided to adjust such numerical values until his results coincided with some
0.20
0.20
0.20
Poisson's ratio (100 days)
(tension) (compression) K-1000-H
(t/m2) principal stresses (100 days)
Fig. 10.56 Stresses (right) and Poisson's ratio (left) in a tunnel lining and in visco- plastic rock mass (after Hagashi, 1968/70 and Jaeger, 1976).
measured curves of deformations. Figure 10.56 illustrates a relatively simple problem of progressive deformation of a 12-m diameter tunnel reinforced with a 1-20-m-thick concrete lining, depending on progress of excavations, for the case k = 0-4, after a period of 100 days. The formation of a compressed rock cylinder around the cavity can be seen on the right of the figure; on the left varying Poisson ratios can be noticed, confirming the remarks made in section 5.3.2. Comparing these stress patterns with the classical analytical
solution, ignoring lining, fundamental differences appear. The classical solution shows, for k = 040, the circumferential stresses to be:
on the horizontal diameter aQ = 2-0p on the vertical diameter ae = 0-2/7 (p = vertical component of residual stress.)
(c) The engineering approach. Several engineers in charge of large tunnel designs advocate an 'engineering approach' to the problem of rock supports, to which Lombardi (1970/74) is an important contributor. They do not accept, without restrictions, either the geophysical classifications, or the results mathematicians obtain with the finite element method. Lombardi (1972) states that a correct description of all boundary conditions at the con- tact of rock and support and within the rock mass, at the edge of the visco- plastic mass, is more important than the method chosen for solving analy- tically or numerically the basic equations of elastic or visco-plastic deforma- tions. Figure 10.57 shows rock deformations near the tunnel heading: a
Fig. 10.57 (after Lombardi). Definition of the 'lines of displacement'. Influence of blasting the mass M on the position of the 'lines of displacement'. 1 = tunnel axis; 2-2' = heading before and after blasting; A-A Control section; S = radial displacement at the excavation edge; I-I line of displacement before blasting; II—II line of displacement after blasting;^ = additional displacement; RA = radius of influence of the heading 2.
three-dimensional problem, which, at some distance before or after the head- ing, is assumed to become two-dimensional. On fig. 10.57, the lines I-I and II—II represent the radial deformations d of the rock surface before and immediately after the blasting of the mass M. The heading itself should be represented as a disc stressed and strained radially.
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Figure 10.58 illustrates the evolution of the deformations with time and the progress of excavations, supposed to be continuous. Part of the rock deformation is said to be 'elastic' and occurs rapidly after blasting, the rest of the so called 'visco-plastic' deformation is due to the progressive loosening of the fissured rock mass. As shown in fig. 10.58, the deformation b of the rock support is only a fraction of the total deformation d, and d is that part of rock settlement which the rock support prevents.
I
(S) (D)
B C^Z i
a
1—t
A
/
1
l 'RA *
" 1
'2 I
,2 (S)
(Q
(D)
V
Fig. 10.58 (after Lombardi). Radial displacements at tunnel edge in a given control section as a function of time. / = time since the heading passed the control section;
d = radial displacement; D-D displacement versus time neglecting viscosity (continuous advance of the heading assumed); S-S displacement versus time including rock viscosity; tRA = time elapsed until the distance from the heading to control section RA on Fig. 10.57; h = installing support structure; t2 = support begins to work after closing of any gap; b = deformation of supports; d — absorbed rock displacement.
The basic idea of the approach of Lombardi and his team has been illus- trated in Fig. 10.50a and b. Characteristic lines of rock deformations d = (p*) are traced. They concern the elastic and the visco-plastic rock deformations. When the characteristic line cuts the ordinate p* = 0, there is no necessity for rock support; the tunnel is self supporting. Otherwise a rock support developing a radial pressure p* > pcr is required, /?* corres- ponding to the radial pressure required for keeping all rock deformations elastic.
Figure 10.59 shows rock characteristics (1) for an excavated gallery (tunnel bore), (2) for an elasto-plastic rock disc cut at the heading, (3) is the charac- teristic line relative to the rock core and (4) to the artificial rock support, d' represents the sum of all rock deformations occurring before the rock support takes over some load and becomes active. This line (4) cuts (1) at point A, causing a radial pressure p* > pcr on the rock surface. Without this support pressure /?*, the tunnel would collapse. Line (3) cuts line (2) at B. If not, the heading would collapse. There are therefore four possible stability or instability cases, depending on whether the tunnel roof or tunnel heading are stable or not.
Pitt
Fig. 10.59 (after Lombardi). Characteristic lines of deformations versus radial load for: (1) excavated gallery (tunnel bore); (2) rock disc at the heading; (3) rock core;
(4) artificial rock support, d' = sum of all rock deformations not absorbed by support; d = deformation of rock support corresponding to load/?*.
Any estimate of rock support based on rock deformations, like the method described in this paragraph, requires extensive in situ measurements of rock deformations. Figure 10.60 shows the arrangement adopted for measuring displacements of the rock inside the Gotthard Road Tunnel, now under construction in the Swiss Alps. Figure 10.60a shows the displacements of the points ex and e2 of the horizontal extensometer. As can be seen on the diagram, displacements begin even before the excavation has reached the relevant measuring cross-section. Diagram b of the same figure refers to the displacements D of the inclined deflectometer. On this diagram too, the deformations D are recorded against the distance from the measuring section.
In Fig. 10.60c the same deformations D are recorded versus time. This is the type of information required for tracing characteristic lines. The information obtained during the construction of the Gotthard Road Tunnel is being used for the design of the projected, far longer, St Gotthard Railway Basis tunnel. Figure 10.61 represents some of the many characteristic curves used for that project on the basis of in situ measurements.
Lombardi and his team have developed in detail a method which allows the correct prediction of expected rock displacements depending on the size and shape of the cavity and the physical characteristic of the rock mass. A rock stress diagram similar to the one shown in fig. 10.55 is obtained by trial and error, assuming boundary conditions for the rock support reaction on the rock surface, and for possible boundary conditions at the limit of the visco-plastic and the elastic zones inside the rock mass. Such assumptions must usually be corrected several times until an acceptable answer is obtained by iteration. The method used for tracing the stress-strain diagrams inside
distance (tunnel bottom-apparatus)
i i i i i i i i
0
- 100 - 80 - 60 5 (mm)
deflectometer 7
6 5 4 3 2L (ằ
1
- 4 0 - 2 0 0 20 40 60 80 100 120
distance (soffit-apparatus)
i i t i t t t t t t t i i
- 60 - 40 - 20 0 20 40 60 80 100 120
stages of construction 3 (mm)
110
Fig. 10.60 (after Lombardi, 1974). St Gotthard Road Tunnel, measurements of rock displacements in granite, (a) Displacement of points e1 and e2 of extenso- meter versus excavation progress; (b) deflectometer displacements versus excavation- progress; (c) same versus time. Progress of excavations: A - B soffit excavation to relevant measuring cross-section; B-C excavation of bottom part of relevant section;
C-D following up with excavation. AI = position of bottom and soffit relative to measuring equipment.
250 r
0 10
Fig. 10.61 (after Rutschmann, 1974). Rock characteristic lines (deformations S versus radial load pressures p*) for a tunnel diameter l l m and an overburden H = 1000 m (Gotthard Railway Basis Tunnel project). Rock parameters: elastic zone: <f> = 40°, c = 1.5 kg/cm2, E=4x 105 kg/cm2; ruptured zone: <£ = 35°, c = 0.5 kg/cm2, E = 2 x 105 kg/cm2. AV= increase in volume of loose rock mass.
1 = characteristic line for half rock core (heading); 2 of the tunnel wall.
the visco-plastic and elastic rock mass can be either the finite element method, or any other less sophisticated mathematical method the designer has evolved for that purpose.
A similar approach has been developed by Egger in a Ph.D thesis of Karlsruhe University (1973) and by Duffaut & Piraud (1975). The 'engineer ing approach' as the method based on rock deformations could be called, is attracting the interest of tunnel designers and is worth developing more systematically.
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