7.4.1 Laboratory tests on clastic and discontinuous models According to Trollope (1968) 'a clastic mass comprises an assembly of units, each unit having a finite physical shape, e.g. spherical, cubical, ellipsoid.
Depending on the shape of the boundary, the units will tend to pack in groups wherein some systematic arrangement will dominate and, in general, the mass is made up of varying arrangements of systematically packed zones.' This clastic model is in direct opposition to the accepted mathematical model of the 'continuum'.
Krsmanovic & Milic (1963) simulated a rock foundation on a model 100 x 100 x 17 cm made up of horizontal layers of parallelepipedic blocks 4-0 x 4-0 cm wide and 16 cm deep. The load, transmitted to the blocks through a slab 15 x 16 cm, was varied from 3 to 30 kg/cm2, the modulus of elasticity of the 'rock' from E = 40 000 to 23 500 and 15 670 kg/cm2. The angle of internal friction of intact 'rock' was about 50° to 56°, the angle of friction in the joints about 30° to 36°. Cohesion of intact hard rock was low, i.e. 0-12 kg/cm2 and compression strength 65 kg/cm2. Stresses and deformations (settlements) were measured for typical points of the model.
Stress distributions (figs. 7.21 and fig. 7.22) were shown to depend largely
b = 15 cm
oy(x •• 0)
1 pằ = 3.0 kg/cm^3^7= 18.0 kg/cm2 2 py = 12.0 kg/cm2 4 py = 30.0 kg/cm2
Fig. 7.21 Normal stresses ay in the joints of 'stratified rock' at different depths for loads py = 3,12,18 and 30 kg/cm2. Concentrated central force applied through an elastic slab (after Krsmanovtt & Milic, 1963).
b = 12 cm
1 Py = 10 kg/cmz 2 p = 20 kg/cm^
= 30 kg/cm2
Fig. 7.22 Normal stresses <ry in the joints of 'stratified rock' at different depths for loads py = 10, 20 and 30 kg/cm2. Load applied through a plate of great rigidity.
Free surface of model loaded uniformly with pi = 3 kg/cm for consolidation before beginning the tests (after Krsmanovid & Milic, 1963).
on: (1) the stiffness of the slab transmitting the load to the 'rock' model, (2) the amount of vertical prestressing a0 given to the jointed rock model.
It was found that the nature of the rupture in a discontinuum composed of jointed rock was much more complex than that in a grained semispace.
Rupture of the model occurs in very different circumstances.
The French laboratory of the ficole Polytechnique (Maury, 1970a) started experimental research on a similar problem of stress transmission through
The clastic theory of rock masses 183 discontinuous media. A first series of tests were based on the transmission of a load P through a plate to an elastic body resting on a stiff base of steel.
The second series concerned a similar load transmission through several elastic bodies piled on top of each other. The tests were carried out using a new photo-elastic material characterized by an elastic modulus E = 29 500 kg/cm2 and a Poisson ratio v == 0-43. This material shows an excellent linear response to stresses. During the tests no lateral stress was applied to the clastic body, which could expand freely in a lateral direction. The purpose of the tests was to measure shear stresses for different angles of friction either between two layers of plastic material or between these plastics and the steel basis.
An angle of friction <f> of only 1-5° was obtained when the smooth surfaces of the plastics were protected with thin layers of 'Teflon (R)' and 'Lanoline'.
Loading and unloading the plastic body up to twenty times did not change this angle at all. Uncoated plastics gave an angle <f> = 24° which was not altered when loading the model up to twenty times. An angle </> = 33° was obtained when using very fine sand between the polished surfaces of the plastics. Using the same sand on the polished surface between plastics and steel resulted in an angle of friction <f> = 37°. The small steel plate trans- mitting the load had a width a = 30 mm and a thickness e = 10 mm. It was very rigid compared to the plastic body of height h.
Other tests, carried out with an elastic plate of the same material as the tested body (e = 20 mm thick), were for varying values hja = 7, or 3 or 1-5 (fig. 7.23a). The shear stress rB on the base was compared to the stress
tensile stresses
/ / / / / / steel plate
(a)
steel plate' (b)
Fig. 7.23
TBO in the continuous half-space, at the same level. Table 7.5 shows the ratios rBjrBo obtained for different conditions. The stress distribution in the test model is more disturbed for small friction factors <f> = 1-5°, than for
<f> = 24° or <f> = 33°.
When the test piece is cut by several planes a similar effect is achieved.
The lower the angle of friction <f>, the greater the disturbance in the stress pattern (fig. 7.23b). With a low <f> value, the compression stresses are more
Table 7.5
* = 1-5°
* = 24°
<t> = 33° T
hja
BjrBo =
B/Tfl0 =
B/TBO =
= 3 / 2 0
1-3 10
\\a = 1 1-9 M 10
concentrated. Tensile stresses also develop which did not exist in the con- tinuous half-space. The use of discontinuous models to represent fissured rock masses has been further developed when testing dam models. Several laboratories represent rock abutments on the models by using a plastics material where the joints and fissures are represented to scale (Fumagalli, 1967). By varying the friction factor along these joints, it can be seen that weakening of the rock mass by jointing depends on the cohesion of the rock mass. More information on this technique will be given in the chapters concerning dam abutments.
These preliminary results show the danger and complexity of tensile stresses which may develop although the theory of the continuous half-space does not indicate such a possibility.
7.4.2 The clastic theory
Figure 7.24a, b, c and d are 'clastic models'. Assuming that there is no friction between the elements, spheres, ellipsoids, cubes, at the points of
*D
I.N.I l /
(a) (c)
Fig. 7.24 Clastic models.
Fig. 7.25 No-arching condition (after Trollope, 1968).
The clastic theory of rock masses 185 contact, it can be shown that for static equilibrium the contact forces /?, q, r in fig. 7.25a need to go through the element's centre of gravity.
Trollope (1957-68) introduces the degree of arching as a major variable in any problem. Figure 7.256 represents the general case where all the forces are greater than zero. In fig. 7.25c the no-arching condition is represented as X± = X2 = 0; in fig. 125d the full-arching condition is W2= Yx = 0.
In addition the a constant is the type of element and 0 the distribution angle. For particles of width D (fig. 7.246), distribution angle 0 and weight w, the average density for y is (depth of the model B = 1)
2w tan 0 y = .
The distance between successive layers is (d/2) cot 0. It may be readily checked that for squares (fig. 7.24c) tan 0 = \ and y = wjD2 (depth of the model B = 1). Expressions can be developed for stresses caused by the self- weight of the elements; others for the transmission of external forces from one layer to the next, assuming different values for k = qjp (fig. 7.25a) or k = YJY2 (fig. 7.256), k being the arching factor. Trollope tried to analyse Krsmanovic experiments using the clastic theory that he had developed for a load P.
Case of no arching (k = 1): In this case the forces are transmitted along 'distribution lines'. There are no horizontal contact forces and no relative movements of the blocks. The magnitude of the forces transmitted along the distribution lines will be P/2 cos 0. In the seventh layer the pressure distribution is as indicated in fig. 7.26.
In the case of 'full arching' (k = 0) forces P/cos 0 and P/2 cos 0 are transmitted along the distribution lines and the vertical stress distribution is different from the previous case (fig. 7.27). Neither distribution corres- ponds exactly to the tests by Krsmanovic, which most probably was for
P P p
1 IA1 /
1 \/N
|/| /|) 1/1/1/
/I/I/I
A A \
1 1 1 1 j