In the early stages of erosion most jointed rocks possess considerable effective cohesion ct of rock masses:
ct = cAJA,
(As = total area of solid rock area, A = total area, c = cohesion of intact rock material) and, as a consequence, they can form vertical or quasi- vertical slopes. The seat of most cohesion of the jointed rock is located in the 'gaps' which interrupt the continuity of the joints. As the height of the side walls of the erosion valley increases the shearing stresses in the rock adjacent to the walls increase correspondingly. When splitting of the gaps occurs cohesion, ci9 decreases. Local stress concentration causes new surfaces of failures to be superimposed on the local system of joints. Depending on the joint pattern, the sides of the valley may vary between 30° and 90°.
Sheeting is a form of joint parallel to the valley, occurring mainly in granite.
Typical types of delayed slope failure are the relatively superficial rock falls, and the more deep-seated rock slides. Falls may be connected with the
weakening effect of frost, whereas slides involve rock masses located below the limit of frost action. The shearing resistance, rr, at a given point, P, of a potential sliding surface in a porous and saturated material is given by the well-established empirical law:
TT = ct + (a — p) tan cf>.
According to Terzaghi, all intact and jointed rock masses with effective cohesion have the mechanical properties of brittle materials. Failure of brittle material slopes starts at the point where the shearing stress, r, becomes equal to rr. Stresses on the surrounding rock masses increase and progressive
failure occurs by brittle shear fracture. If the rock has a random pattern of jointing the shear resistance equation is valid for any section in any direction.
The rock behaves in an analogous manner to an unjointed stiff clay. It has been established that the steepest stable rock slopes are S-shaped, similar to the profile through root and tongue of a clay slide. The critical slope angle
<f>'c decreases with increasing height of the slope, but </>'c remains larger than cf>.
In regularly jointed rock the value of <f> depends on the type and degree of interlock between the blocks on either side of the sliding surface. The effec- tive cohesion, cu of rock masses is very much smaller than the cohesion, c, of rock material. Because of progressive failure cx tends towards zero and it is safe to assume that cx = 0.
9.1.1 Slopes in unstratified jointed rock
Rocks in this category (granite, marble) are divided by continuous random joints into irregularly-fitting blocks locally interconnected. This macro- structure of the rock mass is a large-scale model of the microstructure of intact crystalline rock. Such rocks were tested by von Karman (1911) and by Ros & Eichinger (1930). More recent tests by Borowicka (1962) revealed that the angle of friction of crystalline rocks varies from <f> = 40° at 100 kg/
cm2 pressure, to <f> = 25° at 1000 kg/cm2 pressure. According to Terzaghi, the critical angle <f>'c for slopes with underlying hard massive rock masses with a random joint pattern is about 70°, provided seepage is not acting upon the walls of the joints.
9.1.2 Slopes in stratified sedimentary rock
Stratified sedimentary rocks have layers varying in thickness between a few inches and many feet. These are separated from each other by thin films of material different from that of the rest of the rock. The bedding planes are almost invariably surfaces of minimum shear-resistance and are likely to be continuous. (In Terzaghi's paper they are referred to as bedding joints.) The cross-joints are generally nearly perpendicular to the bedding joints, and they are commonly staggered at these joints. The cohesive bond along the walls of the cross-joints is equal to zero. The intersections between the cross-joints and the bedding planes may be more or less parallel in one or more directions.
Less frequently they may be nearly randomly orientated.
Because of the almost universal presence of bedding- and cross-joints stratified sedimentary rock with no effective cohesion (c{ = 0) has the mechanical properties of a body of dry masonry composed of layers of more or less prismatic blocks which fit each other. The cohesion across the joints between all the blocks of each layer is zero. The stability of a slope will depend primarily on the orientation of the bedding planes with reference to the slope. This relationship is illustrated by figs. 9.1 and 9.2. Cross-joints
206 Rock slopes and rock slides
are assumed to be staggered and perpendicular to the bedding joints. The angle, <j>f of friction along the walls of all the joints is assumed to be 30°.
If the bedding planes are horizontal, no slide can occur, and the critical slope is vertical: (f>c = 90°.
9.1.3 Bedding planes dipping into the mountain
In fig. 9.1 the bedding planes dip into the mountain at an angle a. The line A-A cuts the rock mass at an angle 90° - a to the horizontal. If 90° - a < <f>f
no failure could occur along planes A-A (<f>f = angle at friction). If the cross-joints are parallel to A-A, but staggered, the position of the critical slope depends on the average value of the ratio CjD between the average length of the offset C between cross-joints and the average spacing D between bedding joints (fig. 9.1). For any value of a smaller than 90 — </>f the critical
90° - a
Fig. 9.1 Diagram illustrating the inclination, <f>0 of the critical slope, B-B, in stratified rock (after Terzaghi, 19626).
slope angle is equal to that of the line B-B in the figure. At any given value of a, the critical slope angle, <f>C9 increases with increasing values of the ratio CjD and at a given value of CjD it decreases with decreasing values 90° — a until 90° - a = <f>f = 30°.
At this point the critical slope angle abruptly increases to 90° because the slope angle of the cross-joints becomes smaller than the angle of friction (f>f = 30° along the joints. However, as 90° — a further decreases and a approaches 90° the danger of a failure by buckling of the layers located between bedding joints increases. Cohesion along the bedding joints increases the critical slope angle for any value of a smaller than 90° — <f>f. If the stratum is steeper, cohesion practically eliminates the possibility of a failure of the exposed stratum by buckling.
9.1.4 Bedding planes dipping towards the valley (fig. 9.2) If the bedding planes dip towards the valley at an angle smaller than the angle of friction, <f>f = 30°, the critical slope is 90°. For values of a greater
than 30° the critical slope angle is equal to a. If the slippage along bedding joints is resisted by effective cohesion ct in addition to friction, the steepest stable slope is no longer plane. Up to a certain height H it will be vertical as shown by fig. 9.2 and above it the slope will raise at an angle a.
Fig. 9.2 Greatest height, H, of a rock cliff. The rock cohesion, cu is not negligible (after Terzaghi, 19626).
Let H be the height of the vertical part of the slope. The force which tends to produce a slip along a bedding joint through the foot of the slope is (yr = unit weight of the rock):
yrH cos a sin a,
per unit of area of the bedding joint and the force which resist the slip is Ci + yrH cos2 a tan <f>f.
Hence the vertical slope as shown in fig. 9.2 will not be stable unless H yr cos a (sin a — cos a tan </>f)
An increase in height of the vertical slope would be immediately followed by a slide along the bedding plane B-B through the foot of the slope.