Discussion of “Placed Rock as Protection against Erosion by Flow down Steep Slopes” by W. L. Peirson, J. Figlus, S. E. Pells, and R. J. Cox

The discusser found the authors’ paper to be an interesting contribution to the Journal and to the advancement of the use of stone in the design of steeply sloping channels. The discusser and various associates have also carried out some studies on the stability of stone on steep slopes, which is the primary focus of this discussion.

Smith and Kells (1995) present the development of a semiempirical relationship for the median size of stone required for stability in a two-dimensional open channel flowwhere $d_m=\text{median}$ size of stone in the stone layer $\left(m\right)$ ; $q=\text{unit}$ discharge $({\mathrm{m}}^3∕\mathrm{s}∕\mathrm{m})$ ; and $S=\text{slope}$ of the channel (m/m). In this development, it has been assumed that the value of Shields parameter for the critical shear stress is equal to 0.060 and that the specific gravity of the stone material is equal to 2.65. Manning’s equation was used as the flow equation, with the following Strickler-type relationship used to quantify Manning’s $n$ For values other than 0.060, 2.65, and 0.049 as referred to above, the coefficient in Eq. (1) will be somewhat different than 1.75. In fact, for the authors’ $76\text{-}\mathrm{mm}$ sandstone particles having a specific gravity of 2.29, the coefficient given in Eq. (1) would be equal to 2.30, which in part is indicative of the sensitivity of stone stability to specific gravity as also noted by Peirson and Cameron (2006).

A similar expression for the required stone size can also be derived from an equation developed by Stephenson (1979), as shown by Smith and Kells (1995)which is remarkably similar to Eq. (1). The coefficient in Eq. (3) was derived on the basis of Stephenson’s $C$ coefficient having a value of 0.245, which is the midpoint of the values reported for smooth pebbles (0.22) and crushed granite (0.27), a stone-specific gravity of 2.65, a stone mass porosity of 40%, and an angle of repose of the stone material of 35°. Using the mean angle of repose or friction angle for the stone material reported by the authors of 48.8°, a stone-specific gravity of 2.29 for their $76\text{-}\mathrm{mm}$ sandstone material, and a mean porosity of 46% for the randomly placed stone material as determined from the authors’ data, the coefficient in Eq. (3) would be equal to 1.41.

Abt et al. (2008) introduce the coefficient of uniformity, $C_u$ , into the stability relationship, which they express as (SI units)where $d_{50}=\text{median}$ stone size (cm), which is the same as $d_m$ when both are defined in terms of stone weight; ${\mathrm{C}}_{\mathrm{u}}$ is the coefficient of uniformity of the stone material (dimensionless); and $q_f=\text{design}$ unit discharge $({\mathrm{m}}^3∕\mathrm{s}∕\mathrm{m})$ . For a uniformly graded stone material with, for example, ${\mathrm{C}}_{\mathrm{u}}=2$ , Eq. (4) is very similar to that given by Smith and Kells (1995) in Eq. (1) and Stephenson (1979) in Eq. (3).

The relationship presented in Eq. (1) can be expressed in more general terms aswhere $K$ is a coefficient which, as shown above, reflects the integration of such parameters as stone-specific gravity, stone mass porosity, angle of internal friction of the stone material, particle roughness, and the coefficient in the Strickler-type relationship for Manning’s $n$ [i.e., Eq. (2)]. In the work reported by Smith and Kells (1995), which draws considerably on earlier work described in Smith and Murray (1975), it was found that the value of $K$ is 1.8 for initial stone movement, 1.5 for initial failure of the stone mass (which thereafter heals itself provided that a sufficient amount of stone material has been placed at the crest of the slope), and 1.2 for ultimate failure of the stone-paved slope. Here, initial stone movement refers to the point at which a single stone is removed from the stone layer and transported to the bottom of the slope, initial failure is the point at which temporary exposure of the underlying slope material occurs (e.g., exposure of the filter layer), and ultimate failure is when permanent exposure of some or all of the underlying slope material or filter layer takes place. The discharge used in the analysis was the total discharge over the crest. The stone material was uniformly graded for all but one test (and no difference was detected in this regard) and was classified as semirounded in shape.

Smith and Kells (1995) indicate that the discharge required for ultimate failure was 25–50% or more than that required to produce initial failure. The larger increase in discharge was required for the flat gradient tests (i.e., $S=4\mathrm{\%}$ to 7%) of Smith and Murray (1975). That a smaller increase in the discharge beyond the initial failure condition was required to produce ultimate failure for the larger slope tested by Kells and Smith (i.e., $S=20\mathrm{\%}$ ) is in accordance with the authors’ findings about the reduced “ductility of the armor instability processes with increasing armor slope.”

Although the work of Smith and Murray (1975) was based on flat-gradient slopes ranging in value between 4% and 7%, Smith and Kells (1995) subsequently showed that the same stability relationship could be applied on much steeper slopes of up to 20%. They concluded in an inferential way that the applicability of the same stability relationship, which was essentially developed on the basis of a flat sloping channel as implied by the use of Shields stability parameter, was due to the increased interparticle forces that occur between the stones placed on a steep slope. Fathalla and Kells (1999) studied slopes as high as 35%, although they developed a somewhat different relationship from that given in Eq. (1). Among others, however, they concluded that it is important to consider the stability of the stone layer rather than simply the stability of the individual stones comprising the layer.

On the assumption that the authors’ values of $q_{\mathrm{tot},\text{fail}}$ shown in their Table 2 reflect the unit discharge on the slope at the failure condition (not fully defined in the paper) for randomly placed stone, the discusser has determined the corresponding $K$ values from Eq. (5) using the authors’ data as shown in Table 1. As indicated, the $K$ values for the authors’ data are substantially less than those given in Smith and Kells (1995), regardless of whether one compares to Smith and Kells’s initial $(K=1.5)$ or ultimate $(K=1.2)$ failure conditions. This finding suggests that the findings of Smith and Kells are more conservative than those given by the authors. Of course, the differences in stone-specific gravity must be considered, but this difference does not explain the difference between the findings based on the authors’ data and those of Smith and Kells. Moreover, the slope dependency that is evident on the basis of the authors’ data was not observed in the work of Smith and Kells.

Interestingly, for large slope values (e.g., $0.40\mathrm{m}∕\mathrm{m}$ ), the findings of Smith and Kells (1995) more or less agree with those of Peirson and Cameron (2006) for the ultimate failure condition (cf. Fig. 4 in Peirson and Cameron 2006), but are less conservative for lower slopes for the same failure condition. Inclusive of some margin of safety, Smith and Kells (1995) suggest that $K=1.8$ in Eq. (5) is a reasonably appropriate value for design purposes.