Design of Stable Concave Slopes for Reduced Sediment Delivery

Although a number of studies have shown that concave slopes produce less sediment than planar slopes, few studies have linked the benefits of concave slopes to mass stability. Utili and Nova (2007) demonstrated, by means of the upper-bound method, that concave log-spiral slopes result in higher FS values than planar slopes of the same height and width, as well as planar slopes of the same soil mass. Based on the slip-line field theory and the associated characteristic equations, Sokolovskiĭ (1960) found a longitudinal concave critical slope contour ($\mathrm{FS}=1$) for plane-strain conditions. Sokolovskiĭ’s solution is the result of a numeric boundary value problem, which for a medium with unit weight $\gamma >0$, cohesion $c$, and friction angle $\varphi >0$ has no analytical counterpart. Recently, Jeldes et al. (2013) proposed an approximate analytical expression to obtain coordinates of the critical contour for any desired combination of $\varphi$, $c$, $\gamma$, and slope height ($H_s$). This expression brings a system of differential equations into a single algebraic expression, which highly simplifies the manipulation and solution aswherewhere $H$ = tensile strength of the soil; $K_a=\left(1-\sin \varphi \right)/\left(1+\sin \varphi \right)$ is the Rankine active coefficient of earth pressure; ${\sigma }_y$ = geostatic vertical stress; and $h_{\mathit{cr}}$ is the height of the tension zone. The equation describes a slope contour in the quadrant with the $x\text{-axis}$ positive to the right and the $y\text{-axis}$ positive downward (Fig. 1), and with $h_{\mathit{cr}}$ lying above the $x\text{-axis}$ from the coordinates (0, 0) to (0, $-h_{\mathit{cr}}$). Note that this $h_{\mathit{cr}}$ tension zone does not contribute to resistance, but only to destabilization; this is an artifact used to obtain a complete mathematical solution of the problem. Whereas this assumption influences the way the characteristics would develop inside the soil medium, it was shown via the limit-equilibrium method (LEM) and FEM analyses that the outcomes are not significantly affected by this assumption (Jeldes et al. 2013). In practice, the sharp bluff at the top of the tension zone may be removed or rounded (as indicated by the dashed line in Fig. 1), adding conservatism to the slope design. A limiting slope height ($h_L$) was also found, defining where the failure mechanism changes. Slopes of heights $H_s<h_L$ indicate a toe-failure mechanism with FS values slightly greater than 1, whereas those with $H_s>h_L$ indicate a face-failure mechanism with a steady $\mathrm{FS}\approx 1$. A linear correlation between $h_L$ and $c/\gamma$ was approximated bywhich can be used as a preliminary estimate of the most likely failure mechanism for a concave slope (Jeldes et al. 2013).

Experimental and computational studies have shown that slopes with longitudinally concave shapes yield less sediment than slopes with planar or other curvilinear shapes. Here, the focus is on the difference in soil loss between longitudinally concave (referred to henceforth as concave) and planar slopes, which are the two shapes of interest in this study. Because erosion rates increase with slope length and steepness, for a concave slope, the increased erosion due to the increased length is partially counteracted by the decreased erosion due to decreasing slope steepness. In examining this, Young and Mutchler (1969) used field-scale plots on a silt soil and measured 10% less sediment loss on concave slopes with the same average inclination as planar ones. D’Souza and Morgan (1976) used laboratory scale models with a sandy soil, and measured 20–25% less soil loss on concave slopes with the same average inclination as planar slopes. Rieke-Zapp and Nearing (2005) reported 75% less soil loss for concave slopes with similar surface area as planar slopes using experiments conducted on laboratory models with silt soil. Reductions in erosion for concave slopes can also be predicted via computational models. Meyer and Kramer (1969), using the universal soil loss equation (USLE), reported less soil loss for concave slopes of the same height and average steepness as planar slopes, and the difference in soil loss increased with horizontal slope distance. Williams and Nicks (1988), employing the chemical, runoff, and erosion from agricultural management systems (CREAMS) field-scale model, computed approximately 50% lower soil loss for a concave slope of the same average inclination as a planar slope; grass cover and filter strips did not alter the ratio of soil losses between concave and planar slopes. Hancock et al. (2003), using the landform evolution model SIBERIA, computed up to 80% less soil loss for concave slopes of the same horizontal length and average inclination as planar slopes and showed that sediment loss decreases as the slope becomes more concave. Lee et al. (2004) simulated moderately intense storm conditions via the Hillslope Erosion Model and found that concave slopes yielded 17, 22, 24, and 28% less sediment loss than planar slopes when sandy clay, clay, silt, and silty clay soils were used in the simulation. Priyashantha et al. (2009), also using SIBERIA, computed approximately 50% reduction in soil loss between planar and concave slopes with the same overall inclinations and again found that the reduction occurs regardless of vegetation and climate type.

Whereas the aforementioned erosional studies reported significantly less soil loss for concave slopes, most compared the difference in soil loss between planar and concave slopes with similar average steepness, and none of them linked the concave geometry to mechanical stability considerations. Accordingly, this study begins with the examination of concave shapes satisfying a desired degree of stability and then investigates their ability to reduce sediment yield. This is accomplished by comparing the soil loss on concave and planar slopes with the same FS. Finally, this study investigates the impact of construction precision on the concave slope stability.

In this section, concave slopes for a given design FS for long-term stability are defined and the undrained shear strength required such that short-term stability is unlikely to govern is determined. This approach is conservative in that it neglects any additional shear strength that may be present due to partial saturation and resulting soil suction.

Soil loss for concave and planar slopes with the same FS values was investigated using the widely recognized revised universal soil loss equation (RUSLE2) (USDA 2008)where the predicted soil loss $A$ ($\text{Mg}/\text{ha}/\text{year}$) is directly proportional to the rainfall erosivity $R$ ($\text{MJ}\cdot \text{mm}/\text{ha}\cdot \text{h}\cdot \text{year}$), quantifying the rainfall’s erosive potential; the soil erodibility $K$ ($\text{Mg}\cdot \text{ha}\cdot \text{h}/\text{ha}\cdot \text{MJ}\cdot \text{mm}$) defining the soil’s susceptibility to that erosivity; the topographic factor $\mathit{LS}$ (dimensionless) representing slope length and steepness effects; the surface cover factor $C$ (dimensionless); and the conservation practices factor $P$ (dimensionless).

Through RUSLE2 the pairs of planar and concave slopes [Eqs. (1)–(6)] that had the same FS under the same values of $\varphi$, $c$, $\gamma$, and $H_s$ (Fig. 3) were investigated. For soil-loss calculations, the concave slopes were idealized by discretizing the surface into 10–20 linear segments. Because RUSLE2 determines soil loss over the horizontal projection of the slope, for slope segments steeper than 45° (which are not defined in RUSLE2), the soil loss was calculated assuming a planar slope of 45° over the total horizontal length occupied by the steeper part of the contour. This is possible because (1) the runoff is accumulated based on the horizontal projection of the slope rather than over the inclined slope surface length; and (2) in most cases, the amount of accumulated turbulent energy at the upper portion will be minor, because sheet erosion is occurring. Three different soil textures were employed for the analyses: silt loam (high erodibility), clay (moderate erodibility), and sandy loam (moderately low erodibility). A range of possible $K$ values for each soil texture (Haan et al. 1994; USDA 2008) was used (Table 1). All the simulations were conducted for two locations with very different climates: Dakota County, Minnesota ($R=\mathrm{2,180}\hspace{0.17em}\text{MJ}\cdot \text{mm}/\text{ha}\cdot \text{h}\cdot \text{year}$) representing dry weather conditions, and Monroe County, Florida ($R=\mathrm{10,500}\hspace{0.17em}\text{MJ}\cdot \text{mm}/\text{ha}\cdot \text{h}\cdot \text{year}$) representing high rainfall conditions. For simplicity, a bare surface ($C=1$) and no conservation practices ($P=1$) were modeled. A representative value of friction angle was assumed for each soil texture (Budhu 2011; Hough 1957). Values of $c/\gamma$ ranging from 0.2 to 3 m were employed for the clay soils and from 0.2 to 2 m for the silt and sandy loam soils. Table 1 summarizes the erosion and mechanical properties of the investigated soils.

The widespread use of high-accuracy autoguidance GPS-based construction equipment for grading and earthwork operations suggests that the construction of more complex, nonplanar slope shapes can become more commonplace. The effects of construction precision on the slope stability was investigated by considering the worst-case scenario: the vertical component of the contour is constructed deeper than designed, resulting in a steeper slope. Although the vertical accuracy of three-dimensional grade-control systems is commonly within 30 mm (Trimble 2013), an accuracy of $T=200\hspace{0.17em}\text{mm}$ was investigated, which may be also achieved by conventional equipment. Because the horizontal accuracy of GPS receivers is usually up to three times greater than the vertical (Berber et al. 2012), the horizontal errors are expected to stay in the millimeter scale, with no significant effects on stability. The effect of construction accuracy on the stability was investigated by (1) lowering by 200 mm the vertical component of various critical concave contours ($\mathrm{FS}=1$) for $\varphi =20,30,\text{and}\hspace{0.17em}40°$ and $c/\gamma =0.2,1,\text{and}\hspace{0.17em}5\hspace{0.17em}\text{m}$ [Eqs. (1)–(6)]; and (2) conducting stability analyses on them via the simplified Bishop’s method.

Results from LEM analyses are shown in Fig. 4 for the case $\varphi =30°$, where the computed FS values are plotted against the dimensionless value $H_s\gamma /c$. Each point represents a stability analysis on a concave slope of height $H_s$, cohesion $c$, and unit weight $\gamma$, and each ${\varphi }^{\ast }$ reflects a different value of ${\mathit{FS}}_D$. Note that $H_s\gamma /c$ is the inverse of Taylor’s stability number $N$ (Taylor 1948). A change in failure mechanism is observed to occur at similar values of $1/N$ regardless of ${\varphi }^{\ast }$. The change in failure mode observed on the critical concave slopes (Jeldes et al. 2013) is not altered by the use of a strength-reduction factor, and Eq. (7) can still be used to estimate the most probable mode of failure with $c^{\ast }/\gamma$ instead of $c/\gamma$. Concave slopes with $H_s>h_L$ show steady computed values of $\mathrm{FS}\approx {\mathit{FS}}_D$, whereas those with $H_s<h_L$ showed FS values higher (more conservative) than the $H_s>h_L$ case, with values increasing as $H_s\gamma /c$ decreases. Similar results were obtained for the $\varphi =20\hspace{0.17em}\text{and}\hspace{0.17em}40°$ cases.

It is important to note that the methodology proposed here is based on the slip-line field theory, which requires a value of $c\ne 0$ for a valid mathematical solution to exist. Most soils encountered in practice contain sufficient fines to provide at least a small amount of long-term cohesion, such as that used in the example problem. Although some design codes may not permit the use of cohesion for long-term analyses, there are many opportunities not governed by these codes, such as mine reclamation, where a small level of long-term cohesion would allow the design of concave slopes. For those cases where cohesion must be neglected, concave slopes with controlled degree of stability can still be depicted by (1) enforcing the upper portion of the slope to be planar and inclined at ${\varphi }^{\ast }={\tan }^{-1}\left(\tan \varphi /{\mathit{FS}}_D\right)$ degrees from the horizontal, (2) locating the critical slip surface using any sound LEM or FEM analysis, and (3) monotonically decreasing the inclination of the slope contour below the critical slip, such that the overall FS remains equal to ${\mathit{FS}}_D$. The rate at which the slope contour decreases can be chosen such that erosion resistance is optimized. Naturally, this approach is limited to slopes with $H_s$ sufficiently large so that face failure is the governing mechanism for the $c=0$ case, and it may not provide the same erosional benefits of continuous concave slopes that can be depicted for a small $c\ne 0$.

It must also be emphasized that Eqs. (1)–(6) constitute an approximate solution in analytical form, so even when $H_s>h_L$ there may not be a perfect match between ${\mathit{FS}}_D$ and the actual FS. As seen in Fig. 4, the best match occurs when ${\varphi }^{\ast }\approx 20°$. Concave slopes with ${\varphi }^{\ast }>20°$ yield FS values higher than the design ${\mathit{FS}}_D$ (4% higher when ${\varphi }^{\ast }=30°\hspace{0.17em}\text{and}\hspace{0.17em}6\%$ higher when ${\varphi }^{\ast }=40°$), which is conservative for design. On the other hand, slopes with ${\varphi }^{\ast }<20°$ will have $\mathrm{FS}<{\mathit{FS}}_{D\text{,}}$ which is not conservative. This is shown in Fig. 4 for the case ${\varphi }^{\ast }=16.1°$, where the resulting $\mathrm{FS}=1.92$ is 4% less than the desired ${\mathit{FS}}_D=2$. Nevertheless, the deviations from the target ${\mathit{FS}}_D$ obtained through the methodology do not compromise its applicability for design, considering that neither the geometry nor the shear strength parameters are usually determined within $\pm 6\%$ accuracy (Duncan 1996). Furthermore, because the FS values calculated by LEMs that satisfy all conditions of static equilibrium can vary as much as 12% (Duncan 1996), an accuracy of $\pm 6\%$ is certainly acceptable. Once the desired shape for the design FS has been obtained, verifications can be conducted employing commercial slope stability software. In the same way, the effects of transient groundwater flow and/or external surcharges on the FS can be investigated, which extends the applications of this solution to other field conditions.

The minimum undrained shear strength $S_u^{\min }$ that the soil must possess for concave slopes to have $\mathrm{FS}>{\mathit{FS}}_D$ in the short term can be obtained from Fig. 5 as a function of the effective strength parameters ${\varphi }^{\ast }$ and $c^{\ast }$. Fig. 5 is a solution chart that assembles the numerical results from Eqs. (1) and (10) and is presented in terms of the dimensionless parameters $y\cdot \gamma /c^{\ast }$ and $S_u^{\min }/\left(c^{\ast }\cdot {\mathit{FS}}_D\right)$ for a range of ${\varphi }^{\ast }$. Here, $y$ is the slope height below the tension zone ($y=H_s-\left|h_{\mathit{cr}}\right|$, $y>0$). The inner chart in Fig. 5 provides higher resolution for low values of $y\cdot \gamma /c^{\ast }$. Note that Fig. 5 defines the conditions under which long-term stability will always govern, so the solutions are conservative. Also, this chart is based on an assumed failure mechanism, where the critical surface exists from or above the toe. Consequently, this solution provides a first check for the short-term stability of concave slopes and does not replace the necessity of conducting stability analyses once the undrained shear strength of the soil has been determined.

Soil loss results obtained from RUSLE2 for the silt loam are shown in Fig. 6. All the analyses were conducted on concave and planar slopes with $\mathrm{FS}={\mathit{FS}}_D=1$ ($\varphi ={\varphi }^{\ast }$ and $c=c^{\ast }$), because they provide generic results that can be extended to any FS. Figs. 6(a and b) compare the predicted total soil loss for concave and planar slopes of a silt loam with $c^{\ast }/\gamma =1$ under wet weather conditions in Monroe County, Florida, and dry conditions in Dakota County, Minnesota. Planar slopes yielded more sediment than equally stable concave slopes for all $K$ and $H_s$ values investigated, regardless of climate. Soil loss increased with $K$ and $H_s$ in both profiles, but the upper surfaces (soil loss from planar slopes) remain constantly above the lower surfaces (soil loss from concave slopes). Fig. 6(c) compares, for the two weather conditions, the variation in $A_p-A_c$ ($A_p$ = soil loss from planar slopes, $A_c$ = soil loss from concave slopes), or the erosion savings incurred using equally stable concave slopes. This illustrates that concave slopes can reduce erosion to a much larger extent in climates with high rainfall erosivity.

Generalized results for the silt loam soil are shown in Fig. 6(d) in terms of the ratio $A_c/A_p$ and the stability number $H_s\gamma /c^{\ast }$. Because the same RUSLE2 $R$ and $K$ were used in both planar and concave slopes, they cancel out when the results are presented in terms of $A_c/A_p$ and the only factor affecting differences in the soil loss is the shape of the slope profile. The use of $H_s\gamma /c^{\ast }$ enables us to represent the results of different slopes with the same degree of stability for a given value of ${\varphi }^{\ast }$. For the range of $H_s\gamma /c^{\ast }$ values investigated, $A_c/A_p$ ranges from 0.85 to 0.60 [Fig. 6(d)], indicating that concave slopes yield 15–40% less sediment than their planar counterparts. Similar results are shown in Figs. 7 and 8 in terms of $A_p-A_c$ and the $A_c/A_p$ ratio for ${\varphi }^{\ast }=20°$ (clay soil) and ${\varphi }^{\ast }=35°$ (sandy loam soil), respectively. The computed $A_c/A_p$ values show soil-loss reduction of the same order and suggest that the concave slopes proposed in this article will reduce sediment delivery by 15–40% regardless of soil erodibility and climate. This finding is extendable to slopes having natural or artificial surface covers ($C<1$), where the erosion is reduced approximately proportionally for both concave and planar slopes.

In Table 2 the relative reduction in soil loss [$1-\left(A_c/A_p\right)$] obtained here is compared with values reported in the literature. Although a wide range of erosion reductions were found there, it should be noted that the concave and planar slopes in those studies likely do not satisfy the equal degree of mechanical stability (FS) purposely enforced here. Similar reductions were obtained among the studies except for those reported by Meyer and Kramer (1969), Hancock et al. (2003), and Rieke-Zapp and Nearing (2005), for which the reductions were significantly higher. Meyer and Kramer (1969) and Rieke-Zapp and Nearing (2005) observed that an important amount of sediment deposition occurred at the lowest part of the concave contours, which may explain the higher overall reduction in sediment yield for those studies. In contrast, sediment deposition was not indicated in the RUSLE2 prediction obtained here. Each segment of the concave slopes proposed in this article is the steepest possible to satisfy mass equilibrium requirements, and sedimentation on the profile will likely not occur, especially at large values of ${\varphi }^{\ast }$.

Fig. 9 shows results obtained from the construction sensitivity analyses for concave slopes with $\mathrm{FS}={\mathit{FS}}_D=1$ ($\varphi ={\varphi }^{\ast }$ and $c=c^{\ast }$) for a range of ${\varphi }^{\ast }$ and $H_s\gamma /c^{\ast }$ values, with results expressed as the percent decrease in FS due to the construction error. Slopes with $H_s\le h_L$ were excluded from the analyses, because it was previously shown that their performing FS values are higher than required (conservative), and they did not present stability variations below equilibrium ($\mathrm{FS}=1$). The results demonstrate that the FS values are not significantly influenced by improper construction within the applied 200 mm of vertical accuracy. The largest difference was found for slopes with $c^{\ast }/\gamma =0.2\hspace{0.17em}\text{m}$, where 5–5.5% lower FS values were obtained. Considering that the analyses were conducted with a vertical accuracy only $1/6$ of that which can be achieved with typical autoguidance equipment, it was concluded that the stability of concave slopes is not significantly compromised by the precision of GPS-guided equipment. In addition, practicality suggests that during construction, the top vertical section will likely not be built as depicted here, but will likely be rounded back to a convex shape with a safe average inclination chosen by the designer; this would create a more complex slope with a more natural looking profile. Note that the vertical section corresponds to the maximum surface loading allowed for stability, and any reduction in this loading by shaping this area back results in a reduced surcharge loading and added conservatism. This zone is assumed to have no contribution to the stability or resistance.

Table 3 summarizes the following initial steps: (1) obtain ${\varphi }^{\ast }$ and $c^{\ast }$ with Eqs. (8) and (9) and (2) estimate the failure mechanism via $h_L$ [Eq. (7)] with $c^{\ast }$ instead of $c$. The $x\text{-coordinates}$ of the concave profile were then obtained by introducing ${\varphi }^{\ast }$ and $c^{\ast }$ in Eqs. (1)–(6) for different values of $y$. For the sandy slope, $8.6\hspace{0.17em}\text{m}=h_L<15\hspace{0.17em}\text{m}$ and the concave slope will have a face failure mechanism with a performing $\mathrm{FS}\approx {\mathit{FS}}_D=1.5$. Verification of this condition was provided via FEM analysis (Fig. 10), which yielded $\mathrm{FS}=1.51$. The equivalent planar slope with $\mathrm{FS}=1.5$ is depicted with dashed lines in Fig. 10. Similarly, the fine-grained slope presents a face failure ($22\hspace{0.17em}\text{m}=h_L<30\hspace{0.17em}\text{m}$). In this case, because ${\varphi }^{\ast }<20°$, a small error was introduced and the FEM analysis indicated a $\mathrm{FS}=1.45$.

To investigate the sensitivity of the mechanical stability on the value of the selected cohesion, the stability of the slope created in Fig. 10 was analyzed by the FEM for values of cohesion less than that used to obtain the concave profile; values of $\left(2/3\right)c$, $\left(1/2\right)c$, and $\left(1/3\right)c$ (10, 7.5, and 5 kPa, respectively) were used in the analysis. The results suggest that the FS would decrease to 1.25, 1.10, and 0.89, respectively. These examples illustrate the role that the small value of the long-term cohesion has on the computed stability, and the importance of determining it very carefully. However, it should be kept in mind that a concave slope with the design ${\mathit{FS}}_D$ could have been obtained for each of these values of $c$. Each unique combination of $\varphi$ and $c/\gamma$ will result in a unique concave shape with $\mathrm{FS}\approx {\mathit{FS}}_D$. This implies that as $\varphi$, $c$, or $\gamma$ change, the overall shape of the profile will change, but the expected FS and the expected range of erosion reduction will remain unchanged.

Note that the horizontal extent of the concave and planar slopes in Fig. 10 is similar, but there is less material involved in the concave slope. This has no effect on the balance of cut-and-fill volumes, because the analytical expression for the concave slope could be incorporated into the calculations. However, in mine reclamation, where a shortage of material is common, the construction of concave slopes could be an advantage and provide the benefit of more natural looking slopes and lower sediment loads.

A conservative verification can be made using Fig. 5. For the sandy slope ($\varphi =35°$ and ${\varphi }^{\ast }=25°$), $y=H_s-\left|h_{\mathit{cr}}\right|=13.4\hspace{0.17em}\text{m}$, and $y\cdot \gamma /c^{\ast }=25.4$. From Fig. 5, $S_u^{\min }/\left(c^{\ast }\cdot {\mathit{FS}}_D\right)=5.5$, and thus the long-term stability of the sandy slope will govern as long as $S_u^{\min }\ge 83\hspace{0.17em}\text{kPa}$. Similarly, for the fine-grained soil $y=26.3$, $y\cdot \gamma /c^{\ast }=19.2$, and long-term stability will govern provided that $S_u^{\min }\ge 150\hspace{0.17em}\text{kPa}$.

Total soil loss was computed for low and high erodibility values for each soil texture (Table 4) for the Monroe County, Florida, weather conditions. The results (Table 4) reveal that the sandy concave slope yields on average 24% less sediment than its planar counterpart. Similarly, the fine-grained concave slope yields on average 35% less sediment than a planar slope with the same ${\mathit{FS}}_D$.