Probabilistic Assessment of Commercial Design Guides for Steep Reinforced Slopes: Implications for Design

Deterministic procedures are still the predominate methods used in reinforced slope design. These procedures generally use limit equilibrium methods based on accepted factors of safety. This paper presents a method of probabilistic analysis which is easily adapted to existing deterministic methods. The method is used to analyze the reliability of a two steep reinforced slopes designed using commercially published guidelines. The reliability of internal failure modes (passing within the reinforced zone) and composite failure modes (passing mostly outside of the reinforced zone) are computed and compared. The results indicate that internal and composite failure modes may be nearly equally likely leading to a total probability of failure of twice that of either mode. The paper present a design process to ensure that the probability of failure is controlled by composite failure modes and the reliability of internal modes is sufficiently high that they do not significantly reduce the total reliability of the slope. SLOPE DESIGNS ANALYZED This paper exams two reinforced slopes designed using typical deterministic methods and analyzes these designs using probabilistic methods. The slopes are 45° and 70° slopes reinforced with geogrids. Both designs consist of uniform cohesionless soil both within and below the slopes. The slopes were designed using guidelines published by Tensar (1988). These design guidelines were developed with a two wedge failure surface satisfying only force equilibrium, Schmertmann et al (1987). For a uniform soil with a given friction angle, φ, and slope angle, β, the key design parameters are: the length of the reinforcing elements, L, chosen so that composite failure modes have a satisfactory factor of safety, and the spacing, S, and allowable reinforcement force, Ra, selected so that internal failure modes have a satisfactory factor of safety. The two slopes designed using the Tensar (1988) guidelines are shown in Figure 1. Figure 1. Cross-sections of two reinforced slopes analyzed showing length and spacing of geogrid reinforcement DETERMINISTIC ANALYSES After the slopes were designed using the Tensar (1988) guidelines they were reanalyzed using UTEXAS3 (Wright 1991). This analysis used a full equilibrium method with circular failure surfaces. This allowed failures extending into base materials below the slopes to be investigated. It also allowed investigation of the relationship between allowable reinforcement force and reinforcement length and how these variables affect the location of the critical surface. Nineteen distinct failure modes were investigated: 18 internal modes passing through the reinforced portion of each slope, and one composite mode passing predominately outside the reinforced portion of each slope but intersecting a few reinforcing elements. The failure modes were numbered 0 (zero) for the composite mode and 1 through 18 for the internal modes, the number indicated the lowest reinforcing elements through which the failure surface passed. Typical failure modes are shown in Figure 2. Figure 2. Failure modes of 45 degree showing numbering scheme For each failure mode a search for the critical slip surface was performed to determine the local minimum factor of safety for that mode. The computed factors of safety for each mode are shown in Figure 3. Figure 3 shows that the Tensar (1988) design guidelines generate slope designs where the composite failure mode controls and the critical surface passes predominately below and behind the reinforced portion of the slope. Figure 3 also shows that the internal failure modes have a relatively uniform factor of safety. Both these observations are in keeping with the original intent of these design guidelines (Schmertmann et al, 1987). Figure 3. Comparison of the factors of safety computed for each failure mode A second series of analyses was done to investigate the relationship between allowable reinforcement force and reinforcement length. In this series of analyses the vertical spacing of geogrids was kept the same, but the reinforcement length and allowable reinforcement force were varied. A large number of combinations of reinforcement length and force were analyzed; for each combination the minimum factor of safety was computed and the location of the critical circle identified. Figure 4, show the results of this analysis for the 45° slope. Figure 4 shows that for relatively long or weak reinforcement (the lower shaded portion of the figure) the failure mode is internal and the factor of safety is a function of only the reinforcement force. Figure 4 also shows that for relatively strong or short reinforcement, the failure mode is composite and the factor of safety is governed by the reinforcement length. Note that the transition from internal to composite failure mode is quite abrupt. It is also worth noting that the Tensar (1998) design guides have a composite failure mode close to the point where the failure surface transition from composite to internal. PROBABILISTIC ANALYSES Using the deterministic analyses as a starting point, the reliability index, β, was computed for each of the 19 failure modes. These analyses used two random variables: the shear strength or the soil, s, in the form of s = tan(φ), and the reinforcement force, R. The reinforcement length and spacing were assumed to have no significant variance. The limit state function was taken to be ( ) , 1 ( , ) g s R F s R = − (1) Where, F is the factor of safety computed using standard limit equilibrium methods. Figure 4. Factor of safety as a function of reinforcement length and force for the 45 degree slope The mean-value first-order reliability index, βMV, because of simple form