The design of many geotechnical structures is based on limit state conditions. For example, in designing gravity walls, the load exerted by the retained soil on the wall is based on an assumed active state of stresses. That is, the wall moves sufficiently outward, enabling the retained soil to mobilize its shear strength along a slip surface.
Although not frequently used, the result also includes the corresponding critical shear surface of the soil wedge retained by the wall. Depending on the soil, this movement could be very small. However, the shearing resistance of the soil reduces the resultant of active load acting on the wall. In design and through force equilibrium at a limit state (i.e., a state in which the soil mobilizes its strength along a slip surface), one seeks the highest value of this resultant. The result then is the active load needed for designing a stable wall.
In addition to the active limit state, passive resistance may develop in front of the embedded portion of the wall as it is displacing into the resisting soil. To develop the full passive resistance, the shear strength of the soil in front of the embedded portion of the wall needs to be mobilized, albeit after larger movement than the active state.
For the passive limit state the designer then seeks the lowest resistance. Limit state equilibrium renders the minimum passive resistance and its associated failure wedge. This retaining wall example demonstrates that conventional design aims to accommodate “worst-case,” extreme-value conditions to ensure a stable gravity wall; driving loads are maximized and resisting loads are minimized.
Slope design abides to a similar mantra. In fact, it is more general and versatile than that of retaining walls. For a given soil profile and field conditions such as water and seismicity, one can assess the maximum mobilization of the shear strength of the soil while considering multiple feasible failure surfaces. Dividing the soil actual strength by its maximum mobilized resistance for a given slope’s many potential failure surfaces renders a minimum factor of safety (FS), the objective of slope-stability analyses. A common technique to assess stability of slopes is limit equilibrium (LE) analysis. The same versatile approach can assess the maximum load that a footing can carry before fully mobilizing the soil strength, the resistance of an anchor embedded in fill, or the maximum active load on a wall. LE analysis has been used successfully for many decades in complex geotechnical designs.
Design of geosynthetic-reinforced soil can be viewed as a subset of LE slope stability analysis. Consequently, rational design of complex reinforced problems can be conducted without the need for empirical distinction between reinforced walls and slopes. In the context of reinforced walls, it can account for facing effects and toe restraint at a limit state. In fact, it can be used to rationally determine the load in a geosynthetic at its connection to the facing.
Technical details of the LE analysis as related to designing geosynthetic-reinforced soil structures have been published recently (Leshchinsky et al. 2016, FHWA-HIF-17-004, “Limit Equilibrium Design Framework for MSE Structures with Extensible Reinforcement”; it can be downloaded from the FHWA Geotechnical page of the Office of Bridges and Structures website). It is an extreme-value problem in which, for a given layout of reinforcement, one seeks the distribution of maximum load in any geosynthetic layer, at any location, including at the connection, so as to uniformly mobilize the soil strength along any surface within the soil mass. The result is termed a “baseline solution,” as it provides the basis for adequate selection of reinforcements and connection specifications. That is, rather than dealing with specific products, the baseline solution enables the designer to economically select geosynthetics and connectors having sufficient long-term strengths.
Current LE design of geosynthetic-reinforced slopes considers the long-term tensile capacity of the reinforcement along its length, as depicted in Figure 1. For simplicity, this figure is applicable to vertical slopes having a surcharge-free horizontal crest.
The reinforcement has long-term intrinsic strength limited by pullout resistance at its rear and front end. At the front, pullout resistance starts with the connection capacity. Moving away from the face, pullout resistance due to interaction between soil and the geosynthetic is added to the connection capacity at the front (see Figure 1). The available tensile capacity at the intersection of an analyzed slip surface is considered in the limit equilibrium equations.
Examining numerous slip surfaces, many of which intersect the specified reinforcement layers, one can identify a critical state. A critical state represents the maximum mobilization of soil strength and its associated slip surface. It is noted that the reciprocal of maximum mobilization of soil strength is the minimum safety factor (FS). However, by examining just the critical result, one misses the perspective of the state of stability of the entire slope. Consequently, Baker and Leshchinsky (2001) introduced the safety map, which considers the calculated safety factor associated with each analyzed surface. This spatial distribution of safety factors can then be plotted as contour lines or as a color-coded map for assessment of regions of instability within a given slope profile. This diagnostic tool indicates the state of stability of a given slope system at a glance, suggesting where, for example, reinforcement is needed or where it is just excessive. Such an objective diagnostic can directly guide production of an optimal design. Figure 2b illustrates the safety map using circular slip surfaces combined with Bishop’s stability analysis.
The simple problem in Figure 2a includes a sloping toe, a broken backslope, and a steep slope. Without reinforcement, the critical slip circle will coincide with the slope face rendering a FS of 0.27. While the slope is clearly unstable, such critical slip surface does not provide a clue as to the zone within which instability exists. After specifying the layout and strength of the reinforcement as shown in Figure 2b, the resultant safety map shows that within a large range the safety factor varies between 1.33 and 1.50, implying an optimal design. The map shows that the safety factor is indeed a variable in space.
Although the safety map has proven to be a useful tool in designing reinforced walls and slopes, it does not provide the baseline solution. That is, it does not produce the connection load and the required distribution of tensile resistance along geosynthetic layers so that FS at each location would be the same. While it is useful for checking the suitability of a specific product, the current approach is not sufficiently generic to identify products (geosynthetic and connectors) that meet or exceed the design requirements. To achieve this generic baseline solution, the methodology presented in FHWA-HIF-17-004, and termed here in retrospective as “Soil Reinforcement 101,” was introduced.
While the computational effort associated with the new methodology is intensive, it is based on simple and tangible principles of LE slope-stability analysis. However, it is the converse of a safety map in a sense that instead of determining a spatial FS, the available strength of reinforcements within a system are adjusted to attain the same soil strength mobilization (i.e., constant FS) for any slip surface within the reinforced mass. In other words, the reinforcement layers are considered as virtual elements having variable strength. The process systematically examines slip surface emerging at the face, starting at the crest and ending at the toe or beyond; it is termed top-down search. In the process, the rear pullout capacity serves as a limit to which a reinforcement layer can mobilize its required force. If the required force is lesser than that afforded by rear pullout capacity, that force is valid. As the top-down process continues, the load in the reinforcement layers above the exit point may increase to ensure that for each slip surface there is sufficient tensile resistance to produce the target FS. While the force in the reinforcement may diminish greatly as it approaches the face of the wall/slope, some front end pullout resistance is needed to ensure that the force in the reinforcement can actually develop. This is done by adjusting the pullout resistance envelope at the front by moving it upward until it is tangent to the calculated required tensile resistance needed to render a prescribed FS. The amount of upward translation of the front pullout resistance signifies the connection load or the minimum required connection strength. This “Soil Reinforcement 101” rationally and robustly produces two important values for each layer needed in design: the maximum load, Tmax, and the connection load, To, needed to ensure that the reinforcement can indeed develop its resistance. These basic values are needed for a limit-state design and are a function of factors such as soil-reinforcement interaction parameters, soil properties, layout of reinforcement, facing strength, water, and seismicity.
Once the reinforcement is selected, the traditional global stability analysis, using the safety map and the specified reinforcement, needs to be conducted to ensure that that FS for global stability considering foundation failures (analog to bearing capacity), compound failures, and sliding all exceeds a minimum value of, say, 1.3 or 1.5, depending on the design requirements.
The following simple design examples are intended to make it easier to understand the methodology as well as appreciate its significance.
Consider the problem shown in Figure 3. It represents a simple wall (facing is ignored in this example); the interaction parameters are F*=0.8 and α=0.8 (Ci=0.64). When calculating front or rear pullout resistance in all examples, its computed value is reduced by a factor of 1.5 to account for uncertainties.
Figure 4 displays the calculated tensile resistance distribution along each layer constrained by the rear end pullout capacity. Top-down calculations are done using circular arc combined with Bishop’s method having a target FS=1.0. The blue lines show the superimposed rear end pullout capacity available along each layer. As seen, the tensile resistance mobilized in the upper four or five layers is affected by pullout capacity. Higher pullout capacity should enable upper layers to mobilize higher loads.
Figure 5 demonstrates how the connection load is computed. It shows the calculated load in the reinforcement of the top layer. The black color envelope shows the pullout capacity without any connection load (i.e., the geosynthetic terminates at the face of the slope). Clearly, the red area above this black envelope cannot be realized or mobilized by the reinforcement. However, translating or shifting the black envelope upward until it is a tangent to the red zone—see the green envelope—ensures that the computed load (the red zone) indeed can develop. The amount of shifting is the connection load required to allow for a limit state to develop.
Figure 6 is a color-coded map showing the distribution of load in each layer. The color-coded map is termed a tension map, in a sense, analog to the safety map. The location of the maximum load, Tmax, among layers is not along a singular circle, as commonly assumed in current analysis.
Figure 7 shows the calculated maximum force as well as the connection load for each layer. In AASHTO, the connection load is arbitrarily taken as 100% of Tmax.
Figure 7 indicates that the connection load in LE analysis framework is substantially smaller than the maximum load in the reinforcement. Note that its values are for limit state conditions where some small movement can develop. The maximum load in the reinforcement in LE is approximately half of that determined from AASHTO’s approach. If a single type of geosynthetics is specified with a combined reduction factor or RF=2.0 for installation damage, durability, and creep, and a factor of safety of 1.5 on its strength, then the required LE ultimate strength, Tult, would be about 2,140 lb/ft whereas AASHTO would be about 4,040 lb/ft.
Now that reinforcement has been selected, global stability needs to be assessed. In essence, this is the same requirement as in AASHTO. Figure 8a is an example of such an analysis. It uses Bishop’s analysis combined with reinforcement’s long-term strength (Tult/2.0) of about 1,070 lb/ft. The calculated FS is 1.31, an acceptable value per AASHTO. The critical circle reflects bearing capacity type of potential failure. The red zone is over a large zone within the reinforced zone, indicating an efficient design. AASHTO requires assessment of bearing capacity using Meyerhof’s eccentric load approach. Following AASHTO procedure, considering allowable stress design (ASD), it can be verified that for the given problem the eccentricity is e=1.02 ft (see Figure 8b for considered force vectors) and the factor of safety is 4.18.
Note that this factor is the ratio between the ultimate load the “foundation” can carry and the actual average load acting over a reduced footing area (L-2e), where L is the length of the reinforcement. However, FS in Figure 8a is associated with mobilization of the foundation soil shear strength. That is, the factors of safety associated with Figures 8a and 8b are defined differently, and therefore, their values are different. The advantage of the LE approach is that it will deal with nearly any foundation soil profile whereas Meyerhof’s is limited to homogeneous soil. Furthermore, the LE approach considers the effects of the reinforcement on potential foundation failure whereas Meyerhof looks at the reinforced mass as a rigid block (which it is not). Finally, the LE analysis is conducted consistently through the design, unlike the AASHTO use of unrelated analyses with the hope that all potential instabilities are addressed.
The problem in Figure 3 represents a “wall” by AASHTO or FHWA characterization. Reinforced slopes are defined as having a batter greater than 20 degrees (inclination less than 70 degrees). Consequently, designs for walls and slopes are disjointed, arbitrarily leading to grossly different
outcomes. The LE methodology does not need a distinction between walls and slopes. Consider the problem in Figure 3; all remain the same except that the slope changes from 82 degrees to 64 degrees; i.e., 2(v):1(h). Figures 9a and 9b illustrate the calculated tension along each reinforcement layer. Compared with the wall, Tmax is about 390 lb/ft, a drop of 55%.
Figure 10 compares the connection and maximum tension loads for the reinforced slope and wall. Clearly, not only the reinforcement loads decrease dramatically when somewhat flatter inclination is analyzed but the connection load also drops significantly. The same tool demonstrates consistently the impact of decreased slope angle.
Despite this, when selecting reinforcement as done
in the previous problem but based on Tmax=390 lb/ft, the global factor of safety is deficient, FS<1.30 (Figure 11). In fact, Figure 11 implies that there are two distinctive zones within which FS is deficient. To overcome the internal zone, it can be shown that increase in reinforcement strength by 3% will resolve the problem.
However, the foundation problem remains. This may appear counterintuitive when considering the adequate results for wall case that has the same foundation soil. The foundation soil is frictional, deriving its strength in part due to high overburden pressures. These pressures are lower under the slope, thus resulting in decrease of soil strength leading to lower FS. LE analysis accounts for such a situation. Lengthening the lower four layers from 14ft to 18ft results in FS=1.30 (Figure 12). Consequently, while reducing the slope inclination results in decreased loads, global stability might decrease too, possibly requiring longer reinforcement layers at the bottom. The LE methodology is an objective tool enabling the designer to optimize the reinforcement layout considering reinforcement interaction as well as soil strata.
Dov Leshchinsky, Ph.D., is professor emeritus of civil and environmental engineering at the University of Delaware and a partner at ADAMA Engineering in Clackamas, Ore.
Ora Leshchinsky, P.E., is a partner at ADAMA Engineering in Clackamas, Ore.
Ben A. Leshchinsky, Ph.D., P.E., is an assistant professor at Oregon State University.
- Baker, R., and Leshchinsky, D. (2001). “Spatial Distribution of Safety Factors,” Journal of Geotechnical and Geoenvironmental Engineering,
ASCE, 127(2), 135–145.
- Leshchinsky, D., Leshchinsky, O., Zelenko, B., and Horne, J. (2016). “Limit Equilibrium Design Framework for MSE Structures with Extensible Reinforcement,” Report FHWA-HIF-17-004, October.
- Part 2 (June/July 2017 Geosynthetics): Additional instructive examples will be provided.