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Revisiting bearing capacity analysis of MSE walls
Geotextiles and Geomembranes 34 (2012) 100e107
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Revisiting bearing capacity analysis of MSE walls Dov Leshchinsky a, *, Farshid Vahedifard b, Ben A. Leshchinsky c a
Department of Civil and Environmental Engineering, 301 DuPont Hall, University of Delaware, Newark, DE 19716, USA Paul C. Rizzo Associates, Inc., Pittsburgh, PA 15235, USA c Department of Civil Engineering & Engineering Mechanics, Columbia University, NY 10023, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 October 2011 Received in revised form 3 April 2012 Accepted 22 May 2012 Available online
Common design of MSE walls determines the layout and strength of reinforcement by using a synergy of various analyses. One such analysis is bearing capacity in which the reinforced mass is considered a rigid body exerting uniform pressure over reduced area due to eccentricity. In some codes, Meyerhof’s method for an eccentrically loaded footing is used to assess bearing capacity and to ensure a sufficient margin of safety. In these codes, the horizontal resultant of lateral earth pressure on the reinforced mass affects eccentricity but is ignored in assessing the bearing capacity coefficients in this analysis; i.e., the analysis does not consider the impact of load inclination. Using rigorous upper bound in limit analysis of plasticity, the critical failure mechanisms of the analyzed equivalent footing are identified. It is demonstrated that for a footing subjected to a vertical eccentric load only, Meyerhof’s approximation is reasonable. However, ignoring the impact of horizontal force on bearing capacity in such an equivalent footing is significantly unconservative. Conversely, it is shown that some compounded conservatism in the equivalent problem stems from ignoring the interface friction between the reinforced and retained soils while implicitly considering an unfeasible failure mechanism for bearing of the footing. With conservative selection of soil shear strength properties and a typical value of factor of safety against bearing failure, the end result is likely conservative. The use of this flawed bearing capacity analysis in design is questioned. Also questioned is the applicability of bearing capacity calculations developed for a rigid footing when dealing with a flexible reinforced mass. It is suggested to replace it with a robust and more realistic, albeit simple analysis that considers failures through the foundation soil. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Bearing capacity MSE walls Meyerhof’s method Eccentricity Limit analysis
1. Introduction Many design codes (e.g., AASHTO, 2010; FHWA, 2001, 2009; NCMA, 1997; BS8006-1, 2010; EBGEO, 2011) require that mechanically stabilized earth (MSE) walls be analyzed considering internal and external stability. When determining internal stability, the strength of the reinforcement and its connection to the facing, as well as its pullout resistive length are checked. The reinforced mass is designed to have a sufficient long-term margin of safety internally to sustain various loads such as self-weight, surcharge, and seismicity. When determining external stability, the reinforced mass is taken as a coherent mass, and implicitly treated as a rigid body that is subjected to loading exerted by the retained soil. The size of this ‘block’ is determined to resist sliding, overturning (or, alternatively, have limited value of eccentricity), and to attain
* Corresponding author. Fax: þ1 302 731 1001. E-mail addresses: [email protected] (D. Leshchinsky), (F. Vahedifard), [email protected] (B.A. Leshchinsky).
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0266-1144/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2012.05.006
a minimum bearing capacity value. The foundation soil controls the bearing capacity and may also affect sliding resistance adversely. The synergy of internal and external analyses should produce a safe layout and required strength of the reinforcement. Not surprisingly, experience indicates that global and compound stability analyses (commonly known as reinforced slope stability analysis) are also needed, as dictated by specific local geotechnical conditions. However, in general such a requirement is vaguely stated in some design codes. The concept of broadly dividing failure modes into internal and external analyses, where each mode is independent of the other, is appealing as it simplifies the design process. That is, the assumed independence of failure mechanisms enables one to utilize simple calculations, a great advantage in design. The bearing capacity analysis usually postulates an equivalent problem, utilizing Meyerhof’s approach to eccentrically loaded footings. However, unlike the German code (EBGEO, 2011), most codes (e.g., FHWA, 2009; AASHTO, 2010) ignore the fact that such a ‘footing’ is actually also subjected to an inclined load. Using only an eccentric but not an inclined load provides marginal further simplification while
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implying that such an ‘approximation’ is applicable and safe. This paper questions whether ignoring the inclined load produces a safe equivalent problem within the framework of the corrected analysis. Bearing capacity of eccentrically loaded foundations and retaining walls have been extensively investigated for the last two decades (e.g., Houlsby and Puzrin, 1999; Bransby, 2001; Yun and Bransby, 2007; Loukidis et al., 2008; Georgiadis, 2010; Zhang et al., 2010). Most studies were based on advanced numerical modeling techniques such as finite element (FE) or finite difference (FD). Identifying failure mechanisms of footings using FE or FD requires certain expertise and data that few designers have. Hence, while using FE and FD does not require an a priori assumed failure mechanism, the state-of-practice for the design of MSE walls relies on classical analytical methods (e.g., using the Meyerhof’s reduced footing size approximation). If properly used, the classical methods yield acceptable results. This work does not intend to imply that Meyerhof’s analysis of a footing subjected to eccentric and inclined loads yields unsafe design. Meyerhof and other adequate methods (e.g., Hansen, 1970) have an extensive record of good performance if properly used. The main contribution of this paper is a demonstration of a flaw associated with the use of Meyerhof’s method in some design codes. A rigorous tool was utilized for such a demonstration. This plasticity-based tool enables one to visualize the impact on failure mechanisms, as well as realize the numerical impact, in order to better understand the flaw in the current design approach used in some codes. In view that MSE walls rarely fail in a bearing capacity mode, a legitimate question that code-writers and designers could ask is whether that flaw is merely of an academic interest, of which it is not for a few reasons. First, evolution of design is such that if it is deemed conservative, design criteria (e.g., factor of safety) are reduced with time to produce more economical structures. If one is not aware of existing flaws, such reduction may unmask the unconservative nature of the analysis, eventually leading to ‘unexplained’ failure. Design should be done correctly without knowingly utilizing wrong approximations. Second, if the correct utilization of Meyerhof’s bearing capacity approach renders an overly conservative outcome, than it is feasible that a failure mechanism appropriate for a rigid footing subjected to eccentric and inclined load is not relevant to flexible MSE walls. In such a case, the analysis turns to be irrelevant to the problem and therefore should be replaced with a more realistic assessment of possible deep-seated failures. Use of irrelevant analysis is wasteful, may lead to unnecessary conservatism in structures, and may add confusion. It is hoped that this paper will result in code-writers revisiting the issue of bearing capacity of MSE walls, as the title of this paper implies. 2. The question Simple geometry is used to explain the postulated problem associated with external stability. Fig. 1, Body A, shows the typical forces acting on a coherent soil block considered in design. This coherent block is defined by the length of the reinforcement and it is frequently referred to as the reinforced mass. In this figure, W is the weight of the reinforced mass, H is the height of the wall, and L is the uniform length of the reinforcement. Per FHWA/AASHTO, the interface friction between the retained and reinforced soil, dwall, is taken as zero, and the retained soil exerts an active lateral earth pressure with a resultant Pa. For horizontal crest, it is common to take Pa ¼ (½gH2Ka) in design (e.g., FHWA, 2001; BS8006-1, 2010), where g is the unit weight of the retained soil, Ka ¼ tan2 (45 f/2) is the Coulomb active lateral earth pressure coefficient with interface angle of friction dwall ¼ 0, and f is the internal angle of friction of the retained soil. The German code (EBGEO, 2011) recommends
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Fig. 1. External stability of MSE walls: statically equivalent problem commonly used in design.
taking dwall ¼ (2/3)f, where f is the smaller value of the reinforced and retained soils, in calculating Ka using Coulomb’s equation. It is noted, however, that dwall depends on the vertical movement of the reinforced soil relative to the retained soil. Hence, while dwall ¼ 0 is considered here conservative (i.e., will render larger eccentricity) compared with dwall ¼ (2/3)f, it is feasible that relative vertical movements will produce dwall ¼ f, possibly rendering either design value unconservative. However, this important issue is beyond the scope of this paper since it is limited to prescribed design methods. Fig. 1 illustrates the breakdown of forces for the case commonly considered equivalent to the actual problem. It is comprised of a self-weight force W that is acting at a distance e from the centerline of the reinforced mass (Body B). By virtue of not using the inclined load at e, it is implied that the active resultant Pa is translated parallel to itself to the elevation of the heel (Body C). The eccentricity e is determined from moment equilibrium of the reinforced mass, thus allowing for a downward translation of Pa by H/3. In fact, Fig. 1 shows a statically correct equivalency: Body A ¼ Body B þ Body C. In a typical design, Body B exerts the vertical force W eccentrically at a distance e from the centerline, requiring that this eccentricity does not exceed a prescribed value (e.g., L/6). Then, it is used to assess the bearing capacity of the rigid body (Body B) utilizing Meyerhof’s (1953, 1963) formulation for eccentric and inclined load. However, as seen in Fig. 1, the inclined load aspect is ignored in Body B. A sufficient margin of safety against bearing failure is required. That is, the bearing capacity equation for a vertically and eccentrically loaded footing, typically presented by the classical equation qu ¼ cNc þ ½g(L 2e)Ng, needs to render a capacity that is at least twice a uniform load acting over the reduced-area footing, defined as q ¼ W/(L 2e). In these equations qu and q are the ultimate bearing pressure and the actual pressure exerted by the footing, respectively; c and g are the cohesion and the unit weight of the foundation soil, respectively; Nc and Ng are the bearing capacity coefficients. The values of Nc and Ng are a function of f for which FHWA (2009) and AASHTO (2010) refer to the classical values produced by Prandtl (1921) and Vesic (1975), respectively. Unlike the German code (EBGEO, 2011), FHWA (2009) and AASHTO (2010) disregard the impact of the horizontal load on the bearing capacity coefficients Nc and Ng. It is noted that the original Meyerhof’s solution is suggested for an eccentric and inclined load. While the impact of Pa is not considered to produce an inclined load, it is used in sliding analysis, a stability aspect that is independent of bearing capacity. Body C is used to assess potential sliding along the base comparing the driving force, Pa, and the resisting force along the interface between the reinforced mass and the foundation soil. The basic premise of external stability is that there exists a coherent mass that can be analyzed separately for each of the aforementioned three criteria. However, the equivalency in Fig. 1, while valid statically, is based on a major assumption: the ‘bearing capacity’ of the reinforced mass, Body A, is equivalent to the bearing
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3.1. c0 ef0 foundation
the failure mechanism for the alternative interface problem. For this case, AF was 1.33. Alternative to AF, the decrease in soil strength needed to generate its full mobilization (i.e., the factor of safety on the soil strength) was Fs ¼ 1.13. Not surprisingly, dwall plays a role, especially when considering the AF. Since such friction is likely to develop, ignoring it may be conservative. Summing the moments about the toe, the eccentricity is calculated to be e ¼ 0.79 m (e/L ¼ 0.113). Hence, Meyerhof’s uniform contact pressure is q ¼ W/(L 2e) ¼ 259 kPa. It can be verified that in the context of common of MSE wall design, Meyerhof’s bearing capacity load for Body B (e.g., FHWA, 2001 or AASHTO, 2010) is qu ¼ 430 kPa (where Nc ¼ 14.65 and Ng ¼ 5.24). Hence, the conventional factor of safety on bearing capacity, Fs_bc ¼ qu/q is 1.66, a significantly larger value than 1.0 for the actual problem. For the selected problem, the resulting bearing capacity based on Body B in Fig. 1, is unsafe compared with the actual problem in Body A. Hence, examining the failure mechanisms associated with the bearing capacity for both Body A and Body B as well as the impact of ignoring the likely interface friction angle, dwall, could be instructive in an attempt to explain the differences. Fig. 3a shows the failure mechanism for Body B, supposedly equivalent to the actual problem shown in Fig. 2a, Body A. To render the problem prescribed in common design (e.g., FHWA, 2001), a vertical resultant force W was placed at eccentricity e to the left of
A simple problem was selected to investigate the bearing capacity problem: The wall height of H ¼ 10 m; the width of the coherent (reinforced) soil mass of L ¼ 7 m; the unit weight of the reinforced, retained and foundation soils are all the same having g ¼ 20 kN/m3; the friction angle and the cohesion of the retained soil are f0 ¼ 30 and c0 ¼ 0, respectively; the interface friction between the retained soil and the reinforced mass is dwall ¼ 0; a rough interface is assumed between the reinforced mass and the foundation soil; the cohesion and the friction angle of the foundation soil are c0 ¼ 10 kPa and f0 ¼ 19.8 , respectively. To comply with the external stability approach, the coherent mass was specified in LimitState:GEO as a rigid body. Two aspects related to the selected shear strength of the foundation soil are noted. First, it is that a ‘more’ rational value of strength, representing the extreme of c0 ¼ 0, was not selected for analysis. This was done to avoid typical numerical problems associated with purely frictional materials. Such problems are related to possible inaccuracies resulting from singularities at the corners of the rigid footing. This potential inaccuracy exists with any numerical analysis. However, this issue is not relevant to the objective of the current work and therefore, it was avoided by introducing a reasonably small value of cohesion. Second, for the selected value of c0 ¼ 10 kPa, the design value of f0 ¼ 19.8 signifies a fully mobilized shear strength of the foundation soil for the ‘actual’ problem, Body A. Such a benchmark renders a straightforward comparison with the equivalent problem, Body B, as it signifies how far it is from a numerically induced failure for the original problem. Fig. 2 shows the failure mechanism for the selected problem, Body A. Fig. 2a is based on Adequacy Factor (AF) of 1.0, meaning the structure is on the verge of collapse for the given data. Formally, AF in Body A indicates the amount by which the unit weights of the reinforced and retained soils should be increased to render a collapse. A value of AF ¼ 1.0 implies that for the given strength of soils, the specified unit weights of 20 kN/m3 are critical. It also implies that the factor of safety on shear strength of the soils, Fs, is 1.0 (i.e., the shear strengths of the retained and foundation soils are fully mobilized). In order to realize the impact of likely friction between the retained soil and the reinforced mass, a simulation was conducted using dwall ¼ f, where dwall is the interface friction between the retained soil and the reinforced mass. Fig. 2b shows
Fig. 2. Critical failure mechanism for Body A, foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 : (a) dwall ¼ 0 (AF ¼ 1.0), and (b) dwall ¼ f (AF ¼ 1.33).
capacity of Body B considering the force vector shown in Fig. 1. Such an assumption implies that the original and the equivalent problems have the same failure mechanism rendering the same bearing capacity. The question is whether the approximate equivalency implied in the current bearing capacity approach is correct. If not, the common current approach is inconsistent within its own simplifying assumption of safely and economically using a coherent mass for bearing capacity. 3. The mechanism To study the equivalency question where the feasibility of the failure mechanism is also examined, the computer program LimitState:GEO (LimitState, 2010) was utilized. It uses the Discontinuity Layout Optimization (DLO; Smith and Gilbert, 2007) linear programming procedure to produce a rigorous upper-bound solution in limit analysis of plasticity. It automatically identifies the critical layout of slip-lines in a soil mass that is at a limit state. Use of the DLO algorithm eliminates the need for guessing the failure mechanism expected to be most critical which is usually required in the alternative limit equilibrium analysis. Hence, it is an ideal tool to investigate the identified potential problem.
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To possibly force a one-sided failure, the surcharge of the retained soil, gH ¼ 10 20 ¼ 200 kPa, was applied to the foundation surface on the right side of rigid body e Fig. 3b. This has affected larger portion of the foundation soil compared with Fig. 2a. Also, AF has increased from 1.85 to 2.12. Consequently, an attempt to model the original problem ‘better’ by considering the surcharge of the retained soil is not a good representation of reality. The adequacy factor, AF, for Body A signifies an increase in the unit weights of the retained soil and the rigid body and not the rigid body alone as in the equivalent problem in Fig. 3. Hence, one may legitimately question the comparison of AF for Body A and Body B. A more applicable comparison would be that of Fs, the factor needed to reduce the soil shear strength to attain a limit state. LimitState:GEO computations for the equivalent problem render this factor to be Fs ¼ 1.26 whereas for the actual problem it is 1.0. Such Fs implies that a foundation soil with f0 ¼ tan1 [tan (19.8)/ 1.26] z16 and c0 ¼ 10/1.26 z8 kPa will render collapse for Body B. The failure mechanism for this case is similar to that in Fig. 3a. It is noted that if f0 ¼ 16 and c0 ¼ 8 kPa is used instead of 19.8 and 10 kPa, AF would be 1.0 while Meyerhof’s approach for these reduced strength parameters will produce AF ¼ Fs_bc ¼ 1.0 (q ¼ 259 kPa, and qu ¼ 259 kPa since Nc ¼ 11.63 and Ng ¼ 3.06). Once again, for Body B alone, very strong agreement between limit analysis and Meyerhof’s approximation exists (i.e., AF equal to 1.0 vs. 1.0 whereas for the previous case, Fig. 3a, it was 1.85 vs. Meyerhof’s 1.66). 3.2. Cohesive foundation
Fig. 3. Critical failure mechanism for foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 and dwall ¼ 0: (a) Body B (AF ¼ 1.85), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 2.12).
the centerline. In this case, the rigid body was taken as weightless thus making W the only (eccentric) vertical force. To attain failure, W was increased by 1.85; i.e., AF ¼ 1.85. In fact, AF in this case is also Fs_bc, the conventional factor of safety in bearing capacity problems, defined as the ratio qu/q. Hence, for the equivalent problem, limit analysis yields AF of 1.85 compared to 1.66 produced by Meyerhof’s approach, approximately a 10% difference. This value may be considered small when one recognizes the simplifications associated with Meyerhof’s approach and the typically large differences in results predicted by the many available analytical models of bearing capacity. Comparing Figs. 2a and 3a, it is evident that the ‘equivalent’ problem has questionable equivalency. The failure mechanism generated by Body B (Fig. 3a) is not similar to the one obtained for Body A (Fig. 2b). In the current problem, Body B causes large mass of soil to move away from the toe and, simultaneously, push soil away from the heel e a two-sided failure. In the context of the actual problem, the equivalent problem signifies an unrealistic failure mechanism as it needs to push the retained soil upwards. A single-sided failure, as in Fig. 2, is the likely mechanism. Clearly, the equivalent problem does not inhibit the development of two-sided failure mechanism.
In order to determine whether the same unconservative trend in postulating the equivalent problem exists also for cohesive foundation soil, the same problem was run with an undrained angle of friction fu ¼ 0. Limit state analysis indicates that an undrained shear strength of cu ¼ 57.8 kPa will render failure for the actual problem. Similar to Fig. 2a and b, Fig. 4a and b shows the failure mechanisms for the cases where the interface friction with the retained soil is dwall ¼ 0 and dwall ¼ f, respectively. While AF is 1.00 for dwall ¼ 0, it is 1.09 for dwall ¼ f. It is notable that the failure through the foundation deepens as dwall goes up. Also, while the AF for the c0 ef0 foundation increases to 1.33, it increased only marginally for the cohesive foundation (AF ¼ 1.09). An alternative to AF, the corresponding decrease in the soils strength needed to generate its full mobilization is Fs ¼ 1.08 whereas for the c0 ef0 case it was 1.13, which is fairly close. In the cohesive foundation soil, dwall plays a minor role unlike for the c0 ef0 foundation, where it has significant influence only when considering the AF, not the mobilization of soil strength (i.e., Fs). Fig. 5a shows the failures corresponding to Body B, which consists of a two-sided mechanism which gets closer to being symmetrical. Failure in the direction of the heel (i.e., into the retained soil) does not appear to be feasible; it is an artifact of the postulated equivalent problem. Meyerhof’s approximation for this case produces AF ¼ Fs_bc ¼ 1.15 (q ¼ 259 kPa and qu ¼ 297 kPa since Nc ¼ 5.14 and Ng ¼ 0). Once more, for Body B, the bearing capacity factor of safety agreement between limit analysis and Meyerhof’s approximation, 1.29 and 1.15, is reasonable. The needed reduction of cu to attain failure while holding W in Body B constant is 1.29 (Fig. 7); i.e., cu of 44.8 kPa. Fig. 5b shows the impact of the retained soil surcharge of 200 kPa. It produces a one-sided mechanism with a marginal difference in the bearing capacity factor of safety; i.e., AF ¼ Fs_bc increases from 1.29 for two-sided to 1.32 for a single-sided mechanism. Similar to the c0 ef0 case, imposing more appropriate failure mechanism for the equivalent problem results in increase in AF, which would not be beneficial.
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Fig. 4. Critical failure mechanism for Body A, foundation soil cu ¼ 57.8 kPa and fu ¼ 0: (a) dwall ¼ 0 (AF ¼ 1.0), and (b) dwall ¼ f (AF ¼ 1.09). Fig. 5. Critical failure mechanism for foundation soil cu ¼ 57.8 kPa, fu ¼ 0 and dwall ¼ 0: (a) Body B (AF ¼ 1.29), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 1.32).
3.3. Impact of interface friction on back of wall While AASHTO (2010) and FHWA (2001, 2009) suggest ignoring the interface friction between the retained soil and the back of the coherent mass (i.e., for horizontal crest dwall ¼ 0 is specified and therefore Pa is horizontal), other methods (e.g., NCMA, 1997) suggest using dwall >0. Hence, it is interesting to redo the bearing capacity calculations for the equivalent footing, Body B, using dwall ¼ f. That is, considering the downdrag force and thus increasing the vertical force W accordingly while modifying the eccentricity due to increase in W and simultaneous decrease in Pa. Note that due to relative vertical movements, it is possible that dwall would be as small as f. Common design methods do not address such a possibility although it may increase eccentricity. Subsequently, such a possibility is not addressed in this paper since its objective is to assess the impact of incorrect analysis while utilizing the recommended values provided by the above design methods. It can be shown that for the example problem, e ¼ 0.22 m for dwall ¼ f, where it was e ¼ 0.79 m for dwall ¼ 0. The corrected vertical force due to downdrag is 1549 kN/m for dwall ¼ f versus 1400 kN/m for dwall ¼ 0. The corrected horizontal component of Pa is 257 kN/m for dwall ¼ f compared with 333 kN/m for dwall ¼ 0. Using the revised data, AF ¼ Fs_bc for Meyerhof’s approximation can be calculated for either soil:
c0 ef0 soil (c0 ¼ 10 kPa, f0 ¼ 19.8 ): AF ¼ Fs_bc ¼ 2.08 (q ¼ 236 kPa, and qu ¼ 490 kPa since Nc ¼ 14.65 and Ng ¼ 5.24) Cohesive soil (cu ¼ 57.8 kPa): AF ¼ Fs_bc ¼ 1.26 (q ¼ 236 kPa, and qu ¼ 297 kPa since Nc ¼ 5.14 and Ng ¼ 0). Figs. 6 and 7 are analogous to Figs. 3 and 5. They show the failure mechanisms for the equivalent problem, without and with 200 kPa surcharge, considering the c0 ef0 foundation soil in Fig. 6 and the cohesive foundation soil in Fig. 7, all when dwall ¼ f. The mechanism is two-sided and nearly symmetrical without an applied surcharge and one-sided with a surcharge. Mobilization of the soil strength for the problem without a surcharge (the equivalent problem), as reflected by Fs, are 1.37 and 1.35 for the c0 ef0 and cohesive foundation soil, respectively. Table 1 summarizes the results for the actual problem and the presumably equivalent one used in design. Note in Table 1 that the Fs_bc by Meyerhof’s bearing capacity equation are marked as N/A. Simply, Meyerhof’s formulation does not address the impact of a surcharge load next to the footing. 4. Commentary It is noted that for two reasons, this study was not conducted using bearing capacity models for eccentric and inclined load. First,
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Fig. 6. Critical failure mechanism for foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 and dwall ¼ f: (a) Body B (AF ¼ 2.15), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 2.23).
there are quite a few semi-empirical models dealing with inclination factors serving as modifiers of the bearing capacity coefficients Nc and Ng. The prototype bearing capacity formulation considered in this work follows FHWA (2001, 2009). While FHWA is explicit about the values of Nc and Ng to be used [i.e., those produced by Prandtl (1921) and Vesic (1975), respectively] in conjunction with Meyerhof’s model, it does not address inclination factors. Hence, it seems that the least biased approach in this study would be to use a robust numerical model. Second, the bearing capacity formula, including those using inclination factors, provides important but limited information; i.e., it produces the ultimate bearing load qu. However, the equivalent problem renders a mechanism that is not feasible; a two-sided failure. Clearly, the retained soil inhibits such a failure, producing a one-sided mechanism. Therefore, the impact of the mode of failure cannot be studied using existing common formula. The numerical tool used in this study enables one to study the effects of failure mechanism. Therefore, it provides more insight possibly leading to more general conclusions that existing closedform models. The agreement between Meyerhof’s approximation and the results from limit analysis (LA) for the eccentrically loaded footing (i.e., Body B) is reasonable from a practical viewpoint e see Table 1. This statement should be considered within the context of the existing closed-form formulae which produce, for the same problem, a large range of bearing capacity values, much more than the 12% difference encountered here.
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Fig. 7. Critical failure mechanism for foundation soil cu ¼ 57.8 kPa, fu ¼ 0 and dwall ¼ f: (a) Body B (AF ¼ 1.36), (b) Body B þ 200 kPa surcharge on right (AF ¼ 1.38).
Table 1 also shows that the interface friction between the reinforced and retained soils, dwall, has some impact (about 16% increase in bearing capacity as dwall goes up) when the soil is c0 ef0 and minor impact (about 5% increase) if it is a f ¼ 0 type soil. The impact stems from reduced eccentricity due to downdrag force offsetting the little increase in vertical load. Based on LA, the increase in bearing capacity looking at Body A (the actual problem) is substantial (about 33%) when c0 ef0 is considered and small (about 9%) when cohesive foundation is considered. For all cases, the equivalent problem had a two-sided failure. Since such a mechanism is not feasible, the formation of slip-lines in the direction of the retained soil was prevented by introducing a surcharge of 200 kPa, equal to that exerted by the retained soil on top of the foundation soil. This indeed created a one-sided mechanism. Table 1 shows that for the c0 ef0 soil this constraint actually increased the bearing load by up to 15%, especially when dwall ¼ 0. For the f ¼ 0 type soil, the increase in bearing capacity was minute. When the cohesive foundation was involved, the bearing capacity results are barely affected by whether a one-sided or two-sided mechanism develops. LA shows that the equivalency between ‘bearing capacity’ of the actual structure (Body A) and that of an eccentrically loaded footing (Body B) is questionable. This equivalency is an unsafe approximation when examined within the framework of the assumptions made in the design of MSE walls; i.e., the reinforced soil mass is considered as a rigid body exerting only vertical load on a footing with an effective width of (L 2e). For c0 ef0 soil it may overestimate
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Table 1 Summary of results. (1) dwall
Actual problem: Body A
0
f Equivalent problem: Body B
0
f Equivalent problem: Body B þ surcharge
0
f
c0 ef0 foundation: c0 ¼ 10 kPa, f0 ¼ 19.8
Cohesive foundation: cu ¼ 57.8 kPa
(2) AF: Limit analysis
(3) AF ¼ Fs_bc: Meyerhof
(4) Fs
(5) Mechanism
(6) AF: Limit analysis
(7) AF ¼ Fs_bc: Meyerhof
(8) Fs
(9) Mechanism
1.00 1.33 1.85 2.15 2.12 2.23
N/A N/A 1.66 2.08 N/A N/A
1.00 1.13 1.26 1.37 1.36 1.39
One-sided One-sided Two-sided Two-sided One-sided One-sided
1.00 1.09 1.29 1.36 1.32 1.38
N/A N/A 1.15 1.26 N/A N/A
1.00 1.08 1.29 1.36 1.32 1.38
One-sided One-sided Two-sided Two-sided One-sided One-sided
(1) dwall ¼ interface friction between retained soil and the reinforced mass (assumed as rigid body). (2) AF ¼ Adequacy factor signifying the magnitude of multiplier applied to weight of rigid body and retained soil in Body A and to weight of rigid body, W, in Body B. It is obtained from limit analysis in the framework of upper bound in plasticity. (3) AF ¼ Fs_bc ¼ same meaning as in (2); however, it is obtained from using Meyerhof’s equation for eccentrically loaded footing and it is equal to qu/q where qu and q are the ultimate bearing load and the applied load, all assumed to be uniform acting over (L 2e). (4) Fs ¼ factor of safety commonly used in slope stability analysis; it signifies the amount of soil shear strength mobilization due to a given surcharge. (5) Mechanism ¼ either single-sided where failure under the footing moves in the direction of the toe or double-sided where failure occurs simultaneously displacing soil in both directions, toe and heel (typical failure of shallow foundations). (6)e(9) have the same meaning as (2)e(5), respectively.
the bearing capacity by a factor of about 2; for f ¼ 0 it is a factor of about 1.3. Meyerhof predictions of bearing capacity for eccentrically loaded footing agree reasonably well with the LA results, thus adding confidence in using the LA numerical tool while indicating that the problem is not with the formula itself but rather with its use. The equivalency appears to be flawed because the footing is actually subjected to an inclined load with two force components: W and Pa, with resultant acting at e. The horizontal component, Pa, is large enough to significantly reduce the equivalent bearing capacity well beyond being just an issue in sliding as currently considered by, for example, FHWA. If bearing capacity is a feasible mode of failure, then the Meyerhof model should be used for an inclined load; e.g., as done in the German code (EBGEO, 2011). The conclusion of unconservative equivalency, however, has to be viewed through the perspective of real-life performance, gained over nearly 40 years. Many thousands of MSE walls were designed using Meyerhof’s approximation for an eccentric load. As far as the first author’s experience indicates, these structures, if designed and constructed properly, function well. Hence, the following ‘forensics-in-reverse’ question is reasonable: If the design of bearing capacity renders actual walls with marginal safety, then why aren’t deep-seated failures more prevalent? Several plausible explanations for this can be argued: 1. Geotechnical engineers tend to underestimate the shear strength of soils used in design. The retained and foundation soils are often stronger than assumed in analysis rendering an undeclared margin of safety. 2. For simplicity, Pa is assumed to act horizontally. However, for MSE structures the interface friction between the reinforced mass and the retained soil, dwall, is that of the retained soil. Table 1 indicates that the impact of this assumption can be significant when dealing with c0 ef0 foundations. 3. In assessing bearing capacity, the reinforced soil zone, having a width of L, is considered as a coherent rigid mass. Adopting Meyerhof’s approach, this mass has an effective width of (L 2e). This approach implies that the mass exerts uniform contact pressure of q ¼ R/(L 2e) where R is the vertical resultant force within the mass. The pressure q is limited to a fraction of Meyerhof’s bearing capacity which also considers a footing with an effective width of (L 2e). Hence, by decreasing the effective width of the footing, eccentricity ‘penalizes’ the design outcome twice; it decreases the bearing capacity and it increases the contact pressure. However, while the assumption of eccentricity under a rigid footing has been
validated, thus leading to models such as Meyerhof, there is no such validation for a flexible composite mass which is made up of soil and horizontal layers of reinforcement. In fact, experimental data (e.g., Ling et al., 2005) show that the pressure exerted by the mass is approximately uniform along L and is equal to q ¼ gH, a smaller value compared to the eccentric case. Effectively, the width of the ‘footing’ in such a case is L, rendering larger bearing capacity than that for the eccentric footing. Consequently, the substantial ‘penalty’ imposed by using eccentricity is not justified for a flexible mass subjected to lateral earth pressures. It follows that bearing capacity of a flexible ‘coherent mass’ is likely higher than that predicted by Meyerhof’s model. 4. The feasible one-sided failure mechanism versus Meyerhof’s use of a two-sided mechanism adds to bearing capacity, especially in c0 ef0 foundations. 5. The factor of safety used in design (typically Fs_bc ¼ 2.0 or 2.5) formally results in more conservative outcome, ensuring that marginal cases will still remain stable. In summary, there are five conservative measures which are commonly applied in design: 1) conservative selection of foundation strength properties, 2) taking dwall ¼ 0 , 3) base of wall is not rigid as used by Meyerhof for actual footings, 4) implicitly considering two-sided mechanism, and 5) use of substantial factor of safety may offset the unconservative nature of the equivalent problem. Multiplying the impact of all five conservative may yield an actual factor of safety of 3 or more, in turn producing generally safe structures. That is, one ‘wrong’ is made up by four or five ‘rights’. However, the immediate question is whether design should count on a balance between ‘wrongs’ and ‘rights’. Simply stated, the equivalent problem is incorrect and should be corrected to consider footing subjected to eccentric and inclined load and not only to eccentric load. Admittedly, the end result could be overly conservative implying that bearing capacity for an eccentric and inclined load is not a realistic mode of failure for flexible MSE walls. A better alternative can then be realized by viewing an MSE wall as an embankment (albeit with a near-vertical slope). In designing such structures stability analysis considering potential deep-seated failures should be checked. Adequate stability analysis (e.g., LE, LA, FE, FD), that treats the reinforced soil as such, can be used to ascertain sufficient safety against deep-seated failures. This is an extension of relevant and sound geotechnical practice. In fact, the AASHTO/FHWA design suggests a limit equilibrium (LE) global stability check, thus implying that such practice is far from being
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revolutionary. It simplifies the design process of MSE walls by removing a design aspect that is inconsistent based on its own assumptions. 5. Conclusions Some codes dealing with the design of MSE walls simplify the process of determining the layout and strength of the reinforcement by using a synergy of various analyses. The problem is divided into internal stability and external stability. In the external stability, the reinforced soil mass is considered as a rigid body which has to satisfy three stability criteria: sliding, overturning (or eccentricity), and bearing capacity. Meyerhof’s method for eccentrically loaded footings is used in assessing the margin of safety against bearing failure. Meyerhof provided his solution for eccentric and inclined loads. However, some design codes consider only the vertical eccentric load in bearing capacity calculations. The horizontal resultant due to lateral earth pressures against the reinforced mass is only used to assess the safety against direct sliding of that mass. Intuitively, ignoring the horizontal resultant may not be a conservative assumption in assessing bearing capacity, especially if it renders a large inclined load vector. Investigating the problem based on a rigorous limit analysis of plasticity where the critical failure mechanisms are identified shows that: 1. When considering only a vertically eccentric load on a rigid footing, Meyerhof’s approximation yields results that are fairly close to those predicted by limit analysis. In common designs of MSE walls, Meyerhof’s approximation is considered as equivalent to the postulated problem of a rigid (coherent) mass which retains soil. This equivalent problem is used as a part of a synergistic design approach. 2. The predicted failure load for the postulated equivalent problem is unconservative when compared with the predicted failure for the actual problem. The main reason for this unconservative result is due to treatment of vertical load only rather than considering an inclined load in Meyerhof’s equation. The reason for such ‘equivalency’ is not clear to the authors. 3. The failure mechanism associated with the equivalent problem implies that failure develops in both directions of the footing. That is, the failure is two-sided, implying that a mass of foundation soil is pushed upwards under the retained soil. The limit state analysis performed in the current study shows that such a mechanism is not likely and therefore, the equivalent problem is a somewhat conservative and unrealistic representation of the actual problem. 4. Furthermore, the same designs of MSE walls consider the friction at the interface between the retained and reinforced soils to be zero. It is shown that such an assumption is conservative, especially as the frictional strength of soil increases. 5. Apparently the compounded impact of conservative assumptions (e.g., failure mechanism, interface friction, typical conservative selection of soil shear strength, and a factor of safety applied to the analysis) makes up for the impact of unconservative equivalent problem where the Meyerhof equation uses only an eccentric load and not an eccentric and inclined load. However, it is not clear how the compounded conservative aspects of design make up for the unconservative use of Meyerhof’s equation.
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Considering the inadequacy of the equivalent bearing capacity problem, its use in actual design is questionable and perhaps meaningless. Instead, a global slope stability analysis could be used. Such analysis of reinforced soil structures is a common requirement in many design codes. Proper use of such stability analyses, including examination of deep-seated failures, instead of using an inconsistent analysis in design is not revolutionary but rather evolutionary. The global stability analysis need not consider the reinforced soil as a rigid mass; it should consider possible failures through the reinforcement as well. The end result may or may not require longer and/or stronger reinforcement as compared with the current bearing capacity analysis. This will depend on local conditions such as strength of the foundation, reinforced and retained soils, surcharge, geometry, seismic loading, water, and other relevant factors. References AASHTO, 2010. LRFD Bridge Design Specifications. with 2010 Interim Revisions, fifth ed. AASHTO, Washington, D.C. Bransby, M.F., 2001. Failure envelopes and plastic potentials for eccentrically loaded surface footings on undrained soil. International Journal of Numerical Analytical and Methods in Geomechanics 25 (4), 329e346. BS8006-1, 2010. Code of Practice for Strengthened/Reinforced Soils and Other Fills. British Standard Institution, ISBN 978 0 580 53842 1, ICS 93.020, 260 p. EBGEO, 2011. Recommendations for Design and Analysis of Earth Structures Using Geosynthetic Reinforcement. German Geotechnical Society, Ernst & Sohn, GmbH & Co. KG (Chapter 7). FHWA, 2001. Mechanically Stabilized Earth Walls and reinforced Soil Slopes d Design and Construction Guidelines. FHWA-NHI-00e043, Authors: Elias, Christopher, and Berg. Federal Highway Administration, Washington, D.C. FHWA, 2009. FHWA-NHI-10e024, Authors: Berg, Christopher, and Samtani. Design of Mechanically Stabilized Earth Walls and Reinforced Soil Slopes, vol. I. Federal Highway Administration, Washington, D.C. Georgiadis, K., 2010. The influence of load inclination on the undrained bearing capacity of strip footings on slopes. Computers and Geotechnics 37 (3), 311e322. Hansen, J.B., 1970. A Revised and Extended Formula for Bearing Capacity Bulletin, vol. 28. Danish Geotechnical Institute, Copenhagen, Denmark. Houlsby, G.T., Puzrin, A.M., 1999. The bearing capacity of a strip footing on clay under combined loading. Proceedings of the Royal Society, London, Series A 455 (1983), 893e916. LimitState, 2010. LimitState:GEO Manual Version 2.0. LimitState Ltd. October 2010 Edition. Ling, H.I., Mohri, Y., Leshchinsky, D., Burke, C., Matsushima, K., Liu, H., 2005. Large-scale shaking table tests on modular-block reinforced soil retaining wall. ASCE Journal of Geotechnical and Geoenvironmental Engineering 131 (4), 465e476. Loukidis, D., Chakraborty, T., Salgado, R., 2008. Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads. Canadian Geotechnical Journal 45 (6), 768e787. Meyerhof, G.G., 1953. The bearing capacity of foundations under eccentric and inclined loads. In: Proc. Third International Conference on Soil Mechanics and Foundation Engineering, Zurich. vol. 1, pp. 440e445. Meyerhof, G.G., 1963. Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal 1 (1), 16e26. National Concrete Masonry Association (NCMA), 1997. In: Collin, J.G. (Ed.), Design Manual for Segmental Retaining Walls, second ed. National Concrete Masonry Association, Herndon, VA. Prandtl, L., 1921. Über die eindingungsfestigkiet (harte) plasticher baustoffe und die festigkeit von schneiden. Zeitschrift fur Angewandte Mathematik und Mechanik 1 (1), 15e20. Smith, C., Gilbert, M., 2007. Application of discontinuity layout optimization to plane plasticity problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463 (2086), 2461e2484. Vesic, A.S., 1975. Bearing capacity of shallow foundations. In: Winterkorn, H.F., Fang, H.-Y. (Eds.), Foundation Engineering Handbook. Van Nostrand Reinhold Co., New York, NY, pp. 121e147 (Chapter 3). Yun, G., Bransby, M.F., 2007. The horizontal-moment capacity of embedded foundations in undrained soil. Canadian Geotechnical Journal 44 (4), 409e424. Zhang, L., Zhao, M., Shi, C., Zhao, H., 2010. Bearing capacity of geocell reinforcement in embankment engineering. Geotextiles and Geomembranes 28 (5), 475e482.
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Revisiting bearing capacity analysis of MSE walls Dov Leshchinsky a, *, Farshid Vahedifard b, Ben A. Leshchinsky c a
Department of Civil and Environmental Engineering, 301 DuPont Hall, University of Delaware, Newark, DE 19716, USA Paul C. Rizzo Associates, Inc., Pittsburgh, PA 15235, USA c Department of Civil Engineering & Engineering Mechanics, Columbia University, NY 10023, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 October 2011 Received in revised form 3 April 2012 Accepted 22 May 2012 Available online
Common design of MSE walls determines the layout and strength of reinforcement by using a synergy of various analyses. One such analysis is bearing capacity in which the reinforced mass is considered a rigid body exerting uniform pressure over reduced area due to eccentricity. In some codes, Meyerhof’s method for an eccentrically loaded footing is used to assess bearing capacity and to ensure a sufficient margin of safety. In these codes, the horizontal resultant of lateral earth pressure on the reinforced mass affects eccentricity but is ignored in assessing the bearing capacity coefficients in this analysis; i.e., the analysis does not consider the impact of load inclination. Using rigorous upper bound in limit analysis of plasticity, the critical failure mechanisms of the analyzed equivalent footing are identified. It is demonstrated that for a footing subjected to a vertical eccentric load only, Meyerhof’s approximation is reasonable. However, ignoring the impact of horizontal force on bearing capacity in such an equivalent footing is significantly unconservative. Conversely, it is shown that some compounded conservatism in the equivalent problem stems from ignoring the interface friction between the reinforced and retained soils while implicitly considering an unfeasible failure mechanism for bearing of the footing. With conservative selection of soil shear strength properties and a typical value of factor of safety against bearing failure, the end result is likely conservative. The use of this flawed bearing capacity analysis in design is questioned. Also questioned is the applicability of bearing capacity calculations developed for a rigid footing when dealing with a flexible reinforced mass. It is suggested to replace it with a robust and more realistic, albeit simple analysis that considers failures through the foundation soil. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Bearing capacity MSE walls Meyerhof’s method Eccentricity Limit analysis
1. Introduction Many design codes (e.g., AASHTO, 2010; FHWA, 2001, 2009; NCMA, 1997; BS8006-1, 2010; EBGEO, 2011) require that mechanically stabilized earth (MSE) walls be analyzed considering internal and external stability. When determining internal stability, the strength of the reinforcement and its connection to the facing, as well as its pullout resistive length are checked. The reinforced mass is designed to have a sufficient long-term margin of safety internally to sustain various loads such as self-weight, surcharge, and seismicity. When determining external stability, the reinforced mass is taken as a coherent mass, and implicitly treated as a rigid body that is subjected to loading exerted by the retained soil. The size of this ‘block’ is determined to resist sliding, overturning (or, alternatively, have limited value of eccentricity), and to attain
* Corresponding author. Fax: þ1 302 731 1001. E-mail addresses: [email protected] (D. Leshchinsky), (F. Vahedifard), [email protected] (B.A. Leshchinsky).
[email protected]
0266-1144/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2012.05.006
a minimum bearing capacity value. The foundation soil controls the bearing capacity and may also affect sliding resistance adversely. The synergy of internal and external analyses should produce a safe layout and required strength of the reinforcement. Not surprisingly, experience indicates that global and compound stability analyses (commonly known as reinforced slope stability analysis) are also needed, as dictated by specific local geotechnical conditions. However, in general such a requirement is vaguely stated in some design codes. The concept of broadly dividing failure modes into internal and external analyses, where each mode is independent of the other, is appealing as it simplifies the design process. That is, the assumed independence of failure mechanisms enables one to utilize simple calculations, a great advantage in design. The bearing capacity analysis usually postulates an equivalent problem, utilizing Meyerhof’s approach to eccentrically loaded footings. However, unlike the German code (EBGEO, 2011), most codes (e.g., FHWA, 2009; AASHTO, 2010) ignore the fact that such a ‘footing’ is actually also subjected to an inclined load. Using only an eccentric but not an inclined load provides marginal further simplification while
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implying that such an ‘approximation’ is applicable and safe. This paper questions whether ignoring the inclined load produces a safe equivalent problem within the framework of the corrected analysis. Bearing capacity of eccentrically loaded foundations and retaining walls have been extensively investigated for the last two decades (e.g., Houlsby and Puzrin, 1999; Bransby, 2001; Yun and Bransby, 2007; Loukidis et al., 2008; Georgiadis, 2010; Zhang et al., 2010). Most studies were based on advanced numerical modeling techniques such as finite element (FE) or finite difference (FD). Identifying failure mechanisms of footings using FE or FD requires certain expertise and data that few designers have. Hence, while using FE and FD does not require an a priori assumed failure mechanism, the state-of-practice for the design of MSE walls relies on classical analytical methods (e.g., using the Meyerhof’s reduced footing size approximation). If properly used, the classical methods yield acceptable results. This work does not intend to imply that Meyerhof’s analysis of a footing subjected to eccentric and inclined loads yields unsafe design. Meyerhof and other adequate methods (e.g., Hansen, 1970) have an extensive record of good performance if properly used. The main contribution of this paper is a demonstration of a flaw associated with the use of Meyerhof’s method in some design codes. A rigorous tool was utilized for such a demonstration. This plasticity-based tool enables one to visualize the impact on failure mechanisms, as well as realize the numerical impact, in order to better understand the flaw in the current design approach used in some codes. In view that MSE walls rarely fail in a bearing capacity mode, a legitimate question that code-writers and designers could ask is whether that flaw is merely of an academic interest, of which it is not for a few reasons. First, evolution of design is such that if it is deemed conservative, design criteria (e.g., factor of safety) are reduced with time to produce more economical structures. If one is not aware of existing flaws, such reduction may unmask the unconservative nature of the analysis, eventually leading to ‘unexplained’ failure. Design should be done correctly without knowingly utilizing wrong approximations. Second, if the correct utilization of Meyerhof’s bearing capacity approach renders an overly conservative outcome, than it is feasible that a failure mechanism appropriate for a rigid footing subjected to eccentric and inclined load is not relevant to flexible MSE walls. In such a case, the analysis turns to be irrelevant to the problem and therefore should be replaced with a more realistic assessment of possible deep-seated failures. Use of irrelevant analysis is wasteful, may lead to unnecessary conservatism in structures, and may add confusion. It is hoped that this paper will result in code-writers revisiting the issue of bearing capacity of MSE walls, as the title of this paper implies. 2. The question Simple geometry is used to explain the postulated problem associated with external stability. Fig. 1, Body A, shows the typical forces acting on a coherent soil block considered in design. This coherent block is defined by the length of the reinforcement and it is frequently referred to as the reinforced mass. In this figure, W is the weight of the reinforced mass, H is the height of the wall, and L is the uniform length of the reinforcement. Per FHWA/AASHTO, the interface friction between the retained and reinforced soil, dwall, is taken as zero, and the retained soil exerts an active lateral earth pressure with a resultant Pa. For horizontal crest, it is common to take Pa ¼ (½gH2Ka) in design (e.g., FHWA, 2001; BS8006-1, 2010), where g is the unit weight of the retained soil, Ka ¼ tan2 (45 f/2) is the Coulomb active lateral earth pressure coefficient with interface angle of friction dwall ¼ 0, and f is the internal angle of friction of the retained soil. The German code (EBGEO, 2011) recommends
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Fig. 1. External stability of MSE walls: statically equivalent problem commonly used in design.
taking dwall ¼ (2/3)f, where f is the smaller value of the reinforced and retained soils, in calculating Ka using Coulomb’s equation. It is noted, however, that dwall depends on the vertical movement of the reinforced soil relative to the retained soil. Hence, while dwall ¼ 0 is considered here conservative (i.e., will render larger eccentricity) compared with dwall ¼ (2/3)f, it is feasible that relative vertical movements will produce dwall ¼ f, possibly rendering either design value unconservative. However, this important issue is beyond the scope of this paper since it is limited to prescribed design methods. Fig. 1 illustrates the breakdown of forces for the case commonly considered equivalent to the actual problem. It is comprised of a self-weight force W that is acting at a distance e from the centerline of the reinforced mass (Body B). By virtue of not using the inclined load at e, it is implied that the active resultant Pa is translated parallel to itself to the elevation of the heel (Body C). The eccentricity e is determined from moment equilibrium of the reinforced mass, thus allowing for a downward translation of Pa by H/3. In fact, Fig. 1 shows a statically correct equivalency: Body A ¼ Body B þ Body C. In a typical design, Body B exerts the vertical force W eccentrically at a distance e from the centerline, requiring that this eccentricity does not exceed a prescribed value (e.g., L/6). Then, it is used to assess the bearing capacity of the rigid body (Body B) utilizing Meyerhof’s (1953, 1963) formulation for eccentric and inclined load. However, as seen in Fig. 1, the inclined load aspect is ignored in Body B. A sufficient margin of safety against bearing failure is required. That is, the bearing capacity equation for a vertically and eccentrically loaded footing, typically presented by the classical equation qu ¼ cNc þ ½g(L 2e)Ng, needs to render a capacity that is at least twice a uniform load acting over the reduced-area footing, defined as q ¼ W/(L 2e). In these equations qu and q are the ultimate bearing pressure and the actual pressure exerted by the footing, respectively; c and g are the cohesion and the unit weight of the foundation soil, respectively; Nc and Ng are the bearing capacity coefficients. The values of Nc and Ng are a function of f for which FHWA (2009) and AASHTO (2010) refer to the classical values produced by Prandtl (1921) and Vesic (1975), respectively. Unlike the German code (EBGEO, 2011), FHWA (2009) and AASHTO (2010) disregard the impact of the horizontal load on the bearing capacity coefficients Nc and Ng. It is noted that the original Meyerhof’s solution is suggested for an eccentric and inclined load. While the impact of Pa is not considered to produce an inclined load, it is used in sliding analysis, a stability aspect that is independent of bearing capacity. Body C is used to assess potential sliding along the base comparing the driving force, Pa, and the resisting force along the interface between the reinforced mass and the foundation soil. The basic premise of external stability is that there exists a coherent mass that can be analyzed separately for each of the aforementioned three criteria. However, the equivalency in Fig. 1, while valid statically, is based on a major assumption: the ‘bearing capacity’ of the reinforced mass, Body A, is equivalent to the bearing
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3.1. c0 ef0 foundation
the failure mechanism for the alternative interface problem. For this case, AF was 1.33. Alternative to AF, the decrease in soil strength needed to generate its full mobilization (i.e., the factor of safety on the soil strength) was Fs ¼ 1.13. Not surprisingly, dwall plays a role, especially when considering the AF. Since such friction is likely to develop, ignoring it may be conservative. Summing the moments about the toe, the eccentricity is calculated to be e ¼ 0.79 m (e/L ¼ 0.113). Hence, Meyerhof’s uniform contact pressure is q ¼ W/(L 2e) ¼ 259 kPa. It can be verified that in the context of common of MSE wall design, Meyerhof’s bearing capacity load for Body B (e.g., FHWA, 2001 or AASHTO, 2010) is qu ¼ 430 kPa (where Nc ¼ 14.65 and Ng ¼ 5.24). Hence, the conventional factor of safety on bearing capacity, Fs_bc ¼ qu/q is 1.66, a significantly larger value than 1.0 for the actual problem. For the selected problem, the resulting bearing capacity based on Body B in Fig. 1, is unsafe compared with the actual problem in Body A. Hence, examining the failure mechanisms associated with the bearing capacity for both Body A and Body B as well as the impact of ignoring the likely interface friction angle, dwall, could be instructive in an attempt to explain the differences. Fig. 3a shows the failure mechanism for Body B, supposedly equivalent to the actual problem shown in Fig. 2a, Body A. To render the problem prescribed in common design (e.g., FHWA, 2001), a vertical resultant force W was placed at eccentricity e to the left of
A simple problem was selected to investigate the bearing capacity problem: The wall height of H ¼ 10 m; the width of the coherent (reinforced) soil mass of L ¼ 7 m; the unit weight of the reinforced, retained and foundation soils are all the same having g ¼ 20 kN/m3; the friction angle and the cohesion of the retained soil are f0 ¼ 30 and c0 ¼ 0, respectively; the interface friction between the retained soil and the reinforced mass is dwall ¼ 0; a rough interface is assumed between the reinforced mass and the foundation soil; the cohesion and the friction angle of the foundation soil are c0 ¼ 10 kPa and f0 ¼ 19.8 , respectively. To comply with the external stability approach, the coherent mass was specified in LimitState:GEO as a rigid body. Two aspects related to the selected shear strength of the foundation soil are noted. First, it is that a ‘more’ rational value of strength, representing the extreme of c0 ¼ 0, was not selected for analysis. This was done to avoid typical numerical problems associated with purely frictional materials. Such problems are related to possible inaccuracies resulting from singularities at the corners of the rigid footing. This potential inaccuracy exists with any numerical analysis. However, this issue is not relevant to the objective of the current work and therefore, it was avoided by introducing a reasonably small value of cohesion. Second, for the selected value of c0 ¼ 10 kPa, the design value of f0 ¼ 19.8 signifies a fully mobilized shear strength of the foundation soil for the ‘actual’ problem, Body A. Such a benchmark renders a straightforward comparison with the equivalent problem, Body B, as it signifies how far it is from a numerically induced failure for the original problem. Fig. 2 shows the failure mechanism for the selected problem, Body A. Fig. 2a is based on Adequacy Factor (AF) of 1.0, meaning the structure is on the verge of collapse for the given data. Formally, AF in Body A indicates the amount by which the unit weights of the reinforced and retained soils should be increased to render a collapse. A value of AF ¼ 1.0 implies that for the given strength of soils, the specified unit weights of 20 kN/m3 are critical. It also implies that the factor of safety on shear strength of the soils, Fs, is 1.0 (i.e., the shear strengths of the retained and foundation soils are fully mobilized). In order to realize the impact of likely friction between the retained soil and the reinforced mass, a simulation was conducted using dwall ¼ f, where dwall is the interface friction between the retained soil and the reinforced mass. Fig. 2b shows
Fig. 2. Critical failure mechanism for Body A, foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 : (a) dwall ¼ 0 (AF ¼ 1.0), and (b) dwall ¼ f (AF ¼ 1.33).
capacity of Body B considering the force vector shown in Fig. 1. Such an assumption implies that the original and the equivalent problems have the same failure mechanism rendering the same bearing capacity. The question is whether the approximate equivalency implied in the current bearing capacity approach is correct. If not, the common current approach is inconsistent within its own simplifying assumption of safely and economically using a coherent mass for bearing capacity. 3. The mechanism To study the equivalency question where the feasibility of the failure mechanism is also examined, the computer program LimitState:GEO (LimitState, 2010) was utilized. It uses the Discontinuity Layout Optimization (DLO; Smith and Gilbert, 2007) linear programming procedure to produce a rigorous upper-bound solution in limit analysis of plasticity. It automatically identifies the critical layout of slip-lines in a soil mass that is at a limit state. Use of the DLO algorithm eliminates the need for guessing the failure mechanism expected to be most critical which is usually required in the alternative limit equilibrium analysis. Hence, it is an ideal tool to investigate the identified potential problem.
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To possibly force a one-sided failure, the surcharge of the retained soil, gH ¼ 10 20 ¼ 200 kPa, was applied to the foundation surface on the right side of rigid body e Fig. 3b. This has affected larger portion of the foundation soil compared with Fig. 2a. Also, AF has increased from 1.85 to 2.12. Consequently, an attempt to model the original problem ‘better’ by considering the surcharge of the retained soil is not a good representation of reality. The adequacy factor, AF, for Body A signifies an increase in the unit weights of the retained soil and the rigid body and not the rigid body alone as in the equivalent problem in Fig. 3. Hence, one may legitimately question the comparison of AF for Body A and Body B. A more applicable comparison would be that of Fs, the factor needed to reduce the soil shear strength to attain a limit state. LimitState:GEO computations for the equivalent problem render this factor to be Fs ¼ 1.26 whereas for the actual problem it is 1.0. Such Fs implies that a foundation soil with f0 ¼ tan1 [tan (19.8)/ 1.26] z16 and c0 ¼ 10/1.26 z8 kPa will render collapse for Body B. The failure mechanism for this case is similar to that in Fig. 3a. It is noted that if f0 ¼ 16 and c0 ¼ 8 kPa is used instead of 19.8 and 10 kPa, AF would be 1.0 while Meyerhof’s approach for these reduced strength parameters will produce AF ¼ Fs_bc ¼ 1.0 (q ¼ 259 kPa, and qu ¼ 259 kPa since Nc ¼ 11.63 and Ng ¼ 3.06). Once again, for Body B alone, very strong agreement between limit analysis and Meyerhof’s approximation exists (i.e., AF equal to 1.0 vs. 1.0 whereas for the previous case, Fig. 3a, it was 1.85 vs. Meyerhof’s 1.66). 3.2. Cohesive foundation
Fig. 3. Critical failure mechanism for foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 and dwall ¼ 0: (a) Body B (AF ¼ 1.85), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 2.12).
the centerline. In this case, the rigid body was taken as weightless thus making W the only (eccentric) vertical force. To attain failure, W was increased by 1.85; i.e., AF ¼ 1.85. In fact, AF in this case is also Fs_bc, the conventional factor of safety in bearing capacity problems, defined as the ratio qu/q. Hence, for the equivalent problem, limit analysis yields AF of 1.85 compared to 1.66 produced by Meyerhof’s approach, approximately a 10% difference. This value may be considered small when one recognizes the simplifications associated with Meyerhof’s approach and the typically large differences in results predicted by the many available analytical models of bearing capacity. Comparing Figs. 2a and 3a, it is evident that the ‘equivalent’ problem has questionable equivalency. The failure mechanism generated by Body B (Fig. 3a) is not similar to the one obtained for Body A (Fig. 2b). In the current problem, Body B causes large mass of soil to move away from the toe and, simultaneously, push soil away from the heel e a two-sided failure. In the context of the actual problem, the equivalent problem signifies an unrealistic failure mechanism as it needs to push the retained soil upwards. A single-sided failure, as in Fig. 2, is the likely mechanism. Clearly, the equivalent problem does not inhibit the development of two-sided failure mechanism.
In order to determine whether the same unconservative trend in postulating the equivalent problem exists also for cohesive foundation soil, the same problem was run with an undrained angle of friction fu ¼ 0. Limit state analysis indicates that an undrained shear strength of cu ¼ 57.8 kPa will render failure for the actual problem. Similar to Fig. 2a and b, Fig. 4a and b shows the failure mechanisms for the cases where the interface friction with the retained soil is dwall ¼ 0 and dwall ¼ f, respectively. While AF is 1.00 for dwall ¼ 0, it is 1.09 for dwall ¼ f. It is notable that the failure through the foundation deepens as dwall goes up. Also, while the AF for the c0 ef0 foundation increases to 1.33, it increased only marginally for the cohesive foundation (AF ¼ 1.09). An alternative to AF, the corresponding decrease in the soils strength needed to generate its full mobilization is Fs ¼ 1.08 whereas for the c0 ef0 case it was 1.13, which is fairly close. In the cohesive foundation soil, dwall plays a minor role unlike for the c0 ef0 foundation, where it has significant influence only when considering the AF, not the mobilization of soil strength (i.e., Fs). Fig. 5a shows the failures corresponding to Body B, which consists of a two-sided mechanism which gets closer to being symmetrical. Failure in the direction of the heel (i.e., into the retained soil) does not appear to be feasible; it is an artifact of the postulated equivalent problem. Meyerhof’s approximation for this case produces AF ¼ Fs_bc ¼ 1.15 (q ¼ 259 kPa and qu ¼ 297 kPa since Nc ¼ 5.14 and Ng ¼ 0). Once more, for Body B, the bearing capacity factor of safety agreement between limit analysis and Meyerhof’s approximation, 1.29 and 1.15, is reasonable. The needed reduction of cu to attain failure while holding W in Body B constant is 1.29 (Fig. 7); i.e., cu of 44.8 kPa. Fig. 5b shows the impact of the retained soil surcharge of 200 kPa. It produces a one-sided mechanism with a marginal difference in the bearing capacity factor of safety; i.e., AF ¼ Fs_bc increases from 1.29 for two-sided to 1.32 for a single-sided mechanism. Similar to the c0 ef0 case, imposing more appropriate failure mechanism for the equivalent problem results in increase in AF, which would not be beneficial.
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Fig. 4. Critical failure mechanism for Body A, foundation soil cu ¼ 57.8 kPa and fu ¼ 0: (a) dwall ¼ 0 (AF ¼ 1.0), and (b) dwall ¼ f (AF ¼ 1.09). Fig. 5. Critical failure mechanism for foundation soil cu ¼ 57.8 kPa, fu ¼ 0 and dwall ¼ 0: (a) Body B (AF ¼ 1.29), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 1.32).
3.3. Impact of interface friction on back of wall While AASHTO (2010) and FHWA (2001, 2009) suggest ignoring the interface friction between the retained soil and the back of the coherent mass (i.e., for horizontal crest dwall ¼ 0 is specified and therefore Pa is horizontal), other methods (e.g., NCMA, 1997) suggest using dwall >0. Hence, it is interesting to redo the bearing capacity calculations for the equivalent footing, Body B, using dwall ¼ f. That is, considering the downdrag force and thus increasing the vertical force W accordingly while modifying the eccentricity due to increase in W and simultaneous decrease in Pa. Note that due to relative vertical movements, it is possible that dwall would be as small as f. Common design methods do not address such a possibility although it may increase eccentricity. Subsequently, such a possibility is not addressed in this paper since its objective is to assess the impact of incorrect analysis while utilizing the recommended values provided by the above design methods. It can be shown that for the example problem, e ¼ 0.22 m for dwall ¼ f, where it was e ¼ 0.79 m for dwall ¼ 0. The corrected vertical force due to downdrag is 1549 kN/m for dwall ¼ f versus 1400 kN/m for dwall ¼ 0. The corrected horizontal component of Pa is 257 kN/m for dwall ¼ f compared with 333 kN/m for dwall ¼ 0. Using the revised data, AF ¼ Fs_bc for Meyerhof’s approximation can be calculated for either soil:
c0 ef0 soil (c0 ¼ 10 kPa, f0 ¼ 19.8 ): AF ¼ Fs_bc ¼ 2.08 (q ¼ 236 kPa, and qu ¼ 490 kPa since Nc ¼ 14.65 and Ng ¼ 5.24) Cohesive soil (cu ¼ 57.8 kPa): AF ¼ Fs_bc ¼ 1.26 (q ¼ 236 kPa, and qu ¼ 297 kPa since Nc ¼ 5.14 and Ng ¼ 0). Figs. 6 and 7 are analogous to Figs. 3 and 5. They show the failure mechanisms for the equivalent problem, without and with 200 kPa surcharge, considering the c0 ef0 foundation soil in Fig. 6 and the cohesive foundation soil in Fig. 7, all when dwall ¼ f. The mechanism is two-sided and nearly symmetrical without an applied surcharge and one-sided with a surcharge. Mobilization of the soil strength for the problem without a surcharge (the equivalent problem), as reflected by Fs, are 1.37 and 1.35 for the c0 ef0 and cohesive foundation soil, respectively. Table 1 summarizes the results for the actual problem and the presumably equivalent one used in design. Note in Table 1 that the Fs_bc by Meyerhof’s bearing capacity equation are marked as N/A. Simply, Meyerhof’s formulation does not address the impact of a surcharge load next to the footing. 4. Commentary It is noted that for two reasons, this study was not conducted using bearing capacity models for eccentric and inclined load. First,
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Fig. 6. Critical failure mechanism for foundation soil c0 ¼ 10 kPa, f0 ¼ 19.8 and dwall ¼ f: (a) Body B (AF ¼ 2.15), and (b) Body B þ 200 kPa surcharge on right (AF ¼ 2.23).
there are quite a few semi-empirical models dealing with inclination factors serving as modifiers of the bearing capacity coefficients Nc and Ng. The prototype bearing capacity formulation considered in this work follows FHWA (2001, 2009). While FHWA is explicit about the values of Nc and Ng to be used [i.e., those produced by Prandtl (1921) and Vesic (1975), respectively] in conjunction with Meyerhof’s model, it does not address inclination factors. Hence, it seems that the least biased approach in this study would be to use a robust numerical model. Second, the bearing capacity formula, including those using inclination factors, provides important but limited information; i.e., it produces the ultimate bearing load qu. However, the equivalent problem renders a mechanism that is not feasible; a two-sided failure. Clearly, the retained soil inhibits such a failure, producing a one-sided mechanism. Therefore, the impact of the mode of failure cannot be studied using existing common formula. The numerical tool used in this study enables one to study the effects of failure mechanism. Therefore, it provides more insight possibly leading to more general conclusions that existing closedform models. The agreement between Meyerhof’s approximation and the results from limit analysis (LA) for the eccentrically loaded footing (i.e., Body B) is reasonable from a practical viewpoint e see Table 1. This statement should be considered within the context of the existing closed-form formulae which produce, for the same problem, a large range of bearing capacity values, much more than the 12% difference encountered here.
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Fig. 7. Critical failure mechanism for foundation soil cu ¼ 57.8 kPa, fu ¼ 0 and dwall ¼ f: (a) Body B (AF ¼ 1.36), (b) Body B þ 200 kPa surcharge on right (AF ¼ 1.38).
Table 1 also shows that the interface friction between the reinforced and retained soils, dwall, has some impact (about 16% increase in bearing capacity as dwall goes up) when the soil is c0 ef0 and minor impact (about 5% increase) if it is a f ¼ 0 type soil. The impact stems from reduced eccentricity due to downdrag force offsetting the little increase in vertical load. Based on LA, the increase in bearing capacity looking at Body A (the actual problem) is substantial (about 33%) when c0 ef0 is considered and small (about 9%) when cohesive foundation is considered. For all cases, the equivalent problem had a two-sided failure. Since such a mechanism is not feasible, the formation of slip-lines in the direction of the retained soil was prevented by introducing a surcharge of 200 kPa, equal to that exerted by the retained soil on top of the foundation soil. This indeed created a one-sided mechanism. Table 1 shows that for the c0 ef0 soil this constraint actually increased the bearing load by up to 15%, especially when dwall ¼ 0. For the f ¼ 0 type soil, the increase in bearing capacity was minute. When the cohesive foundation was involved, the bearing capacity results are barely affected by whether a one-sided or two-sided mechanism develops. LA shows that the equivalency between ‘bearing capacity’ of the actual structure (Body A) and that of an eccentrically loaded footing (Body B) is questionable. This equivalency is an unsafe approximation when examined within the framework of the assumptions made in the design of MSE walls; i.e., the reinforced soil mass is considered as a rigid body exerting only vertical load on a footing with an effective width of (L 2e). For c0 ef0 soil it may overestimate
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Table 1 Summary of results. (1) dwall
Actual problem: Body A
0
f Equivalent problem: Body B
0
f Equivalent problem: Body B þ surcharge
0
f
c0 ef0 foundation: c0 ¼ 10 kPa, f0 ¼ 19.8
Cohesive foundation: cu ¼ 57.8 kPa
(2) AF: Limit analysis
(3) AF ¼ Fs_bc: Meyerhof
(4) Fs
(5) Mechanism
(6) AF: Limit analysis
(7) AF ¼ Fs_bc: Meyerhof
(8) Fs
(9) Mechanism
1.00 1.33 1.85 2.15 2.12 2.23
N/A N/A 1.66 2.08 N/A N/A
1.00 1.13 1.26 1.37 1.36 1.39
One-sided One-sided Two-sided Two-sided One-sided One-sided
1.00 1.09 1.29 1.36 1.32 1.38
N/A N/A 1.15 1.26 N/A N/A
1.00 1.08 1.29 1.36 1.32 1.38
One-sided One-sided Two-sided Two-sided One-sided One-sided
(1) dwall ¼ interface friction between retained soil and the reinforced mass (assumed as rigid body). (2) AF ¼ Adequacy factor signifying the magnitude of multiplier applied to weight of rigid body and retained soil in Body A and to weight of rigid body, W, in Body B. It is obtained from limit analysis in the framework of upper bound in plasticity. (3) AF ¼ Fs_bc ¼ same meaning as in (2); however, it is obtained from using Meyerhof’s equation for eccentrically loaded footing and it is equal to qu/q where qu and q are the ultimate bearing load and the applied load, all assumed to be uniform acting over (L 2e). (4) Fs ¼ factor of safety commonly used in slope stability analysis; it signifies the amount of soil shear strength mobilization due to a given surcharge. (5) Mechanism ¼ either single-sided where failure under the footing moves in the direction of the toe or double-sided where failure occurs simultaneously displacing soil in both directions, toe and heel (typical failure of shallow foundations). (6)e(9) have the same meaning as (2)e(5), respectively.
the bearing capacity by a factor of about 2; for f ¼ 0 it is a factor of about 1.3. Meyerhof predictions of bearing capacity for eccentrically loaded footing agree reasonably well with the LA results, thus adding confidence in using the LA numerical tool while indicating that the problem is not with the formula itself but rather with its use. The equivalency appears to be flawed because the footing is actually subjected to an inclined load with two force components: W and Pa, with resultant acting at e. The horizontal component, Pa, is large enough to significantly reduce the equivalent bearing capacity well beyond being just an issue in sliding as currently considered by, for example, FHWA. If bearing capacity is a feasible mode of failure, then the Meyerhof model should be used for an inclined load; e.g., as done in the German code (EBGEO, 2011). The conclusion of unconservative equivalency, however, has to be viewed through the perspective of real-life performance, gained over nearly 40 years. Many thousands of MSE walls were designed using Meyerhof’s approximation for an eccentric load. As far as the first author’s experience indicates, these structures, if designed and constructed properly, function well. Hence, the following ‘forensics-in-reverse’ question is reasonable: If the design of bearing capacity renders actual walls with marginal safety, then why aren’t deep-seated failures more prevalent? Several plausible explanations for this can be argued: 1. Geotechnical engineers tend to underestimate the shear strength of soils used in design. The retained and foundation soils are often stronger than assumed in analysis rendering an undeclared margin of safety. 2. For simplicity, Pa is assumed to act horizontally. However, for MSE structures the interface friction between the reinforced mass and the retained soil, dwall, is that of the retained soil. Table 1 indicates that the impact of this assumption can be significant when dealing with c0 ef0 foundations. 3. In assessing bearing capacity, the reinforced soil zone, having a width of L, is considered as a coherent rigid mass. Adopting Meyerhof’s approach, this mass has an effective width of (L 2e). This approach implies that the mass exerts uniform contact pressure of q ¼ R/(L 2e) where R is the vertical resultant force within the mass. The pressure q is limited to a fraction of Meyerhof’s bearing capacity which also considers a footing with an effective width of (L 2e). Hence, by decreasing the effective width of the footing, eccentricity ‘penalizes’ the design outcome twice; it decreases the bearing capacity and it increases the contact pressure. However, while the assumption of eccentricity under a rigid footing has been
validated, thus leading to models such as Meyerhof, there is no such validation for a flexible composite mass which is made up of soil and horizontal layers of reinforcement. In fact, experimental data (e.g., Ling et al., 2005) show that the pressure exerted by the mass is approximately uniform along L and is equal to q ¼ gH, a smaller value compared to the eccentric case. Effectively, the width of the ‘footing’ in such a case is L, rendering larger bearing capacity than that for the eccentric footing. Consequently, the substantial ‘penalty’ imposed by using eccentricity is not justified for a flexible mass subjected to lateral earth pressures. It follows that bearing capacity of a flexible ‘coherent mass’ is likely higher than that predicted by Meyerhof’s model. 4. The feasible one-sided failure mechanism versus Meyerhof’s use of a two-sided mechanism adds to bearing capacity, especially in c0 ef0 foundations. 5. The factor of safety used in design (typically Fs_bc ¼ 2.0 or 2.5) formally results in more conservative outcome, ensuring that marginal cases will still remain stable. In summary, there are five conservative measures which are commonly applied in design: 1) conservative selection of foundation strength properties, 2) taking dwall ¼ 0 , 3) base of wall is not rigid as used by Meyerhof for actual footings, 4) implicitly considering two-sided mechanism, and 5) use of substantial factor of safety may offset the unconservative nature of the equivalent problem. Multiplying the impact of all five conservative may yield an actual factor of safety of 3 or more, in turn producing generally safe structures. That is, one ‘wrong’ is made up by four or five ‘rights’. However, the immediate question is whether design should count on a balance between ‘wrongs’ and ‘rights’. Simply stated, the equivalent problem is incorrect and should be corrected to consider footing subjected to eccentric and inclined load and not only to eccentric load. Admittedly, the end result could be overly conservative implying that bearing capacity for an eccentric and inclined load is not a realistic mode of failure for flexible MSE walls. A better alternative can then be realized by viewing an MSE wall as an embankment (albeit with a near-vertical slope). In designing such structures stability analysis considering potential deep-seated failures should be checked. Adequate stability analysis (e.g., LE, LA, FE, FD), that treats the reinforced soil as such, can be used to ascertain sufficient safety against deep-seated failures. This is an extension of relevant and sound geotechnical practice. In fact, the AASHTO/FHWA design suggests a limit equilibrium (LE) global stability check, thus implying that such practice is far from being
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revolutionary. It simplifies the design process of MSE walls by removing a design aspect that is inconsistent based on its own assumptions. 5. Conclusions Some codes dealing with the design of MSE walls simplify the process of determining the layout and strength of the reinforcement by using a synergy of various analyses. The problem is divided into internal stability and external stability. In the external stability, the reinforced soil mass is considered as a rigid body which has to satisfy three stability criteria: sliding, overturning (or eccentricity), and bearing capacity. Meyerhof’s method for eccentrically loaded footings is used in assessing the margin of safety against bearing failure. Meyerhof provided his solution for eccentric and inclined loads. However, some design codes consider only the vertical eccentric load in bearing capacity calculations. The horizontal resultant due to lateral earth pressures against the reinforced mass is only used to assess the safety against direct sliding of that mass. Intuitively, ignoring the horizontal resultant may not be a conservative assumption in assessing bearing capacity, especially if it renders a large inclined load vector. Investigating the problem based on a rigorous limit analysis of plasticity where the critical failure mechanisms are identified shows that: 1. When considering only a vertically eccentric load on a rigid footing, Meyerhof’s approximation yields results that are fairly close to those predicted by limit analysis. In common designs of MSE walls, Meyerhof’s approximation is considered as equivalent to the postulated problem of a rigid (coherent) mass which retains soil. This equivalent problem is used as a part of a synergistic design approach. 2. The predicted failure load for the postulated equivalent problem is unconservative when compared with the predicted failure for the actual problem. The main reason for this unconservative result is due to treatment of vertical load only rather than considering an inclined load in Meyerhof’s equation. The reason for such ‘equivalency’ is not clear to the authors. 3. The failure mechanism associated with the equivalent problem implies that failure develops in both directions of the footing. That is, the failure is two-sided, implying that a mass of foundation soil is pushed upwards under the retained soil. The limit state analysis performed in the current study shows that such a mechanism is not likely and therefore, the equivalent problem is a somewhat conservative and unrealistic representation of the actual problem. 4. Furthermore, the same designs of MSE walls consider the friction at the interface between the retained and reinforced soils to be zero. It is shown that such an assumption is conservative, especially as the frictional strength of soil increases. 5. Apparently the compounded impact of conservative assumptions (e.g., failure mechanism, interface friction, typical conservative selection of soil shear strength, and a factor of safety applied to the analysis) makes up for the impact of unconservative equivalent problem where the Meyerhof equation uses only an eccentric load and not an eccentric and inclined load. However, it is not clear how the compounded conservative aspects of design make up for the unconservative use of Meyerhof’s equation.
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Considering the inadequacy of the equivalent bearing capacity problem, its use in actual design is questionable and perhaps meaningless. Instead, a global slope stability analysis could be used. Such analysis of reinforced soil structures is a common requirement in many design codes. Proper use of such stability analyses, including examination of deep-seated failures, instead of using an inconsistent analysis in design is not revolutionary but rather evolutionary. The global stability analysis need not consider the reinforced soil as a rigid mass; it should consider possible failures through the reinforcement as well. The end result may or may not require longer and/or stronger reinforcement as compared with the current bearing capacity analysis. This will depend on local conditions such as strength of the foundation, reinforced and retained soils, surcharge, geometry, seismic loading, water, and other relevant factors. References AASHTO, 2010. LRFD Bridge Design Specifications. with 2010 Interim Revisions, fifth ed. AASHTO, Washington, D.C. Bransby, M.F., 2001. Failure envelopes and plastic potentials for eccentrically loaded surface footings on undrained soil. International Journal of Numerical Analytical and Methods in Geomechanics 25 (4), 329e346. BS8006-1, 2010. Code of Practice for Strengthened/Reinforced Soils and Other Fills. British Standard Institution, ISBN 978 0 580 53842 1, ICS 93.020, 260 p. EBGEO, 2011. Recommendations for Design and Analysis of Earth Structures Using Geosynthetic Reinforcement. German Geotechnical Society, Ernst & Sohn, GmbH & Co. KG (Chapter 7). FHWA, 2001. Mechanically Stabilized Earth Walls and reinforced Soil Slopes d Design and Construction Guidelines. FHWA-NHI-00e043, Authors: Elias, Christopher, and Berg. Federal Highway Administration, Washington, D.C. FHWA, 2009. FHWA-NHI-10e024, Authors: Berg, Christopher, and Samtani. Design of Mechanically Stabilized Earth Walls and Reinforced Soil Slopes, vol. I. Federal Highway Administration, Washington, D.C. Georgiadis, K., 2010. The influence of load inclination on the undrained bearing capacity of strip footings on slopes. Computers and Geotechnics 37 (3), 311e322. Hansen, J.B., 1970. A Revised and Extended Formula for Bearing Capacity Bulletin, vol. 28. Danish Geotechnical Institute, Copenhagen, Denmark. Houlsby, G.T., Puzrin, A.M., 1999. The bearing capacity of a strip footing on clay under combined loading. Proceedings of the Royal Society, London, Series A 455 (1983), 893e916. LimitState, 2010. LimitState:GEO Manual Version 2.0. LimitState Ltd. October 2010 Edition. Ling, H.I., Mohri, Y., Leshchinsky, D., Burke, C., Matsushima, K., Liu, H., 2005. Large-scale shaking table tests on modular-block reinforced soil retaining wall. ASCE Journal of Geotechnical and Geoenvironmental Engineering 131 (4), 465e476. Loukidis, D., Chakraborty, T., Salgado, R., 2008. Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads. Canadian Geotechnical Journal 45 (6), 768e787. Meyerhof, G.G., 1953. The bearing capacity of foundations under eccentric and inclined loads. In: Proc. Third International Conference on Soil Mechanics and Foundation Engineering, Zurich. vol. 1, pp. 440e445. Meyerhof, G.G., 1963. Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal 1 (1), 16e26. National Concrete Masonry Association (NCMA), 1997. In: Collin, J.G. (Ed.), Design Manual for Segmental Retaining Walls, second ed. National Concrete Masonry Association, Herndon, VA. Prandtl, L., 1921. Über die eindingungsfestigkiet (harte) plasticher baustoffe und die festigkeit von schneiden. Zeitschrift fur Angewandte Mathematik und Mechanik 1 (1), 15e20. Smith, C., Gilbert, M., 2007. Application of discontinuity layout optimization to plane plasticity problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463 (2086), 2461e2484. Vesic, A.S., 1975. Bearing capacity of shallow foundations. In: Winterkorn, H.F., Fang, H.-Y. (Eds.), Foundation Engineering Handbook. Van Nostrand Reinhold Co., New York, NY, pp. 121e147 (Chapter 3). Yun, G., Bransby, M.F., 2007. The horizontal-moment capacity of embedded foundations in undrained soil. Canadian Geotechnical Journal 44 (4), 409e424. Zhang, L., Zhao, M., Shi, C., Zhao, H., 2010. Bearing capacity of geocell reinforcement in embankment engineering. Geotextiles and Geomembranes 28 (5), 475e482.