A Study on Reliability Analysis for Reinforced Earth Retaining Walls

Transcription

1 A Study on Reliability Analysis for Reinforced Earth Retaining Walls Byung Sik, Chun Department of Civil Engineering, Hanyang University, Seoul, Korea(Rep.) Kyung Min, Kim Department of Civil Engineering, Hanyang University, Seoul, Korea(Rep.) Deok Ki, Min Department of Civil and Environmental Engineering, Ulsan University, Ulsan, Korea(Rep.) ABSTRACT: Traditionally, the reliability of retaining walls is achieved through the use of safety factors or margins and adopting conservative assumption in the process of design, that is, by ascertaining that a minimum supply condition will remain adequate under a maximum demand condition. However that is often defined on the basis of subjective judgments. Such a traditional design methods are difficult to quantify and lack the logical basis of describing uncertainty. Especially, reinforced walls consider not only soil properties but soil-reinforcement interaction uncertainties. There has been much emphasis recently the use of probabilistic method in the geotechnical engineering. The most effective applications of probabilistic methods are involving relative probabilities of failure or illuminating the effects of uncertainties in the parameter. This thesis described how probabilistic description of soil parameters and soil-reinforcement interaction parameters were applied to the stability analysis. The first-order, second moment approach was explored and applied to the design of reinforced retaining walls. An example illustrated the relative contribution of uncertainties about different parameters to the reliability of the reinforced retaining walls. The results obtained from this study were follows; the reliability of the soil-reinforcement interface friction angle, was highly sensitive to the coefficient of variation. However, when the reinforced fill unit weight γ r, and the reinforcement length, L were lower than the limited values, the probabilities of failure were increased. The reliability of the retained backfill soil unit weight, γ f in the unreinforced area was lowly sensitive to the coefficient of variation. 1 INTRODUCTION This system used steel strips for reinforcement and as geosynthetic reinforcement has developed, this method is widely applied in various sites. Reinforced earth retaining walls are generally less expensive and more convenient than conventional concrete retaining walls. Various analyses and design methods were proposed for reasonable design of reinforced earth retaining wall. Limit equilibrium analysis is mostly used to decide the safety factor for failure of retaining wall. This design method shows different results according to differences for assumptions and application of safety factor because equilibrium for stresses and moment is analyzed assuming stress condition and shape of active failure plane and pullout resistance of geosynthetic reinforcements. Therefore, in this study, reliability analysis is carried out under consideration of uncertainty of assumptions to compensate for limit equilibrium analysis. 248

2 2 DESIGN CASE OF GEOSYNTHETICS REINFORCED EARTH RETAINING WALL The two kinds of design methods by limit equilibrium analysis are used for suitable design of reinforced earth retaining wall. Firstly, tie-back analysis method is generally used. This method represents that lateral active pressure is almost equal to lateral pressure. The second method is similar to traditional slope stability analysis method. In this method, the effect of reinforcement is considered when analyzes force of active surface of failure or moment equilibrium. In this study, 'tie-back' method Figure 1. Final section of design case combined with tensile-resistant has chosen for carrying out sensitivity and reliability analysis precisely. Design parameters are as follows. Reinforced fill properties γ r = 1.9tonf/m 3, c r = 0.0tonf/m 2, φ r = 32 Retained backfill properties γ f = 1.85tonf/m 3, c f = 0.0tonf/m 2, φ f = 30 Foundation ground properties γ b = 1.85tonf/m 3, c b = 0.0tonf/m 2, φ b = 30 Geosynthetic reinforcement is two kinds of geogrids each of which maximum tensile strength is 10tonf/m and 6tonf/m. The frictional characteristic between soil and geosynthetic reinforcement are c * = 0.0tonf/m 2, = 30. The height of retaining wall is 8m and design length of geosynthetic reinforcement is 6m. Because slope of wall is vertical, is zero, and slope angle is 25. Figure 1 shows the vertical spacing of geosynthetic reinforcement and type of geogrids by layers. Figure 1. Design example 3 STABILITY ANALYSIS BY LIMIT EQUILIBRIUM METHOD Figure 2 illustrates limit equilibrium state of soil mass acting on the sliding surface for vertical direction. In this case, the total sum P R of weight of soil mass V 1, vertical and horizontal factor of earth pressure F V, F H, reaction force on active surface R, and resisting friction force acting on geosynthetic reinforcement layers 249

3 placed on active surface are acting. So, angle of sliding failure planes is ϕ, equilibrium formula is described below(korean Geotechnical Society, 1998). P R = F H +(V 1 +V 2 +F T sin )tan(ϕ- ) (1) Resisting friction force P R of Each geosynthetic reinforcement is as follows. where, P R = 2L e (c*+ 'γtanφ*) (2) L e = Effective length of geosynthetic reinforcement, i.e. the length of geosynthetic reinforcement in the zone of resistance (beyond failure planes in Fig. 2) c * = Cohesion between soil and geosynthetic reinforcement (not considered sandy soil with little fine fraction) δ= Interface friction angle between soil and geosynthetic reinforcement σ' v = Effective vertical stress acting on geosynthetic reinforcement (=γ r z + L tanβ γ f /2, z : penetrated depth of geosynthetic reinforcement in upper surface of retaining wall) Figure 2. Stability analysis of geosynthetic reinforced earth retaining wall 4 SENSITIVITY ANALYSIS Sensitivity analysis was carried out to confirm the significant design parameters which could have influence on stability analysis. For this study, basic data used are provided below. L e =0.485m, c * = 0.0tonf/m 2, =30, γ r =1.9tonf/m 3, z=5.4m, H=8m, L=6m, =25, γ f =1.85tonf/m 3 Frictional resistance of geosynthetic reinforcement is 7.35tonf/m under the above condition. Table 1 shows that the most sensitive tendency was found for interface friction angle between soil and geosynthetic reinforcement and also unit weight of reinforced fill soils γ r was same. On the other hand, retained backfill soil 250

4 unit weight γ f, length of geosynthetic reinforcement L, and cohesion between soil and geosynthetic reinforcement c * had low sensitivity level. Table 1. Results of sensitivity analysis Baseline Coefficient of Variation, Parameter Value CV(%) New Value P R Change P R δ γ r 1.9tonf/m c * 0.16tonf/m L 6m γ f 1.85tonf/m RELIABILITY ANALYSIS Resistance conditions or load conditions for design of structures are diverse and these design parameters have uncertainty. So it is required that basic design parameters which affect the stability of structures are represented by general formulas in order to analyze probability analysis. General formula which consists of basic design parameters is as shown in Eq. (3). where, X term is vector value for X 1,X 2,X 3,...,X n. g(x)=g(x1, X2, X3,..., Xn) (3) For Eq.(3), the state of g(x)=0 is limit state, and if g(x) > 0 and g(x) < 0, then, each state is safe state, failure state. Generally, there are existing resisting force (R) and sliding force(s) in various structures and condition of maintaining those structures is R>S. This condition can be expressed as probability for the state P(SM < 0) which structures come to failure (Safety Margin, SM = R-S, Fig. 3). So, Eq. (3) becomes SM = R - S = g(x 1, X 2, X 3,..., X n ) < 0 (4) 251

5 In this study, reliability analysis was focused on interface friction characteristics and evaluated the probability of failure of reinforced earth retaining wall. For the characteristic of interface friction between soil and geosynthetic reinforcement, g-function is able to be obtained from the concept of the force acting on active area within retaining wall(s) and resisting force with frictional resistance between soil and geosynthetic reinforcement(r). SM = R - S = ΣP R - [F H +(V 1 +V 2 +F T sinβ)tan(φ-ψ)] = g(γ r, c *, L, β,δ, γ f ) (5) From probability Density Function by using μ M and σ M, probability can be obtained(rosenblueth, 1975). Reliability is dependent on kinds of structures, but generally allowable design standard is required over 95%(Smith, 1986). Also, Meyerhof (1982) suggested the limit value of β=3.1, maximum P f = 10-3 for earth retaining structures. When probability of failure is calculated, Equation of Reliability Index, β is used. This equation is defined as ratio of average of SM(μ M ) over root mean square deviation(σ M ), and from this Eq.(6), probability of failure can be written as Eq. (7). β = μ M / σ M (6) P f = P f FORM = Φ(-β) (7) Φ in Eq. (6) means Cumulative Normal Distribution Function in formula (6), and Reliability Index is decided by FORM, then probability of failure can be obtained by Φ-diagram. Figure 3. PDF of safety margin, SM 6 RESULTS OF RELIABILITY ANAYSIS Results of reliability analysis of design variables for proposed design case are shown in Fig 4, 5, 6, 7. Figure 4 shows that if interface friction angle is under 24 with respect to CV = 5%, 10%, 15%, probability of failure would be over 50%. As unit weight of reinforced fill within reinforced earth is increasing, probability of failure decreases in Figure 5. If CV = 15%, γ r = 1.9tonf/m 3 then, probability of failure was , and if CV = 20%, γ r = 1.9tonf/m 3 then, probability of failure was These values were evaluated larger than limit value 10-3 which was proposed by Meyerhof(1982). Figure 6, 7 show probability of failure according to length of geosynthetic reinforcement L, and retained backfill soil unit weight of retaining wall. For sensitivity analysis carried out already, effects on probability of failure are little, but influences of those factors are shown. The result shows Probability of failure depends on Length of geosynthetic reinforcement and unit weight γ f. 252

6 Figure 4. Probability of failure according to interface friction angle δ, P f Figure 5. Probability of failure according to reinforced fill unit weight γ r, P f Figure 6. Probability of failure according to 1ength of geogrid L, P f Figure 7. Probability of failure according to retained backfill soil unit weight γ f, P f 253

7 7 CONCLUSION In this paper, owing to uncertainty of design parameters, reliability analysis concentrates upon the frictional characteristic between soils and geosynthetic reinforcements. As a result, interface friction angle between soil and geosynthetic reinforcement and unit weight of reinforced fill have more influence on probability of failure than other design parameters. Besides, results showed that the length of geosynthetic reinforcement and unit weight of retained backfill soils have little influence on probability of failure. REFERENCES Korean Geotechnical Society, Geosynthetics, Geotechnical Series 9(1998). Meyerhof, G. G., "Limit state designs in geotechnical engineering", Structure Safety 1, No. 1, pp (1982). Rosenblueth, E., "Point estimates for probability moments", Proceedings of the National Academy of Science of the United State America, Vol. 72, pp (1975). Smith, G. N., "The use of probability theory to assess the safety of propped embedded cantilever retaining walls", Géotechnique, Vol. 35, No. 4, pp (1985). 254