Design Of Retaining Walls : System Uncertainty & Fuzzy Safety Measures J. Oliphant *, P. W. Jowitt * and K. Ohno + * Department of Civil & Offshore Engineering, Heriot-Watt University, Riccarton, Edinburgh. EH14 4AS. Email: johno@civ.hw.ac.uk + Department of Bioresources, Mie University, 1515 Kamihama, TSU 514 Japan. Abstract This paper explores some of the aspects of embedded retaining wall design, in particular the interrelationship between system uncertainty and the use of factors of safety in the various design procedures. System uncertainty reflects the mismatch between idealised retaining wall behaviour and what is observed in practice. This feature is modelled using the concept of fuzzy sets where the degree of support for the lateral stress state at failure is considered and related to current limit equilibrium design methods, measured lateral pressures and the results of finite element analyses. The proposed model is shown to have some advantages over current design practice. 1 Introduction This paper explores some of the aspects of embedded retaining wall design, in particular the interrelationship between system/model uncertainty and the use of factors of safety in the various design procedures. Data from field trials, laboratory models and from finite element models suggest that actual lateral stresses behind and in front of
embedded retaining walls do not coincide with those predicted by the idealised stress distributions based on classical theory. Nonetheless, the basis of most design procedures is still rooted in these classical approaches and their variants. As is common in geotechnical engineering, and in other branches of engineering, factors of safety are used to provide a margin between the forces applied (S) and the capacity provided to resist them (R). A factor of safety may simply reflect parameter uncertainty within a well-defined and accepted behavioural model. For example, this parameter uncertainty could relate to material properties and/or variability. In principle, such parameter uncertainty could be handled probabilistically, and, as is often the case, re-interpreted for day to day use as a suggested (partial) safety factor. On the other hand, a factor of safety could be used to reflect uncertainty in the model itself, which might be termed system or model uncertainty. Here a probabilistic interpretation might be more difficult to grasp, but nonetheless the perceived uncertainty in the model is often real. With the retaining wall problem, such model or system uncertainty might reflect the perceived mismatch between idealised retaining wall behaviour and previous experimental/model results. This aspect is examined, and an interpretation suggested which is based on the concept of fuzzy sets; that is to say the perceived confidence in the lateral soil stress state at failure is considered, and related to experimental data reported in the literature and related to recommended factors of safety. 2 Analysis Of Embedded Walls 2.1 Overall Stability Consider now a simple vertical retaining wall such as that depicted by AB in Figure 1a, which represents an embedded cantilever retaining wall. The limiting condition governing the overall stability of the wall is usually assumed to correspond to incipient rotation about point O of the wall. The corresponding earth pressure distributions correspond to the idealised forms, with the soil in an active and passive states as shown in Figure 1b. The required embedment depth can be found by establishing that value of d which simultaneously provides both moment and horizontal equilibrium. This calculation method is complicated
arithmetically and known as the full method. In practice, a simpler representation is adopted, as shown in Figure 1c, in which the effect of that part of the wall below point O is replaced by an equivalent force R acting at point O, and the embedment depth determined solely from consideration of moment equilibrium and rotation about O. To allow for this simplification, the depth d o so determined is then increased by an empirical amount of 20% to give the design embedment depth d ( Padfield & Mair 1 ). It might be noted that the value of R is not explicitly considered. 2.2 Calculation Methods 2.2.1 Traditional Methods For design purposes, the analysis is generally undertaken with a factor of safety employed to ensure that the resisting moment exceeds by some margin the overturning moment. A variety of different design procedures have been developed from the simplified (and classical) condition shown in Figure 1c, with the result that the factor of safety is applied in slightly different ways and to different aspects of the lateral stress diagram. For example, for the Gross Pressure Method (GPM ; Code of Practice 2 2 ), the factor of safety (F GPM ) is simply applied to the whole of the passive earth pressure zone in front of the wall. The Strength Method (SM) is based on the soil pressure distribution of the GPM but with a factor of safety (F SM ) on shear strength (this method was used to produce the results for d for the full calculation method given in Table1). Other methods apply the safety factor to a different component of the earth pressure distribution. For example, in the Burland, Potts & Walsh method (Burland et al 3 ) the factor of safety (F BPW ) is applied to the net resisting moment, while in the British Steel method (British Steel 4 ) the factor of safety (F BS ) is applied to the net passive lateral stress component only. The purpose of these factors of safety is to allow for uncertainties associated with, at least, the following : (i) variability in material strengths from their estimated and variability in loadings applied to the structure (parameter uncertainty) ; (ii) inaccuracies in behavioural prediction due to system uncertainty ; and
(iii) limiting the movements of the structure. A full discussion of these traditional calculation methods has been carried out by Potts & Burland 5. These methods are now out of date, having been overtaken by BS8002 6, but because of their popularity will continue to be used and justified for a long time. 2.2.2 BS8002 Approach For cantilever embedded walls, the idealised condition of Figure 1b forms the basis of the limiting equilibrium calculation approach recommended in BS8002 where the following procedures need to be applied : the classical limiting active and passive lateral pressures are changed to working lateral pressures by reducing the peak soil strengths by a mobilisation factor M a minimum uniform surcharge of 10kN/m 2 to act on the retained soil surface. a minimum unplanned excavation depth of 0.5m. The mobilisation factor M is in effect a factor of safety on soil strength which has the effect of uniformly increasing the active pressures and uniformly reducing the passive pressures on a wall. Detailed discussions on BS8002 have been carried out by Akroyd 7, Bolton 8 and Puller & Lee 9. 3 System Uncertainty & Design 3.1 Introduction It is now interesting to examine some of the various experimental and field data relating to lateral soil pressures on propped and cantilevered embedded retaining walls e.g. data due to Rowe 10, Tschebotarioff 11, Carder and Symons 12, and the results of finite element analyses by Potts and Fourie 13 and Fourie and Potts 14. In all cases, there are obvious departures from the idealised behaviour of Figure 1; these differences are not due especially to experimental variability. The state of lateral stress in a soil will depend on the following factors : (a) the nature and magnitude of wall movement (b) relative stiffness of the wall and prop
(c) shear strength of the soil (d) at-rest pressure (e) construction form and sequence. The simple design assumptions of Figure 1 are hence a considerable simplification of some very complex processes. There is an obvious and significant mismatch between the design assumptions and what is actually observed in practice with system uncertainty dominating the issue. 3.2 Proposed Model For any particular wall there will be some variation in the confidence with which particular sections of the wall can be assumed to be at either full active or full passive pressure at failure. For example, whilst the wall might be assumed to be rigid and to rotate, producing conditions broadly in accordance with Figure 1b, the actual rotation and the corresponding conditions near the toe of the wall are less precisely defined. In this sense, there is some system uncertainty which ought to have some bearing on the design procedure. However, the design procedures such as those referred to above, use a single factor F or M) to scale up (or down) uniformly the complete portion of the stress distribution. This is perhaps an oversimplified view, and an exploration of how this system uncertainty might be described and incorporated into design is detailed below. For simplicity, the present discussion is confined to the case of a cohesionless, dry soil with no wall friction. Consider a particular location down the wall. The earth pressure coefficient is required to lie between the two values K a and K p (this range could be modified to reflect the nature of the problem where it may be more appropriate to choose different bounds e.g. a lower bound could be K o while an upper bound could be defined in terms of a fraction of K p ). If the likely kinematics of the wall are such that the position under consideration is likely to be a region of passive pressure then the coefficient K p will be supported; if an active condition is likely, then the active coefficient K a will be supported. In the simplified soil pressure distribution shown in Figure 1c, the retained side of the wall is nominally active down to the toe; the resisting side of the wall is nominally passive.
The degree of support (belief) for a particular stress condition can be described by a fuzzy set membership function, as shown in Figures 3a and 3b. Figure 3a represents an active stress state: the support value represents the degree of support for a particular pressure coefficient K being a member of the fuzzy set A = Active Stress. In this paper, a simple triangular membership function is used, so that for α = 1 (full support) the membership of the active stress set is just the coefficient K a. If α = 0.5 (representing much weaker support), then all values down to (K a +K p )/2 are members of the active stress state with support α = 0.5 and with α = 0, all values down to K p are admitted. The active support value for a measured or calculated pressure coefficient K can be represented by (K p -K)/(K p -K a ) where K a K K p. Similarly, Figure 3b describes the fuzzy set P describing Passive Stress. This time, at a level of support α = 1, then only K = K p is admitted; the membership of P when α= 0.7 is confined to all values of K down to Ka + 0.7(K p -K a ). The passive support value for a measured or calculated pressure coefficient K can be represented by (K-K a )/(K p -K a ) where K a K K p. Clearly, the membership function need not be confined to a triangular form, but is used herein for simplicity to illustrate the use of the model. In effect, the level of support or belief in the magnitude and shape of the pressure distribution can be reflected though the assignment of α values down each side of the wall ( e.g. α 1 applied to the back of the wall and α 2 applied to the front of the wall). This then allows, in the design situation, the required depth of the wall d to be determined; for an existing wall, the margin of safety against failure can be determined. This can be done by expressing the limiting condition on the behaviour of the wall by the following limit state equation α 2.R* α 1.S* (1) where R* and S* are the design values for resistance and demand, and the safety factors applied to them relate to parameter uncertainty, while the α values allow for the mismatch between the observed lateral pressure distributions and those idealised pressure distributions assumed in current design, and relate to system uncertainty. Here distinction can be made between system uncertainty and parameter uncertainty and
appropriate values of α can be calculated to satisfy both the serviceability and ultimate conditions. The proposed model is now best illustrated through its use in limit equilibrium methods and later through a comparison with measured/calculated lateral pressures. 3.2.1 Use of the Model in Limit Equilibrium Methods Consider first the simple situation shown in Figure 2, which shows a simple cantilevered wall of height 4m above the base level. The soil is assumed to have a φ angle of 25, K a = 0.41, K p = 2.46 and unit weight γ = 20kN/m 3. The nominal areas of passive and active stress at failure are as shown in Figure 4. If the α level is taken to be unity, then the situation is just as in Figure 1c, with full support (belief) in the passive coefficient K p on the resisting side and full support for K a on the active side. Under these conditions of limiting equilibrium the required depth of wall is d o = 4.90m. Two cases are now explained. 3.2.1.1 Case 1 : α = 0.8 in both active & passive regions Suppose now that the α level is taken to be α = 0.8 in both the active and passive regions. The corresponding fuzzy sets for this α level are: Active [0.41 to 0.82]; Passive [2.05 to 2.46]. So far as the stability of the wall is concerned, the least favourable condition on the active side will be at a coefficient K = 0.82 and on the passive side when K = 2.05. Under these conditions the required depth of wall is d o = 11.19m, which compares to d o = 4.90m in the normal case. The variation in the required values of d o with α is shown in the two left-hand columns of Table 1. Table 1 Variation in α with F GPM and d o for Figure 4 α d o (m) F GPM 1.0 4.90 1.0 0.9 7.43 1.64 0.8 11.19 2.39 0.7 18.56 3.33 It is possible to relate these α values with the traditional factors of safety. For example, if a particular design approach is adopted, for example the GPM, then the variation of α with F GPM is shown in Table 1, together with the corresponding values of d o.
However, the simple analysis above assigns the same value of support - the α value - for both active and passive zones at all depths, whereas it is more appropriate that the support level α should be varied in the light of experience and data. For example, α values towards the toe might be expected to be rather less than unity reflecting the uncertainty that exists with regard to stress conditions in this area of the wall. 3.2.1.2 Case 2 :Equivalent Cases Corresponding to GPM (F=2), SM (F=1.5) & BS8002 (M=1.2) By referring to Figure 4 once more, and assuming that α 1 varies over the entire depth of the back of the wall and α 2 varies over the entire embedment depth, it is again possible to relate these α - values with the traditional calculation methods and the calculation approach recommended in BS8002. For example, consider the GPM in which the factor of safety is effectively applied as a reduction factor on the passive resistance. The pressure distributions at equilibrium then correspond to full active pressures acting on the back of the wall and reduced passive pressures on the front of the wall. If current practice is followed by taking F GPM = 2, then this would be equivalent to α 1 = 1.0 and α 2 = 0.60 giving a depth of embedment of 9.0m. In this case, the α - values would be modelling parameter uncertainty as well as system uncertainty. In the strength method (SM) the shear strength parameters of the soil are reduced by a factor of safety, F SM. The pressure distribution corresponds to pressures above the full active values acting on the back of the wall, and to pressures below the full passive values in front of the wall. Current practice for the SM is to assume F SM = 1.5, which is equivalent to α 1 = 0.95 and α 2 = 0.70 giving a depth of embedment of 8.0m. Again, the α - values would cover both parameter and system uncertainties. In the BS8002 approach the shear strength parameters of the soil are reduced by a mobilisation factor, M. The pressure distributions therefore correspond to pressures above the limiting active values and to pressures below the limiting passive values. Adopting an M = 1.2 is equivalent to α 1 = 0.97 and α 2 = 0.84 based on the pressure distribution alone and does not allow for an unplanned excavation and a minimum uniform surcharge. 3.2.2 Comparison with Measured Lateral Pressures
A field study was conducted on a section of cantilevered diaphragm walling of the A329 (M) Reading to Bracknell road (Carder & Symons 12 ). The section of wall was instrumented to monitor the total horizontal stress and pore water pressure both in front and at the back of the wall. The best estimates of the total lateral stress distribution prior to retaining wall construction and about 14 years after its construction are shown in Figure 5. The estimates have been made from spade cell readings using an empirical correction together with self-boring pressuremeter and dilatometer measurements. For assessing the stability of the wall, Carder and Symons produced upper and lower bound strength values ( see Table 2a ). The relationship between the lateral pressure coefficients, K, measured 14 years after construction, and the back-calculated α - values for the back of the wall and both lower bound and upper bound shear strength values, is given in Table 2(a). The distribution of α - values indicates that the level of support for the pressure distribution reduces with depth below ground level, ending in weak support at approximately 15m. Some increase of pressure occurs towards the base of the wall, which corresponds to a decrease in pressure in front of the wall at this depth. This observation is consistent with limit equilibrium assumption of a point of rotation at some depth below excavation level. A weaker support reflects the uncertainty associated with demonstrating this point of rotation. The more conservative, and hence less certain, lower bound value approach, as expected, results in lower α - values. For practical design purposes the α - values averaged over the full height would correspond to α = 0.74 for the lower bound values and α = 0.79 for the upper bound values. These α - values are lower than those calculated from the traditional limit equilibrium methods. Table 2a Relationship Between the Measured Coefficient K ( Figure 5 ) and the Calculated α - Values for the Back of the Wall Depth Below G.L. (m) K (measured) Alpha - Values Lower Bound Values Upper Bound Values φ = 21, c = 0, δ = φ φ = 25, c = 0, δ = φ (Ka =0.4 ; Kp = 3.2) (Ka = 0.34 ; Kp = 4.06) 5 0.877 0.83 0.86 7 1.001 0.79 0.82 9 1.062 0.76 0.81
11 1.041 0.77 0.81 13 1.161 0.73 0.78 15 1.515 0.60 0.68 The results for the soil in front of the wall below excavation level are given in Table 2 (b). For the lower bound values, most of the measured K s are higher than the calculated values with α being undefined. However, this clearly cannot happen unless there has been an increase in shear strength to enlarge the lateral pressures. There may be some element of cohesive strength or the soil may be capable of mobilising much higher angles of shearing resistance than determined from the conventional compression tests. For the upper bound values, the level of support for the pressure distribution reduces to a depth of 6m below excavation level, where it then increases and at 8m below excavation level the measured K value is greater than the calculated value. The yielding of the wall is sufficient to fully mobilise the passive pressure near to excavation level (even under working conditions) and hence the level of support is strong. However, at the lower end of the wall the yielding is low and not sufficient to mobilise the full calculated shear strength. This behaviour reflects actual practice as the soil at the base of the excavation is likely to fail passively as a result of vertical stress relief. Table 2b Relationship Between the Measured Coefficient K ( Figure 5 ) and the Calculated α - Values for in Front of the Wall Depth K Alpha - Values Lower Bound Values Upper Bound Values Below (measured) φ = 21, c = 0, δ = φ φ = 25, c = 0, δ = φ E.L. (m) (Kp = 3.2 ; Ka = 0.4) (Kp = 4.06 ; Ka = 0.34) 3.4 4.06-1.0 4.0 3.71-0.91 5.0 3.28-0.79 6.0 3.12 0.97 0.75 7.0 3.56-0.87 8.0 4.41 - - 3.2.3 Comparison with Finite Element Analysis Fourie and Potts 14 carried out a comparison between the results of finite element and limit equilibrium analyses for an embedded cantilever retaining wall. They considered a 1m thick, 20m deep concrete retaining wall installed in a linearly elastic-perfectly plastic soil. The study was
restricted to fully drained conditions with the following strength parameters : c = 0 and φ = 25. The influence of the initial stress regime was investigated by adopting K o = 0.5 and 2.0. The horizontal effective stresses developed on the back and on the front of the wall for excavation depths of 5.17m, 8.85m and 10.6m are shown in Figures 6 and 7 respectively. The unbroken lines on the figures indicate the distribution of at-rest pressure and fully mobilised active and passive earth pressure for the condition δ = φ. The corresponding alpha values for the above conditions are shown in Tables 3 and 4. Table 3 shows the variation in α with depth on the back of the wall for the limiting condition (F BPW = 1, corresponding to an excavation depth of 10.6m) and K o = 0.5 and 2.0. The fully active condition is reached only above excavation level and hence there is strong support for the calculated values. Below the excavation level the pressure increases rapidly, particularly for the condition of K o = 2.0. However, the support for an active pressure distribution is much stronger in the soil with K o = 0.5 ( ranging from 0.91 to 1.0 ) than the soil with K o = 2.0 ( ranging from 0.57 to 0.99 ) indicating a dependence on the initial stress state. Table 3 Alpha Values ( Back of the Wall, F BPW = 1, K o = 0.5 & 2.0 ) Depth Below G.L. (m) K (Calculated) K o = 0.50 K o = 2.0 Alpha - Values (K a = 0.33, K p = 4.19) K o = 0.50 K o = 2.0 2.0 0.33 0.33 1.0 1.0 4.0 0.33 0.33 1.0 1.0 6.0 0.33 0.33 1.0 1.0 8.0 0.33 0.33 1.0 1.0 10 0.33 0.36 1.0 0.99 12 0.33 0.45 1.0 0.96 14 0.35 0.65 0.99 0.92 16 0.40 1.0 0.98 0.83 18 0.50 1.55 0.96 0.68 20 0.68 2.0 0.91 0.57 Table 4 shows the variation in α with depth below the excavation level in front of the wall for the limiting condition and K o = 0.5 and 2.0.
The mobilisation of full passive resistance in front of the wall is also highly dependent on the initial stress state. For example, passive failure defined by K p = 4.19 is mobilised over approximately two thirds of the embedded wall depth for K o = 2.0. The support for a passive pressure distribution is therefore stronger in the soil with K o = 2.0. Table 4 Alpha Values ( Front of the Wall, F BPW = 1.0, K o = 0.5 & 2.0 ) Depth Below E.L. (m) K (calculated) K o = 0.50 K o = 2.0 Alpha Values (K a = 0.33, K p = 4.19) K o = 0.50 K o = 2.0 1.4 3.25 4.10 0.76 0.98 3.4 3.70 4.19 0.87 1.0 5.4 2.25 3.98 0.50 0.95 7.4 1.22 3.64 0.23 0.86 9.4 0.47 3.35 0.04 0.78 4 Summary & Conclusions Parameter and system uncertainty are inherent in geotechnics, but system uncertainty is dominant. System uncertainty reflects the mismatch between idealised retaining wall behaviour and what is actually observed in practice. This aspect has been modelled using the concept of fuzzy sets where the degree of support for the lateral stress state at failure was considered and related to current limit equilibrium calculation methods, measured lateral pressures and the results of finite element analyses. The assumption of a triangular, fuzzy membership function was used to illustrate the proposed model. Levels of support to give results similar to those of existing design practice were indicated and related to experimental and finite element calculated data. The results indicated that system uncertainty is dependent primarily on the initial stress state of the soil, the perceived movement of the wall, and the shear strength of the soil. The model clearly demonstrated that the degree of support for a pressure distribution is depth dependent, highlighting the sections of wall where the behaviour is less certain. Hence, the limiting active and passive pressures at failure normally used in limit equilibrium methods are an over-simplification of the real behaviour of an embedded retaining wall. The proposed model therefore has considerable advantages over
current practice. A clear recognition is given to the differences between the pressures acting on a wall in service and the limiting active and passive pressures normally assumed in limit equilibrium calculations. 5 References 1. Padfield, C. J. & Mair R. J. (1984). Design of retaining walls embedded in stiff clay, CIRIA Report 104, London. 2. Code of Practice 2 (1951). Earth Retaining Structures, Civil Engineering Code of Practice, Institution of Structural Engineers, London. 3. Burland J.B., Potts,D.M. & Walsh, N.M. (1981). The Overall Stability of Free and Propped Embedded Cantilever Walls, Ground Engineering, 14, 5, 23-38. 4. British Steel (1988). Piling Handbook, Sixth edition, British Steel plc, Scunthorpe. 5. Potts, D. M. & Burland, J. B. (1983). A parametric study of the stability of embedded earth retaining structures, TRRL SR813, Crowthorne, England. 6. BS8002 (1994). Code of Practice for Earth retaining structures, BSI, Milton Keynes, England. 7. Akroyd, T. N. W. (1996). Earth-retaining structures: introduction to the Code of Practice, The Structural Engineer, V74, N 21/5, pp 360-364. 8. Bolton, M. D. (1996). Geotechnical design of retaining walls, The Structural Engineer, V74, N 21/5, pp 365-369. 9. Puller, M. & Lee, C. K. T. (1996). A comparison between the design methods for earth retaining structures recommended by BS8002: 1994 and previously used methods, Proc. I. C. E., Geotechnical Engineering, 119, Jan., 29-34. 10. Rowe, P.W. (1952). Anchored Sheet-Pile Walls, Proceedings, I.C.E., 1, Part 1, Paper No. 5788, 27-70. 11. Tschebotarioff, G.P. (1973). Foundations, Retaining and Earth Structures, McGraw-Hill, New York. 12. Carder, D.R. & Symons, I.F. (1989). Long-Term Performance of an Embedded Cantilever Retaining Wall in Stiff Clay, Geotechnique, 39, 1, 55-75. 13. Potts, D.M. & Fourie, A.B. (1984). The Behaviour of a Propped Retaining Wall : Results of a Numerical Experiment, Geotechnique, 34, 3, 352-383. 14. Fourie, A.B. & Potts, D.M. (1989). Comparison of Finite Element and Limiting Equilibrium Analyses for an Embedded Cantilever Retaining Wall, Geotechnique, 39, 2, 175-188.