Partial factors: where to apply them?

Transcription

1 LSD2000: International Workshop on Limit State Design in Geotechnical Engineering Melbourne, Australia. 18 November 2000 Partial factors: where to apply them? Brian Simpson Arup Geotechnics, London, UK ABSTRACT: The application of partial factor design in geotechnics has been under debate for several decades. Although the approach has some obvious, and generally accepted, benefits, it has proved very difficult to reach agreement on where in the calculation process partial factors should be applied. In this paper, it is argued that four issues should determine the point in calculations at which factors are applied: the factors should directly cover the principal uncertainties involved in the calculation, whilst also adequately covering secondary uncertainties the factors should lead to designs in which both geometry and strength are adequate and compatible the factors should not be applied to quantities which may become negligible as a result of a balance between loads (actions) the factors must improve the safety of the design, which will generally mean changing the design from what it might otherwise have been. These concepts are illustrated by examples. 1 INTRODUCTION The application of partial factor design in geotechnics has been under debate for several decades. Although the approach has some obvious, and generally accepted, benefits, it has proved very difficult to reach agreement on where in the calculation process partial factors should be applied. Although some engineers would still argue for single global factors of safety, the main alternative views now relate to applying factors either to primary variables material properties and actions, or to some variable obtained part way through the calculation resistances and action effects. A related question is the application of factors to models, both resistance and action effect models. A partial resolution of these problems was proposed in the 1994 publication of Eurocode 7, ENV (eg BSI DD ENV1997 (1995)). This required that the designer considered three cases A, B and C, which were sets of partial factors for both actions and materials (or alternatively resistances in some cases). The factors of Case B were derived from normal structural design, and those of Case C from geotechnical design; Case A related to situations where loads were in balance with little involvement of material strength. The ENV required, in effect, that all designs should comply with all three cases in all respects, both geotechnical and structural; that is, complete designs of the geometry and structural strength were to be checked separately against all three sets of factors. It was noted that in many situations the critical case might be obvious by inspection, in which case it would not be necessary to produce formal calculations for all three cases. More recently, because there was no overall agreement on the system to be adopted, further cases have been added to be used as alternatives. The purposes of these have been: a) to reduce the number of calculations required b) to introduce factors on resistances and action effects rather than on material properties and actions c) to introduce model factors. Table 1 shows the values of partial factors proposed by ENV1997-1, with the action factors for Case A modified as required by more recent thinking. In Case A, permanent actions for a single source are split into their unfavourable or favourable components, whilst in Case B the same factor, 1.35 or 1.0, is applied to any permanent action from a single source. This scheme is identical with 1

2 that used in structural design. It is usually, but not invariably, found that Case C governs the geometry of the design of a geotechnical structure. The structural strength required is often governed by Case B, but sometimes by Case C. Case A governs rarely, but is none the less important. Table 1. Partial factors C ultimate limit states in persistent and transient situations (ENV1997-1) Actions Ground Properties Case Permanent Variable Unfavourable Favourable Unfavourable tan f c' c u q u 1) Case A Case B Case C ) Compressive strength of soil or rock. Table 2 shows a new nomenclature for three alternative Design Approaches, briefly summarised from the latest draft of EN Whilst the ENV required that all three cases be checked, the draft EN offers the three design approaches as alternatives. Table 2. Three design approaches in pren Design approach Brief description 1 (as ENV1997-1) Generally, apply all three cases A, B and C, checking both geometry and strength, geotechnical and structural. For piles and anchors, apply resistance factors in place of the materials factors. Where application of action factors is physically unreasonable, the factors may be applied to action effects (particularly applies to water pressure effects on retaining walls). 2 Check Case A for stability but not for strength. Apply Case B action factors to action effects in the structure. Simultaneously, apply factors to ground resistances such as passive pressure and bearing resistance. For slope stability problems, check Case C. 3 Check Case A for stability but not for strength. Apply Case B action factors either to actions or to action effects (to be determined nationally for particular situations), together with Case C materials factors (possibly slightly reduced). 2 PURPOSE OF THE FACTORS Before discussing where in the calculation process factors should be applied, it is first necessary to consider their overall purpose. It is unlikely that any system will perfectly satisfy the requirements, even if they are clearly understood, and there is a great danger that attempts to get a perfect system lead to complexity and confusion, which could very easily cause designers to make dangerous errors. Three main purposes for partial factors may be distinguished: a) to allow for uncertainty in material properties, actions or calculation models; b) to ensure that deformations are acceptable. Essentially this means using limit equilibrium calculations to achieve serviceability limit state design; c) to achieve compatibility with past practice which has been shown to be safe. The first two of these represent fundamentally different approaches and lead to much heated debate. However, in application they are often very similar, and when tempered by (c), it may be difficult to determine whether (a) or (b) actually determines the selected values of factors. Both must be achieved in an acceptable design, and it is often found that direct calculation of deformation or 2

3 working stresses, which would provide a more fundamental approach to serviceability design, is not sufficiently reliable. Item (b) tends to suggest that factors on the materials side should be applied directly to material strength, since it is the gradual mobilisation of strength that causes deformation. However, an alternative approach of applying factors to resistance can be developed in some cases. 3 WHERE TO APPLY FACTORS SOME BASIC POINTS Design calculations include many sources of uncertainty: actions (loads), derivation of action effects by an action effect model, material strengths, and resistances of structural sections or zones of ground, derived using a resistance model. The effect, later in a calculation, of an uncertainty at one point in a calculation is not easily foreseen and may be highly disproportionate. Hence, factors should be applied to the uncertainties themselves, where possible, so that their effects on derived quantities may result from the calculation, rather than being assumed by code-drafters. It is important, however, that the number of factors be kept as small as possible, in order to minimise the risk of confusion, which could cause mistakes in calculation. Most schemes have therefore chosen to factor only some of the uncertain parameters, with the intention of giving a sufficient margin to cover those not factored. In this case it is important that the principal uncertainties are directly factored, especially any which might have disproportionate, or non-linear effects elsewhere. In geotechnical design, material strength is often the principal uncertainty, suggesting that the strength of the ground should be factored at source. 3.1 Non-linear effects The Eurocode on Basis of Design, EN1990, notes an important point in relation to actions, which is relevant to item (a) above: For non-linear analysis (i.e. when the relationship between actions and their effects is not linear), the following simplified rules may be considered in the case of a single predominant action : a) When the action effect increases more than the action (as line A in Figure 1), the partial factor γ F should be applied to the representative value of the action. b) When the action effect increases less than the action (as line B in Figure 1), the partial factor γ F should be applied to the action effect of the representative value of the action. In other words, the factor should be applied to the basic variable, or to a quantity derived later in the calculation, according to where its effect will be most severe. If the same approach is applied to materials, particularly that of non-linear, frictional materials, it will generally require that factors are applied to material strength, rather than to calculated material resistances. For example, bearing capacity increases more, in proportion, than the angle of shearing resistance (or tan φ ) from which it is calculated, so it is appropriate in this case to apply the factor to the material property, tan φ, rather than to bearing resistance itself. Another example arises in the design of embedded retaining walls, where the magnitude of the maximum bending moment increases more than linearly with the applied earth pressures, which themselves change more than linearly with soil strength, expressed as tanφ. The EN1990 rule would here imply that factors should be applied directly to earth pressures, or better still to tanφ, rather than to the action effect, bending moment. 3.2 Geometric parameters and water pressure Often ground water pressure is also an important uncertainty, and it is difficult to make any sensible allowance for this except by modifying the design water pressure or water level itself. Similarly, 3

4 designs for embedded retaining walls are very sensitive to the ground level assumed in front of them. This may be uncertain, either because of imperfect workmanship or because of natural or human processes causing erosion in the long term. It is not possible to allow for these geometric uncertainties in a rational way be applying factors to a derived variable later in the calculation, so it is appropriate to make direct allowances in the design values of the geometric parameters, ground level or ground water level. Figure 1. Relationships between actions and action effects 3.3 Model factors In some situations, significant uncertainty may lie in the calculation models for either resistances or action effects. ENV did not contain detailed calculation models, so could not give values for partial factors on specific models. Instead, it required that all calculation models must either be conservative or they must be modified (eg factored) to ensure that they would either be accurate or err on the side of safety. In the specific cases of piles and anchors, a different approach was taken, as discussed below. In structural design, steel is generally a much more reliable material than concrete. However, steel is used in thin sections and small imperfections in manufacture may have big effects. It is therefore logical that in design of reinforced concrete partial factors are applied to the material strength of concrete, but in steel design the factors are applied to steel sections and connections, effectively a factor on the resistance model. This has led to different approaches in Eurocode 2, for concrete and Eurocode 3, for steel structures. 3.4 Factors must improve safety One consideration is obvious: the purpose of the partial factors is to improve safety, and for this they must lead to a modification in the design. In particular, there will be no purpose in applying factors to a derived quantity which, due to the balance of other variables, may turn out to be close to zero. Some examples which help to indicate where factors should be applied will now be considered. Most of these have been debated by people involved in Eurocode 7, but without overall agreement on the outcomes. 3.5 Finite element analysis The application of factors in finite element and other numerical analyses is discussed in this workshop by Bauduin, M. De Vos and B. Simpson (2000), in relation to ultimate limit state design. The main 4

5 conclusion of this study is that partial factors should in most cases be applied to soil strength rather than to resistances derived from the strength. Figure 2. The brick on table problem 4 SOME EXAMPLES 4.1 The brick on table problem Figure 2 shows a very simple conceptual problem which illustrates the significance of Case A, and also the fact that all features of a design must comply with a given set of partial factors. A brick is balanced on the edge of a table. The characteristic design of the brick assumes that it is uniform, each half of the brick having characteristic weight W k. So with no allowances for uncertainties it is balanced, without any external force applied at point A. However, any small deviation from this uniform state, or any small extra load on its right hand side, would make it unstable. Suppose it is decided to restrain the brick, allowing for uncertainties in loading, by a vertic al anchor at point A; this could be structural anchor bolt or a ground anchor. Then what should be the required design resistance, R d, for the anchor? Design approach 1, as in the ENV, would require that the Case A factors be applied to the two halves of the brick. Taking moments about point O, this gives a design anchor resistance of R d = (1.1W k 0.5a 0.9W k 0.5a)/a = 0.2W k In Design Approach 2, the stability is first checked using Case A, but the anchor is designed to Case B, and not to Case A. For Case B, the same factor is applied to all components of the permanent action from a single source, so: R d = (1.35 W k 0.5a 1.35 W k 0.5a)/a = 0 If, alternatively, the characteristic anchor force is first calculated, this would be R k = (W k 0.5a W k 0.5a)/a = 0 Hence R d = γ F R k = 0 It would clearly be incompatible to design the stability of the brick to Case A but the strength of the anchor to Case B. Case A would require an anchor for stability, but Case B would lead to a design 5

6 with no strength for the anchor. Although Case A is intended to deal with situations, like this one, in which forces are nearly in balance, it is clear that if it results in a requirement for structural or geotechnical strength, then this strength must also be checked for Case A. An alternative approach would be to find the anchor force to Case A, using ULS design values for the weights of the half-bricks, then treat this as a characteristic force for design of the anchor, applying Case B factors to the action effect, the anchor force. This would give a non-zero anchor force equal to W = 0.27W, which is 35% more severe than the 0.2W required by Approach 1. An objection to this approach is that it takes design action effects, then treats them as characteristic values for the next calculation, so increasing conservatism. 4.2 Tied retaining wall derived quantities near zero Figure 3a shows a situation in which a sheet pile retaining wall is required. Using unfactored soil strengths, it could provide equilibrium, as a cantilever, with a length of 11.6m. However, any system of factoring would indicate that it requires a longer length to work safely as a cantilever. For example, Design Approach 1 above would give a length of 14m for a cantilever, as shown in Figure 3b. (a) Situation for which a retaining wall is required. (b) Cantilever: Design Approach 1. (c) Tied retaining wall: Design Approach 1. (d) Tied retaining wall, characteristic state. Figure 3. Tied cantilever wall 6

7 Now suppose that, for reasons external to the design calculations, it is decided that the length of the wall will be 12m and further safety is to be provided by an anchor acting at 1m from the top of the sheet pile as shown in Figure 3a. This could occur because sheet piles of 12m length are readily available, or possibly because the sheet pile wall is already in place when the required depth of excavation in front of it becomes known. The design requirements now are to check that the wall is sufficiently stable with a tie at 1m depth, and to find the required design resistance for the tie. For economy, the designer wants to adopt the minimum allowable design tie force. Relevant calculations are summarised in Table 3. Calculations to Case C confirm that a length of 11.9m will be sufficient provided the tie has a design resistance of 75 kn/m, as shown in Figure 3c. The calculation for Case B is less severe, so Case C determines the design in Design Approach 1. Since a wall length of 11.6m was sufficient to give equilibrium as a cantilever using unfactored characteristic soil properties, the minimum tie force calculated using characteristic properties is 0.0, as shown in Figure 3d. Hence an approach which calculates this characteristic action effect, then applies a factor to it, will require a minimum design tie force of 0. As with the brick on table example, it is important to place the factors near the source of the uncertainty they represent; factors applied to the action effects, such as the tie force in this case, come too late in the calculation. An alternative approach could be considered in which both the ground effects and the structural action effects are factored. For this purpose, factors could be applied either to the ground strengths (tanφ ) or to the ground resistance available passive pressure. However, it has been found that systems of this type which are sufficiently safe in all cases tend to be uneconomic in situations such as the one described here. Table 3. Summary of calculations for tied retaining wall. Case C without anchor Case C with anchor Characteristic state γ kn/m φ k γ φ φ d δ/φ active δ/φ passive K ad K pd Design anchor force kn/m Length m Data B-2CC B-2CP B Design of piles importance of model uncertainty In some situations, the properties of materials are well understood, but the design resistance offered by a structural element or zone of ground is nevertheless uncertain. It was suggested above that steel sections offer one example of this, the strength of the section being more uncertain than that of the base material. In the geotechnical field, the design resistances of piles also lie in this category. For these, it was reckoned in drafting ENV that the major uncertainty was not the strength of the in situ ground but the way the construction would interact with it. The partial factor required is largely a factor on the resistance model, rather than on the strength of material. In such cases, it is appropriate to factor resistance rather than material strength. It was therefore considered that factors should be applied to the overall resistance given by a pile or anchor rather than the material strength of the ground; the resistance could, with advantage, be divided into the base and shaft components, considered separately. This approach was adopted in ENV1997 and is followed also in draft EN

8 4.4 Cantilever retaining wall compatibility of length and strength Figure 4 shows the characteristic situation for design of a cantilever retaining wall. Relevant calculations are shown in Table 4, all based on simple active and passive pressures for this cantilever situation. Figure 4. Characteristic situation for an embedded cantilever retaining wall. Table 4. Calculations for cantilever retaining wall Case CIRIA Fs=1.5 CIRIA F=1 EC7 Case C overdig Column γ φ =F s φ d δ/φ active δ/φ passive ½ ½ 2 3 ½ CIRIA F=1 overdig K a K p Overdig (m) Surcharge (kpa) Data CANTB6A CANTB3 CANTB5 Length (m) 15.2 (10.0) BM (knm/m) (822) (= ) BM factor ?? ULS BM (knm/m) (822) ?? Factor (Z el /Z pl ) ULS BM for Z el (822) Present British design is represented by CIRIA Report 104 (Padfield & Mair (1984), in which this is Example B2). This uses separate calculations of the geometry (length) and strength of the wall. 8

9 The length is first calculated using factors applied either to soil strength, as in Column 1 of Table 4, or to passive resistance. Then a separate calculation is used to determine the bending moment, with no factors on soil strength or resistance; a factor of 1.5 is applied to the derived bending moment to obtain a ULS design value (Column 2). The first two columns of Table 4 show that if the full length of the wall were ever needed, the bending moment generated (822 knm/m in column 1) is likely to far exceed the ULS bending moment for which the wall is designed (455 knm/m in column 2), but this feature is disregarded in the British design. Perhaps as a result of this, comparisons have shown that British designs require long walls, and for many retaining walls built in Britain the steel section is actually determined by driving capacity rather than calculated bending moment. Column 3 shows the design required by EC7 Approach 1, for which Case C is critical here. EC7 specifies that an allowance shall be made in the geometry of the problem, lowering the design level of the supporting soil by 0.5m. As noted above, uncertainty in this level has a major effect on the design, and the only way to make a rational allowance for it is to change the ground level used in the calculation. Nevertheless, the EC7 design requires a shorter wall than the CIRIA design, though a significantly bigger bending moment is calculated. The EC7 Approach 1 design has compatible length and strength in the wall; that is, its length and strength are both adequate for the most severe design situation to which it is subjected. This is not achieved by methods in which characteristic bending moments are multiplied by factors to obtain ULS values. The CIRIA Report 104 falls into this category, requiring insufficient strength to use the design length. The same problem exists, in the writer s opinion, with EC7 Design Approach 2 The bigger bending moment calculated for EC7 is largely a result of its requirement of lowering the passive ground level by 0.5m. Column 4 of Table 4 shows that if the same allowance had been applied in the CIRIA calculation, the bending moment would have increased by 70%. The latest draft of EC7 qualifies its requirement for lower design ground level on the passive side. It is sanguine to note, however, what a large effect on cantilever bending moment a comparatively small variation in ground level can have. Although it is beyond the scope of this paper, it is noted that the final choice of steel section for this sheet pile wall will depend on the requirements of the structural code to which it is designed. Simpson and Driscoll (1998) point out that Eurocode 3 Part 5 allows the full plastic moment resistance to be used for sheet piles, whereas previous codes have used the elastic resistance. Hence the calculation of a higher ultimate limit state bending moment to EC7 does not imply undue cost. The trends shown by this example are typical of other situations, as illustrated by further examples presented by Simpson and Driscoll. 5 CONCLUDING REMARKS This paper has argued that: partial factors should be applied as directly as possible to the principal uncertainties is in a design, rather than to quantities derived later in the calculation; partial factors should not be applied to quantities which may become near zero due to the balancing of various actions; the system adopted should lead to compatible length and strength in design of structural elements uncertainties in geometric parameters, such as ground levels and water levels, are best covered by direct adjustment of these parameters, rather than by safety factors applied elsewhere in the calculation. In most cases, this leads to the conclusion that partial factors should be applied to the strength of the ground and to actions, rather than to geotechnical resistances and action effects. Some exceptions to this have been noted. A similar conclusion is reported by Bauduin, M. De Vos and B. Simpson (2000), in relation to the use of finite element analysis for ultimate limit state design in geotechnics. 9

10 REFERENCES Bauduin, C, De Vos, M & Simpson, B (2000). Some Considerations on the Use of Finite Element Methods in Ultimate Limit State Design. LSD2000: Int. Workshop on Limit State Design in Geotechnical Engineering, ISSMGE, TC23, Melbourne. BSI DD ENV1997 (1995) Eurocode 7: Geotechnical Design (Part 1: General Rules; together with United Kingdom National Application Document) British Standards Institution Draft for Development. ENV : Padfield, C.J. & Mair, R.J. (1984) Design of retaining walls embedded in stiff clay. CIRIA Report 104. Simpson, B & Driscoll, R (1998) Eurocode 7 - a commentary. Construction Research Communications Ltd, Watford, UK. 10