Design of deep excavations with FEM - Influence of constitutive model and comparison of EC7 design approaches

Transcription

1 Reference: H.F. Schweiger Design of deep excavations with FEM - Influence of constitutive model and comparison of EC7 design approaches Proc. of the 21 Earth Retention Conference (Finno,R.J., Hashash, Y.M.A., Arduino, P., eds.) Bellevue, Washington, USA, August 21, ASCE,

2 Design of deep excavations with FEM - influence of constitutive model and comparison of EC7 design approaches H.F. Schweiger Computational Geotechnics Group, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology, Rechbauerstr. 12, 81 Graz, Austria; PH (+4331) ; helmut.schweiger@tugraz.at ABSTRACT Numerical analyses are performed on a routine basis in practical geotechnical engineering to assess the deformation behaviour of deep excavations under service load conditions, but it becomes increasingly common to use results from numerical analysis for ultimate limit state design (ULS). When doing so, compatibility of the design with relevant standards and codes of practice, valid in the respective country, has to be assured but there are no clear guidelines how this can be achieved. In this paper two aspects are addressed. First the influence of the constitutive model employed for modelling the mechanical behaviour of the soil on calculated structural forces of retaining walls is discussed and secondly the possibilities and limitations of introducing the partial factor concept as established in EC7 in combination with numerical analysis are highlighted. INTRODUCTION Numerical analyses are widely used in practical geotechnical engineering to assess the deformation behaviour of deep excavations, in particular when the influence on existing infrastructure such as buildings or adjacent tunnels has to be evaluated. In addition it becomes increasingly common to use results from numerical analysis as basis for the design. When doing so, compatibility of the design with relevant standards and codes of practice, valid in the respective country, has to be assured. In general this is a well established procedure when employing conventional design calculations based e.g. on limit equilibrium methods, but there are no clear guidelines how this can be achieved when numerical methods are used. Thus not much literature is available on this issue although some attempts have been made (e.g. Bauduin et al. (2), Schweiger (25, 29), Simpson (2, 27)). An additional difficulty arises, namely the appropriate choice of the constitutive model for the soil, which has a direct consequence for the design because different constitutive models will lead to different design forces. Both aspects are addressed in this paper by means of a benchmark example. Finally, results form the analysis of a real case history, where a diaphragm wall with prestressed ground anchors is used as a retaining system are briefly presented. It follows from these examples that the choice of the constitutive model and the design approach has an influence on the results, but given the uncertainties inherent in any analysis in geotechnical engineering the differences due to the different design approaches seem acceptable provided a suitable constitutive model is employed.

3 BENCHMARK EXAMPLE - INFLUENCE OF CONSTITUTIVE MODEL Problem definition and calculation steps The basic geometry of the investigated deep excavation is depicted in Figure 1. In order to study the effect of different constitutive models for various ground conditions two different (homogeneous) soil conditions are assumed, namely dense sand and a soft soil. For simplicity the wall (EA = 2.53E kn/m, EI = 3.2E4 knm 2 /m) and the strut (EA = 1.5E kn/m) have been assumed the same for both ground conditions, only the length of the wall and drainage conditions vary. Wall friction was taken as 2/3 of the friction angle of the soil. The soil parameters have been determined based on experimental results which can be considered to be representative for the respective soil. The following calculation steps have been performed, but only results for the final stage are presented in the following. Step : Initial stress state ( ' v =.h, ' h = K ' v, K = 1 - sin ') Step 1: Apply surcharge load (permanent load of 1 kpa) Step 2: Activate wall (wished-in-place), set displacements to zero Step 3: Excavation to level -2. m Step 4: Activate strut at level -1.5 m Step 5: Lowering of GW-Table to -. m inside excavation (only for dense sand) Step : Excavation to level -4. m Step 7: Excavation to level -. m Step 8: Apply variable load of 15 kpa (only for comparison of design approaches) Figure 1. Geometry of benchmark problem for sand and clay layer For the excavation in the sand layer a deep hydraulic barrier is assumed at the level of the base of the wall and thus no seepage flow is considered (see Figure 1). The analysis considering the soft soil layer has been performed under undrained conditions and it has been assumed that the water is excavated simultaneously with

4 the soil and no modifications to the groundwater conditions are made. The original GW-table is assumed to be at -3.5 m for the sand and at -2. m for the clay (Figure 1). The clay above the water table has been modelled as drained material. CONSTITUTIVE MODELS AND PARAMETERS Sand layer For the excavation in the sand layer three different constitutive models have been employed, namely the simple Mohr-Coulomb failure criterion (MC), the standard Plaxis Hardening Soil model (HS), which is a double hardening plasticity model, and the Hardening Soil Small model (HSS), which is the extension of the latter to account for small strain stiffness (Benz, 27). The parameters are listed in Table 1. Strength parameters are the same for all models but stiffness parameters are different. They are stress dependent in the HS and HSS model (values in Table 1 are reference values) but constant in the Mohr-Coulomb model. The average value of loading and unloading stiffness which follows from the HS model at the base of the retaining wall has been assigned as stiffness in the latter. Table 1. Parameters for dense sand for Hardening Soil Small Model (HSS) Parameter Meaning Value [kn/m³] Unit weight (unsaturated) 18 r [kn/m³] Unit weight (saturated) 2 [ ] Friction angle 41 c [kpa] Cohesion [ ] Angle of dilatancy 15 ur [-] Poisson s ratio unloading-reloading.2 ref E 5 [kpa] Secant modulus for primary triaxial loading 3 ref E oed ref E ur [kpa] Tangent modulus for oedometric loading 3 [kpa] Secant modulus for un- and reloading 9 m [-] Exponent of the Ohde/Janbu law.55 p ref [kpa] Reference stress for the stiffness parameters 1 nc K [-] Coefficient of earth pressure at rest (NC) 1-sin( ) R f [-] Failure ratio.9 Tension [kpa] Tensile strength G [kpa] Small-strain shear modulus 112 5,7 [-] Reference shear strain where G sec =.7G.2 Clay layer For the clay layer the Plaxis Soft Soil model (SS) has been used in addition to the Hardening Soil models and the Mohr-Coulomb model. The Soft Soil model is a modification of the well known Modified-Cam-Clay model incorporating a Mohr-

5 Coulomb failure criterion and allowing for a modification of the volumetric yield surface in order to improve K -predictions. The parameters for the HSS model and the SS model are listed in Tables 2 and 3. All models are implemented into the finite element code Plaxis (Brinkgreve et al. 2), which is used for all analyses presented in this paper. For the MC-model the same assumption with respect to the Young's modulus has been made as for the dense sand. Table 2. Parameters for soft clay for Hardening Soil Small Model (HSS) Parameter Meaning Value [kn/m³] Unit weight (unsaturated) 15 sat [kn/m³] Unit weight (saturated) 1 ' [ ] Friction angle (Mohr-Coulomb) 27 c [kpa] Cohesion (Mohr-Coulomb) 15 [ ] Angle of dilatancy ur [-] Poisson s ratio unloading-reloading.2 ref E 5 [kpa] Secant modulus for primary triaxial loading 4 3 ref E oed ref E ur [kpa] Tangent modulus for oedometric loading 1 8 [kpa] Secant modulus for un- and reloading 14 4 m [-] Exponent of the Ohde/Janbu law.9 p ref [kpa] Reference stress for the stiffness parameters 1 [-] Coefficient of earth pressure at rest (NC) 1-sin( ) R f [-] Failure ratio.9 t [kpa] Tensile strength G [kpa] Small-strain shear modulus 25.7 [-] Reference shear strain where G sec =.7G.3 K nc Table 3. Parameters for soft clay for Soft Soil Model (SS) Parameter Meaning Value [kn/m³] Unit weight (unsaturated) 15 r [kn/m³] Unit weight (saturated) 1 [ ] Friction angle 27 c [kpa] Cohesion 15 [ ] Angle of dilatancy ur [-] Poisson s ratio.2 * [-] Modified swelling index.125 * [-] Modified compression index.55 nc K [-] Coefficient of earth pressure at rest 1-sin( )

6 RESULTS Sand layer Figure 2 (left) shows the lateral displacement of the sheet pile wall for the final excavation stage. It is observed that the MC model predicts the smallest maximum displacement but of course this strongly depends on the chosen elasticity modulus. HS and HSS model show similar behaviour but including small strain stiffness effects reduces the maximum displacement slightly. It should be mentioned at this stage that the results from the HSS model may be quite sensitive on the choice of the parameter.7 (which is the shear strain at which the maximum small strain shear modulus is reduced to 7%) but reasonable values based on literature data have been chosen in this study. A similar trend is observed for bending moments (Figure 2, right). The notable difference between the simple and the advanced models become apparent when examining surface settlements behind the wall (Figure 3). The MC model shows unrealistic heave whereas the advanced models show the expected settlement, the maximum values being approx. 5% of the maximum horizontal displacement. Although this is not an issue from a design point of view it emphasizes the well known fact that simple elastic perfectly plastic models are not capable of representing the stress strain behaviour of soils correctly and therefore it remains questionable whether they should be used for design purposes. Strut forces obtained are -78 knm/m for the MC model and -12 and -17 knm/m for the HS and HSS model respectively. horizontal wall displacement [mm] bending moments [knm/m] HS HSS MC 1 HS HSS MC depth below surface [m] depth below surface [m] Figure 2. Comparison of wall deflection and bending moments - sand layer

7 surface displacement [mm] distance from wall [m] HS HSS MC Figure 3. Comparison of surface displacements - sand layer Clay layer Figure 4 depicts lateral wall displacements and bending moments for the wall, now 11 m long, in the soft soil. The difference between HS and HSS models are similar as in the previous case but again this depends to a large extent on the value chosen for.7. The SS model gives the smallest displacements and the MC model shows a different shape of wall deflection, namely an almost parallel movement of the bottom half of the wall, which is in contrast to the other models. This behaviour also leads to differences in the bending moments. horizontal wall displacement [mm] bending moments [knm/m] HS HSS MC SS 1 HS HSS MC SS depth below surface [m] depth below surface [m] Figure 4. Comparison of wall deflection and bending moments - clay layer

8 For the settlement trough behind the wall (Figure 5) the same can be observed as in the previous section, namely that the MC model produces significant heave adjacent to the wall and in this case due to undrained conditions settlements in the far field (the lateral model boundary for this analysis was placed at a distance of 75 m from the wall). The calculated settlement troughs can be generally considered as too wide with the exception of the HSS model which is a consequence of taking into account small strain stiffness effects. surface displacement [mm] distance from wall [m] HS 4 HSS MC 3 SS Figure 5. Comparison of surface displacements - clay layer BENCHMARK EXAMPLE - INFLUENCE OF EC7 DESIGN APPROACH EC7 design approaches The same examples as discussed in the previous section is used to demonstrate the applicability of using the finite element method for ULS-design in accordance with Eurocode7. In Eurocode7 the partial factor of safety concept is introduced replacing the global factor of safety concept employed until now. Three different design approaches DA1 to DA3 have been specified which differ in the application of the partial factors of safety on actions, soil properties and resistances. They are given in Tables 4 and 5 for all three approaches. It is noted that 2 separate analyses are required for design approach 1. The problem which arises for numerical analyses is immediately apparent because DA1/1 and DA2 require permanent unfavourable actions to be factored by a partial factor of safety, e.g. the earth pressure acting on retaining structures. This is of course not possible because in numerical analyses the earth pressure is not an input but a result of the analysis. However, EC7 allows for the alternative of putting the partial factor on the effect of the action instead on the actions itself, e.g. bending moments or strut forces. This is commonly referred to as DA2*. In this way finite elements can be used because the analysis is performed with characteristic loads and characteristic parameters introducing the relevant partial factor at the end of the analysis. It is beyond the scope of this contribution to elaborate on the advantages and disadvantages of each of the approaches in detail but some discussion can be found e.g. in Simpson (2, 27), Bauduin et al. (23)

9 and Schweiger (25). However the differences in results with special emphasis on the constitutive model will be shown. Table 4. EC7 partial factors for actions design permanent approach unfavourable variable DA1/ DA1/ DA DA3-Geot Table 5. EC7 partial factors for soil strength properties and resistances design undrained passive tan ' c' approach shear strength resistance DA1/ DA1/ DA DA3-Geot The calculation steps are the same as in the previous section but an additional variable load of 15 kpa extending to a width of 5 m is added as a final calculation step in order to have the influence of a variable load taken into account (Figure 1). Again a sand and a clay layer are considered, but only two constitutive models, the HSS-model and the MC-model are compared. The (characteristic) parameters are the same as listed in Tables 1 and 2. For the clay layer an additional aspect is addressed, namely the consequences of performing the undrained analysis in terms of effective strength parameters ' and c, or in terms of the undrained shear strength c u because this does not only involve a difference in the method of analysis but different partial factors apply to effective strength parameters and the undrained shear strength respectively (Table 5), namely 1.25 versus 1.4. For the analysis in terms of undrained shear strength c u, the following assumptions have been made. The distribution of c u with depth has been worked out based on the Mohr-Coulomb criterion and this distribution has been also used for the HSS-model. It is noted at this stage that by doing so, some of the advanced features of the HSS-model are lost. It should also be mentioned that in the analysis in terms of effective strength parameters the undrained shear strength obtained from the HSS-model depends on a number of input parameters (not only strength) and is therefore different to the undrained shear strength in the MC-model. This approach yields the following distribution of characteristic undrained shear strength as used in DA2: c u at depth -2. m below surface: 23.9 kpa with an increase of 2.1 kpa/m For DA3 the strength parameters have to be reduced by the partial factors listed in Table 5 resulting in values for the effective friction angle, the effective cohesion and

10 the undrained shear strength as given in Table. The dilatancy angle is also reduced by the partial factor which is however not explicitly mentioned in EC7. Finally a decision with respect to initial stresses has to be made. Here the value for K has been kept the same for DA2 and DA3, i.e. it is based on the characteristic value for the friction angle (1 - sin ' char ) although an alternative would be to have it based on the design value in DA3. (For certain conditions K based on ' char may however violate the yield function). Table. Strength parameters used in DA3 (partial factors applied) Parameter Meaning Value sand [ ] Friction angle 34.8 c sand [kpa] Effective cohesion sand [ ] Angle of dilatancy 12 clay [ ] Friction angle 22.2 c clay [kpa] Effective cohesion 12 clay [ ] Angle of dilatancy c u, clay [kpa] Undrained shear strength at 2. m depth 17.1 c u, clay [kpa/m] Increase of undrained shear strength 1.5 RESULTS In this section the differences in design strut forces and bending moments obtained from utilizing design approaches DA2 and DA3 (DA1 is basically a combination of the two) are presented. Sand layer Figure shows a comparison of design bending moments (envelope over all construction stages) obtained for the two constitutive models for DA2 and DA3. The design moments for DA2 are obtained by the following procedure: characteristic bending moments are calculated without (M 1 ) and with (M 2 ) the variable load applied and from these the design bending moments are calculated by applying the appropriate partial factors. The same procedure is used for calculating design strut forces. It should be noted that this is an approximation only, due to the nonlinear behaviour of the soil. M design, DA2 = M 1 x (M 2 M 1 ) x 1.5 In DA3 results from the analysis are directly design values because the partial factors on soil strength and the variable load (15 kpa > 19.5 kpa) are taken into account in the input of the analysis. It follows from Figure that differences coming from the different design approaches are more pronounced for the MC-model than for the more advanced HSS-model. DA3 leads to significantly higher design moments for the MC-model

11 whereas for the HSS-model very similar values are obtained. The same holds for strut forces where for the HSS-model actually a slightly lower design force is obtained from DA3 (Table 7). The reason for this behaviour is that a reduction in strength has a different effect in a linear elastic-perfectly plastic model than in an advanced hardening plasticity model due to the different stress paths followed. bending moments [knm/m] HSS-DA3 MC-DA3 HSS-DA2 MC-DA depth below surface [m] 7 8 Figure. Comparison of design bending moments - sand layer 9 Table 7. Comparison of design strut forces - sand layer DA2 Strut force after Strut force Design strut excavation due to load force MC HSS DA3 Strut force after Strut force Design strut excavation due to load force MC HSS

12 Clay layer Figure 7 shows design bending moments for both design approaches and both constitutive models for analyses in terms of effective strength parameters (denoted "A" in Figure 7) and undrained shear strength parameters (denoted "B"). The following can be observed: for the HSS-model again DA2 and DA3 yield similar results (DA2 slightly higher in this case) for analyses with effective strength parameters. For the MC-model the difference is higher and in contrary to the excavation in dense sand DA2 design bending moments are higher than the ones obtained from DA3. For analyses in terms of undrained strength parameters (c u ) it is different because the partial factor on undrained strength is higher than for drained strength parameters. Thus DA3 results in higher bending moments than DA2 for both models. For the MC-analysis with characteristic parameters (DA2) the differences in method A and B are negligible, which can be expected. For the HSS-model this is not the case because the analysis in terms of effective parameters will lead to a different undrained strength as the one specified in method B. Strut forces are summarized in Table 8. design bending moments [knm/m] depth below surface [m] HSS_DA2-A MC_DA2-A HSS_DA2-B MC_DA2-B HSS_DA3-A MC_DA3-A HSS_DA3-B MC_DA3-B Figure 7. Comparison of design bending moments - clay layer

13 Table 8. Comparison of design strut forces - clay layer DA2 strut force after strut force design excavation due to load strut force MC HSS MC_B HSS_B DA3 strut force after strut force design excavation due to load strut force MC HSS MC_B HSS_B PRACTICAL EXAMPLE The benchmark examples presented in the sections above indicate that both design approaches DA2 and DA3 and consequently also DA1 can be applied in combination with the finite element method. It also follows from these examples that the differences in results due to the choice of the constitutive model are at least in the same order (or larger) than differences coming from the design approaches. However, for real practical problems details of the design may have more severe consequences for the choice of the design approach as compared to the simplified examples presented above. This will be illustrated as an example by considering a diaphragm wall with three rows of prestressed anchors. The excavation is about 17 m deep in a reasonably homogeneous layer of medium dense sand (Figure 8). The details of the analysis will not be discussed here because the goal of this section is only to highlight a particular aspect, namely the resulting design anchor forces. As described in the previous section, analyses were performed with characteristic soil strength parameters (DA2) and with design strength parameters (DA3). The permanent action is the earth pressure, there are no variable loads. Again DA2 is used in form of DA2*, i.e. the partial factor is applied to effects of actions rather than on the action itself. The resulting design anchor forces obtained from the two approaches are summarized in Table 9 and it can be seen that DA3 leads to significantly lower forces. The reason for this difference is the following: if anchors are highly prestressed, as it is the case here, a reduction in soil strength does not change calculated anchor forces significantly as compared to the analysis with characteristic soil strength. Thus in DA2 the result is multiplied by the partial factor for actions (= 1.35) whereas in DA3 the calculated forces are already design forces. It should be pointed out that in DA2 the effects of the water pressure are fully factored whereas they are not in DA3. It is acknowledged that, strictly speaking, an

14 uncertainty in the water table should be considered in DA3 as a "geometric" factor, but this would not bring forces near the values obtained for DA2. Figure 8. Layout of practical example Table 9. Comparison of design anchor forces - practical example anchor force layer 1 (kn/m) anchor force layer 2 (kn/m) anchor force layer 3 (kn/m) characteristic DA2* (= char. x 1.35) DA CONCLUSION In the first part of this contribution the influence of the constitutive model on the results of finite element analyses of deep excavations has been demonstrated. The results clearly emphasize the well known fact that elastic-perfectly plastic constitutive models such as the Mohr-Coulomb model are not well suited for analysing this type of problems and more advanced models are required to obtain realistic results. Although reasonable lateral wall movements may be produced with simple failure criteria with appropriate choice of parameters, vertical movements behind the wall are in general not well predicted, obtaining heave in many cases instead of settlements. Strain hardening plasticity models including small strain stiffness behaviour are in general a better choice and produce settlement troughs being more in agreement with expected behaviour. As the goal of the study presented here was to qualitatively highlight the differences in results with respect to the constitutive model no quantitative comparison with in situ measurements has been provided. The second part of the paper addressed the ULS-design of deep excavations by means of numerical methods. It has been shown that the concept of partial factors of

15 safety as established in Eurocode7 can be applied, but differences have to be expected depending on how this is done in the respective design approaches. Although more experience is needed in performing such analyses for practical examples, it emerges from this study that the differences in results depending on the design approach used are less pronounced for more advanced constitutive models as compared to simple elastic-perfectly plastic failure criteria. On first sight this seems to be in contradiction to practical experience but can be explained by different stress paths soil elements follow when different models are used. Postulating that advanced models are more reliable in describing the stress strain behaviour of soils for stress levels ranging from working load conditions up to failure it could be argued that advanced models have advantages not only for predicting displacements and stresses for working load conditions but have their merits also in ULS-design. Finally it should be mentioned that the models used in this study should be seen as representatives for certain classes of models and conclusions can be transferred to other constitutive models of similar type. REFERENCES Bauduin, C., De Vos, M. and Simpson, B. (2). "Some considerations on the use of finite element methods in ultimate limit state design." Proc. Int. Workshop on Limit State Design in Geotechnical Engineering, Melbourne. Bauduin, C., De Vos, M. and Frank, R. (23). "ULS and SLS design of embedded walls according to Eurocode 7." Proc. XIII ECSMGE, Prague (Czech Republic), Vol. 2, Benz, T. (27). "Small-Strain Stiffness of Soils and its Numerical Consequences." Publication No. 55, Institute for Geotechnical Engineering, University of Stuttgart. Brinkgreve, R.B.J., Broere, W. and Waterman, D. (2). "Plaxis, Finite element code for soil and rock analyses, users manual." The Netherlands. Schweiger, H.F. (25). "Application of FEM to ULS design (Eurocodes) in surface and near surface geotechnical structures." Proc. 11th Int. Conference of IACMAG, Turin, Italy, June 25. Bologna: Patron Editore Schweiger, H.F. (29). "Influence of constitutive model and EC7 design approach in FEM analysis of deep excavations." Proc. ISSMGE Int. Seminar on Deep Excavations and Retaining Structures (Mahler & Nagy, eds.), Budapest, Simpson, B. (2). "Partial factors: where to apply them?" Proc. Int. Workshop on Limit State Design in Geotechnical Engineering, Melbourne, Simpson, B. (27). "Approaches to ULS design The merits of Design Approach 1 in Eurocode 7." First International Symposium on Geotechnical Safety & Risk, Oct , 27, Shanghai, Tongji University, China.