Application of FEM to ULS design (Eurocodes) in surface and near surface geotechnical structures

H.F. Schweiger Computational Geotechnics Group, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology, Austria

Published in: Proc. 11th Int. Conf. Computer Methods and Advances in Geomechanics (G. Barla, M. Barla eds.), Patron Editore, Bologna, 2005, Vol.4, 419-430

Keywords: numerical methods, ultimate limit state, Eurocode

ABSTRACT: The finite element method is generally accepted as a tool for assessing the serviceability limit state for geotechnical structures whereas the factor of safety at the ultimate limit state is more commonly determined by conventional limit equilibrium methods. In this paper some examples for application of the finite element method to determine the factor of safety are presented. It is however not always clear whether these results comply with respective standards and codes and therefore the design approaches defined in Eurocode7 are discussed with respect to their compatibility with numerical methods.

1 Introduction Finite element analyses have been widely accepted as a powerful tool to solve boundary value problems in geotechnical engineering. The serviceability limit state (SLS), i.e. prediction of deformations, stresses and internal forces under working load conditions, for complex soil-structure interaction problems can be reliably assessed only by numerical modelling. Significant experience has been gathered in using numerical models and due to a number of user-friendly codes being available this is not only true for the research environment but also in practice. However, once it comes to the determination of factors of safety (FOS) and ultimate limit state design (ULS) the question arises to what extent numerical models can be applied in a standard way and whether results from numerical analyses comply with the definitions of the factor of safety as used in standards and codes. In Eurocode7 (EC7) for example various design approaches have been identified. They differ in the way the partial factors of safety are applied to soil strength, resistance and different types of loads (actions). Although it was the original aim of EC7 to ensure an unified design approach throughout Europe it is unfortunately left to the national Annex which of the suggested approaches will be relevant for a particular country. In addition it seems that, although numerical methods are mentioned in EC7 as a possible design tool, not all of the design approaches can be applied in a straightforward way within the framework of numerical modelling. So far the problem is not extensively discussed in the literature. Exceptions are e.g. Bauduin et al. (2000), Simpson (2000) and Bauduin et al. (2003). In this paper it is first shown that numerical methods are indeed capable of evaluating the factor of safety for various types of problems. Following a definition of the FOS the topic is introduced by presenting results from a simple slope stability problem where FOS values from numerical analyses

compare well with calculations based on limit equilibrium methods. Three-dimensional analyses for investigating tunnel face stability emphasise the advantages of numerical tools because no simplified failure mechanisms have to be postulated as in limit equilibrium analysis. Unreinforced and reinforced tunnel faces are considered. Finally a deep excavation problem is discussed in more detail. The implications of modelling structural elements as elastic or elastic-perfectly plastic material are highlighted. A brief summary of the design approaches as defined in EC7 is given and problems arising when using numerical methods to comply with EC7 are addressed. However, conclusions are somewhat preliminary because the emphasis here is on deep excavation problems with one level of struts and more analyses for different types of problems are required to arrive at definite recommendations. All analyses presented in this paper have been performed with the finite element code Plaxis (Brinkgreve & Vermeer, 2001; Brinkgreve, 2002).

2 Definition of FOS using FEM The safety factor resulting from a finite element analysis assuming a Mohr-Coulomb failure criterion can be obtained by reducing the strength parameters incrementally, starting from unfactored values ϕavail and cavail, until no equilibrium can be found in the calculations. The corresponding strength parameters can be denoted as ϕfailure and cfailure and the safety factor ηfe is defined as

failure

avail

failure

availfe c

ctanφtanφη == (1).

There are basically 2 possibilities to arrive at the FOS as defined in Equ. (1). Method 1: An analysis is performed with unfactored parameters modelling all construction stages required. The results represent the behaviour for working load conditions at the defined construction steps. This analysis is followed by an automatic reduction of strength parameters of the soil until equilibrium can be no longer achieved in the calculation. The procedure can be invoked at any construction step. This approach is commonly referred to as ϕ/c-reduction technique. Method 2: The analysis is performed with factored parameters from the outset, i.e. strength values are reduced, again in increments, but a new analysis for all construction stages is performed for each set of parameters. If sufficiently small increments are used the factor of safety is again obtained from the calculation where equilibrium could not be achieved. It is worth noting that in this approach the calculation for the serviceability limit state has to be performed in an additional analysis using unfactored design parameters. Equ. (1) implies that the same reduction factor is imposed on friction angle and cohesion and although this is not necessarily required it is assumed to be valid for the sake of simplicity in all analyses presented in this paper. In addition it corresponds to Eurocode7.

3 Introductory example – slope stability analysis As an introduction the determination of the factor of safety for a slope is considered. Calculations of this type have been presented in the literature and are repeated here briefly for the sake of completeness (e.g. Griffiths, 1980; Brinkgreve & Bakker, 1991; Dawson et al., 2000). The example is taken from Griffiths & Lane (1999) and is shown in Figure 1. Depending on the ratio of cu1 and cu2 failure occurs in the slope or in the underlying layer. Of course a finite element analysis will produce the appropriate failure mechanism (Figures 2 and 3) and it can be easily verified that obtained values of FOS compare well with limit equilibrium analysis.

20,00 20,00 20,00

10,0

010

,00

60,00

10,0

0

21cu1

cu2

cu1

cu2

Figure 1. Example for slope stability problem.

Figure 2. Predicted failure mechanism through base layer (cu2 / cu1 = 1.0), ηfe = 1.43.

Figure 3. Predicted failure mechanism through slope (cu2 / cu1 = 2.0), ηfe = 1.94.

4 Tunnel face stability Another area where numerical methods become increasingly important is the assessment of tunnel face stability, both for tunnels constructed with an open face as well as for shield tunnels. Whereas in the latter case the goal of the analysis is to calculate the support pressure at the face to ensure the desired factor of safety, in the former case emphasis is on the decision whether reinforcement at the face is required. In particular for large openings and/or very unfavourable ground conditions this is of significant concern because excavation in small cross sections is disadvantageous from a construction point of view and thus excavation of larger cross section with support from face reinforcement may provide an economical solution. Although a number of analytical solutions have been presented in the literature (a good summary can be found e.g. in Mayer et al., 2003) numerical methods again prove to have advantages because no a priori assumption on the failure mechanisms have to be made and therefore project specific conditions such as soil stratigraphy and support measures are easily included in the analysis. A comprehensive investigation of tunnel face stability problems by means of the finite element method has been presented by Ruse (2004) and summarized e.g. by Vermeer et al. (2002). Simple formulae have been developed based on a large number of 3D FE-calculations which allow for a quick estimate of the required support pressure in order to secure a stable tunnel face. However, these analyses did not explicitly model face reinforcement e.g. by means of rock bolts or glasfibre rods. Provided the structural elements are modelled as elastic perfectly plastic materials the factor of safety of a reinforced tunnel face can be readily determined by the ϕ/c-reduction technique and various aspects can be studied, such as the influence of the length of the face reinforcement. An example of such an analysis is shown in Figure 4 (Schweiger & Mayer, 2004). The short reinforcement is not able to provide the required support and therefore the FOS is just larger than 1.0 whereas the long reinforcement prevents the failure mechanism reaching the surface and thus increases the FOS.

a) b)

Figure 4. Contour lines of total displacements ϕ/c-reduction for different length of face reinforcement; a) L = 4 m, ηfe = 1.01; b) L = 12 m; ηfe = 1.19.

5 Design approaches in Eurocode7 Eurocode7 allows for three different design approaches DA1 to DA3 which differ in the application of the partial factors of safety on actions, soil properties and resistances. EC7 states: "It is to be verified that a limit state of rupture or excessive deformation will not occur with the sets of partial factors" as given in Tables 1 and 2 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 also 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 however not readily taken into account in numerical analyses because the earth pressure is not an input but a result of the analysis.

Table 1. Partial factors for actions according to EC7.

Actions γF Permanent

unfavourable 1)

Variable 2)

Design

approach

γG γQ DA1/1 1.35 1.50 DA1/2 1.00 1.30 DA2 1.35 1.50

DA3 Geot.3): 1.00 Struct.4):1.35

1.30 1.50

Table 2. Partial factors for soil properties and resistances according to EC7.

Soil properties γM Resistances

tanϕ’ c’ cu Unit weight Passive Anchor

Design

approach

γϕ γc γcu γF γR;e γa DA1/1 1.00 1.00 1.00 1.00 1.00 1.10 DA1/2 1.25 1.25 1.40 1.00 1.00 1.10 DA2 1.00 1.00 1.00 1.00 1.40 1.10 DA3 1.25 1.25 1.40 1.00 1.00 1.00

1) Favourable permanent action: γG = 1.00 2) When unfavourable; favourable action should not be considered 3) Geotechnical action: action by the ground on the wall 4) Structural action: action from a supported structure applied directly to the wall

6 Example: Single strutted excavation This simple excavation problem serves as an example to demonstrate that the ϕ/c-reduction technique can be applied to determine the factor of safety of deep excavation problems. However, as will be shown the development of internal forces in structural elements has to be observed in order to obtain realistic FOS-values. The example is also used to demonstrate possibilities and limitations of numerical methods in the design process of deep excavations with respect to Eurocode7.

6.1 Geometry and parameters Figure 5 depicts the simple example of a sheet pile wall, embedded 4.0 m into a homogeneous soil layer. Excavation depth is 8.0 m below surface and a single strut is placed at a depth of 1.0 m below ground level. The finite element mesh consists of approximately 850 15-noded elements (Figure 6). The so-called Hardening Soil model, an elastic-plastic constitutive model including deviatoric and volumetric hardening, as implemented in the code Plaxis, has been used for the analysis. The main feature of this model is a stress dependent stiffness and a distinction in stiffness between primary loading and unloading/reloading. The failure strength is described by a Mohr-Coulomb failure criterion and it is pointed out that for the determination of the factor of safety the Hardening Soil model is actually reduced to a Mohr-Coulomb failure criterion. However, all soil parameters used are given for completeness in Table 3. Structural elements have been assumed to behave as linear elastic-perfectly plastic materials (Table 4).

1.0 m 1.5 m

6.5 m

4.0 m

sheet piletyp AZ-18

Figure 5. Geometry of example for single strutted excavation.

Table 3. Parameters for Hardening Soil model. Parameter Symbol Unit Value

Unit weight γ kN/m3 17.0 Deviatoric stiffness E50

ref kPa 45 000 Compression stiffness Eoed

ref kPa 45 000 Unloading stiffness Eur

ref kPa 135 000 Cohesion c kPa 0.1

Friction angle ϕ ° 35 Dilatancy angle ψ ° 0

Poisson ratio for unloading ν - 0.2 Reference pressure pref kPa 100

Power for stress dependent stiffness m - 0.5

Coefficient of lateral earth pressure for normal consolidation K0

nc - 0.43 Interface strength reduction Rinter - 0.7

Table 4. Parameters for structural elements.

Parameters for sheet pile wall Symbol Unit Value

Normal stiffness EA kN/m 3.008E6 Bending stiffness EI kNm2/m 6.84E4 Moment capacity Mpl kNm/m 505

Parameters for strut

Elastic modulus E kPa 3.0E7 Cross section area A m2 0.24

Horizontal strut distance d m 1.0

Figure 6. Finite element mesh for example for single strutted excavation.

6.2 Results The following excavation sequence is modelled in the analysis:

1. Initial stress state (σv = γh, σh = K0σv, K0 = 1- sinϕ) 2. Sheet pile wall wished-in-place and excavation to 1.5 m below surface 3. Installation of strut and excavation to final level of 8.0 m below surface 4. Reduction of strength parameters until failure occurs (ϕ/c-reduction technique)

Results are shown after the final calculation step, i.e. after ϕ/c-reduction. Two different analyses were performed. In the first one the wall was assumed to behave as linear elastic material whereas in the second analysis the capacity of the wall in bending was limited to 505 kNm/m. However a partial factor for the wall capacity of γwall = 1.5 was introduced and thus the moment capacity in the analysis was limited to 337 kNm/m. It should be noted that the wall was modelled as elastic-perfectly plastic material, i.e. the moment capacity was retained at this value. Figure 7 shows the deformed mesh and the contours of incremental plastic shear strains after ϕ/c-reduction, i.e. at failure, for the case of an elastic wall. The factor of safety obtained is ηfe = 2.02.

Figure 7. Deformed mesh and contours of incremental plastic shear strains after ϕ/c-reduction for elastic wall.

Figure 8. Deformed mesh and contours of incremental plastic shear strains after ϕ/c-reduction for elastic-perfectly plastic wall.

Figure 8 shows the same for the elastic-perfectly plastic wall. The calculated factor of safety is ηfe = 1.75 and considerable less than in case of an elastic wall. Figure 8 indicates the development of a plastic hinge in the wall and it is clearly observed that the distinct failure surface as shown in Figure 8 is now replaced by a more diffuse failure mechanism due to the limited bearing capacity of the wall. This simple example highlights two important aspects: firstly the potential of numerical methods in taking into account soil-structure interaction when investigating failure mechanisms and secondly that one could be misled by a high factor of safety when structural elements are treated as elastic material, an assumption which is often acceptable for investigating working load conditions.

6.3 Influence of partial factor of safety on internal forces

In this chapter the influence of the partial factor of safety (γsoil) on soil strength on calculated strut forces (F) and bending moments (M) is investigated by performing analyses according to method 2 as defined in chapter 2. Table 5 summarizes the results by comparing the applied partial factor of

safety on the soil with the resulting factor of safety on the structural elements, which is defined as the force or bending moment obtained from the analysis with the respective γsoil divided by the force or bending moment from an analysis with γsoil = 1.0. As expected this relationship is nonlinear and in this case the increase in structural forces is higher than the corresponding partial factor on the soil, but for other examples the opposite has been observed (Schweiger, 2003). These analyses are the basis for the next chapter where compatibility of numerical methods and Eurocode7 is addressed.

Table 5: Influence of partial factor of safety on soil (γsoil) on strut forces and bending moments

F Fγsoil / Fγsoil=1.0 Mmax

Mγsoil / Mγsoil=1.0 γsoil

[kN/m] [-] [kNm/m] [-]

1.00 -90 1.00 -105 1.00

1.25 -117 1.30 -149 1.42

1.50 -147 1.63 -220 2.10

1.90 -199 2.21 -393 3.74

7 FEM and design approaches in Eurocode7 for single strutted excavation Looking at the design approaches summarized in Tables 1 and 2 it is immediately apparent that for the given problem DA1/1 and DA2 require partial factors of safety on actions and resistances which are results and not input from a numerical analysis, namely the active earth pressure for both and in addition the passive earth pressure for DA2. DA3 (geotechnical action) and DA1/1 are in principle no problem for numerical methods because it simply implies the input of factored strength parameters. For staged construction problems again methods 1 and 2 (see chapter 2) may be applied, i.e. the complete analysis is performed with factored strength parameters (method 2) or the analysis is performed in terms of characteristic values and the ϕ/c-reduction technique (method 1) is performed at each relevant construction stage (see also Bauduin et al., 2003). One way of dealing with DA2 could be that the analysis is performed in terms of unfactored strength parameters for the soil and the resulting bending moments, anchor forces and the passive resistance is factored by the respective partial factor of safety in order to arrive at design values (this approach has been denoted by DA2* by Bauduin et al., 2003). However, it follows implicitly from the previous section that due to nonlinear soil behaviour this is not quite what DA2 is intended to be, although differences are probably not very significant for internal forces in wall and strut but may be significant in terms of passive resistance. The example of the previous section will be used to illustrate the differences in approaches. Two sets of analyses have been performed:

1. Example as defined in section 6 a. with reduced strength parameters (DA3, method 2) b. with ϕ/c-reduction technique for final excavation stage (DA3, method 1) c. unfactored strength parameters (DA2*)

2. Example as defined in section 6 but with a surcharge load of 25 kPa behind the wall a. with reduced strength parameters and partial factor for surcharge, γsurcharge = 1.3

(DA3, method 2) b. with ϕ-c-reduction technique for final excavation stage, γsurcharge = 1.3 (DA3,

method 1) c. unfactored strength parameters, γsurcharge = 1.5/1.3 (DA2*)

The results are summarized in Tables 6 and 7 and it follows that methods 1 and 2, applicable to DA3 (and to DA1/1), lead to very similar results and also DA2* results in very similar design bending moments and strut forces (the design forces are obtained by multiplying internal forces obtained from analysis with unfactored values by the partial factor γG = 1.35). Only in the case with the surcharge load some discrepancies are observed with respect to the maximum bending moment. However, it remains to be further investigated whether this is a coincidence in this particular example and whether it holds to the same extent for multipropped walls (Schweiger, 2003). It follows also from these results that the design values for active and passive earth pressures based on DA2*, i.e. imposing the partial factor on the results of the analysis with unfactored parameters, do not correspond to the results from analysis DA3. The changes in earth pressure distributions for both examples are depicted in Figures 9 and 10 and it can be seen that due to the reduction in soil strength the passive pressure increases slightly in the analysis due to a higher mobilization caused by increasing wall deflection.

Table 6: Comparison of bending moments, strut forces and earth pressures for different approaches for case without surcharge

F M Ea, res Ep, res DA

[kN/m] [kNm/m] [kN/m] [kN/m]

DA3 Method 1 -114 -144 -435 318

DA3 Method 2 -117 -149 -450 322

DA2* -90 > 122 -105 > 142 -410 308

Table 7: Comparison of bending moments, strut forces and earth pressures for different approaches for case with surcharge

F M Ea, res Ep, res DA

[kN/m] [kNm/m] [kN/m] [kN/m]

DA3 Method 1 -225 -180 -601 369

DA3 Method 2 -227 -181 -602 370

DA2* -166 > 224 -120 > 162 -518 342

Effective normal stresses [kN/m2]

050100150200 50 100 150 200

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

DA3, method 2DA3, method 1DA2*

Figure 9. Calculated earth pressures for case without surcharge.

Effective normal stresses [kN/m2]

050100150200 50 100 150 200

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

DA3, method 2DA3, method 1DA2*

Figure 10. Calculated earth pressures for case with surcharge.

8 Conclusions Some examples from two and three-dimensional numerical analyses for ultimate limit state design in geotechnical engineering have been presented. For simple slope stability problems numerical methods yield essentially the same results as limit equilibrium analyses but for more complex problems numerical methods have the advantage that the failure mechanism has not to be assumed a priori and that soil-structure interaction can be taken into account. Application of the finite element method to ULS design of a single strutted retaining wall investigating the different design approaches proposed in Eurocode7 has been discussed. It could be shown that results are reasonably consistent with respect to design bending moments and strut forces, at least for this particular example. Design values for passive resistance are however more difficult to assess when different approaches are compared. However, further work is required in order to evaluate possibilities and limitations of the various approaches for more complex multipropped retaining walls.

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