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Although time-domain analysis using acceleration (velocity or displacement) time histories generally provides the most accurate tool for the simulation of the response of geotechnical structures under dynamic loading, this type of analysis is typically very time-consuming. Moreover, the quality of the predictions may depend on the ability of the constitutive models to reproduce complex dynamic phenomena, which, in turn, may depend on the availability of a significant amount of experimental data for its proper calibration. Due to its simplicity and relative inexpensiveness, finite element pseudo-static analysis may provide a reliable tool, at least for the preliminary assessment of the stability of geotechnical structures (such as embankments, retaining walls and tunnels) against earthquake-induced failure.
When performing a force-based pseudo-static analysis, the effects of a dynamic loading (such as earthquake-induced loading) are represented by equivalent inertial forces, which are approximated as constants body forces, with magnitude proportional to the horizontal and/or vertical accelerations imposed by the dynamic loading:
Fx = kx W
Fy = ky W
Fz = kz W
where Fx, Fy and Fz are the components of the body force along x-, y- and z-direction, kx, ky and kz are the corresponding pseudo-static acceleration coefficients, which are input parameters, and W is the weight of the mass. Note that these forces are applied to the whole mesh.
As schematically illustrated in Figure 1 for a 2D plane strain model, when a horizontal pseudo-static force is applied to the model, the soil will tend to move in the opposite direction of the applied horizontal pseudo-static force as if the model would be rotated. It is important to highlight that, in reality, the model is not rotated, only equilibrium is modified due to the application of the horizontal pseudo-static force.
Figure 1 – Schematically representation of the effect of the pseudo-static forces on equilibrium: (a) application of body force and (b) “fictitious” equilibrium situation.
Three examples described in the literature (Loukidis et al.,2003; Kontoe et al., 2013) are reproduced using PLAXIS 2D CONNECT Edition V20 Update 2 to illustrate the fundamentals of the pseudo-static approach. In all examples, 2D plane-strain conditions are considered.
The first example consists of a level deposit of soil with 12.0 m of thickness overlying a perfectly rigid bedrock. The mechanical response of the soil deposit is modelled with Mohr-Coulomb model, having the material properties indicated in Table 1. The water level is located at the soil-rock interface, with the soil deposit being considered fully dry. A horizontal pseudo-static acceleration coefficient of kx = −0.50 is applied to the model, as shown in Figure 2. The allowable tolerance is set to 0.001, while failure is assumed to occur once the first unloading step is undertaken, to capture the moment that instability is initiated. Conventional static displacement boundary conditions are considered, consisting of restraining the horizontal displacements along the lateral boundaries of the model and both horizontal and vertical displacements along the bottom boundary of the model.
Table 1 – Properties of the deposit of soil.
Figure 2 – Level ground deposit model phases and conditions.
As illustrated in Figure 3, a slope mechanism developing tangentially along the soil-rock interface (i.e. along the bottom boundary of the model) and extending up to the left boundary of the model is obtained in the numerical analysis (henceforth referred to as “layer mechanism”). As pointed out by Loukidis et al. (2003) and Kontoe et al. (2013), although theoretically justified, this layer mechanism has little physical meaning, since this type of failure has not been observed in the field. Nevertheless, this example illustrates the main principles of pseudo-static analysis and brings awareness of one type of failure mechanism that may be observed in this type of analysis, as further explored later.
Figure 3 – Incremental shear strains at the last step of the pseudo-static analysis of a level ground deposit.
In this example, failure is observed to occur when the phase multiplier, SMstage, which controls the application of the out-of-balanced forces during the numerical analysis, reaches a value of approximately 0.889 (Figure 4). By using Equation 1, the critical horizontal yield acceleration coefficient, kx,crit, can be estimated in 0.444.
where kx corresponds to the horizontal pseudo-static acceleration used in the analysis.
Figure 4 – Plastic points at the last step of the pseudo-static analysis of a level ground deposit.
The obtained value (kx,crit ≈ 0.444) is very close to that obtained by Equation 2, proposed by Loukidis et al. (2003) to define the limit equilibrium at the interface between soil deposit and rigid bedrock.
where DH = 12.0 m is the thickness of the slope deposit.
The second example concerns the stability of a homogeneous dry slope when subjected to a horizontal pseudo-static acceleration of −0.30 g, as shown in Figure 5. Once again, the Mohr-Coulomb model with the material properties indicated in Table 1 is employed in the numerical analysis. The only difference in relation to the previous example consists of the use of an associated flow rule, meaning that a dilatancy angle, y, of 20° is adopted in this case. As highlighted by Loukidis et al. (2003), this assumption is required to compare directly the results obtained in FE analysis with those obtained in limit analysis, being also adopted by Kontoe et al. (2013). Particular attention is also given to the element size and strategy used to generate the initial stress state. Regarding the former aspect, and following the guidelines suggested by Kontoe et al. (2013), an element size of about 0.4 is adopted in the vicinity of the slope, corresponding to 5 % of the slope height, H = 8.0 m.
In relation to the generation of the initial stress state, a strategy similar to that reported by both group of authors is adopted, comprising the following two phases: (1) generation of the stress state in a level ground deposit using a K0-procedure; and (2) simulation of the excavation required to shape a 8 m-high slope (i.e. deactivation of the soil cluster on the right side of the model presented in Figure 5). Subsequently, a pseudo-static analysis is performed to evaluate the critical horizontal pseudo-static acceleration coefficient, kx,crit.
Figure 5 – Homogeneous dry slope model phases and conditions.
As illustrated in Figure 6 and Figure 7, a slope failure mechanism is fully developed when the phase multiplier, SMstage, reaches a value of approximately 0.810. Since kx = −0.30 was used in the pseudo-static analysis, kx,crit can be estimated in 0.243 (Equation 1). As expected, this value is significantly smaller than that obtained in the previous example concerning the level ground deposit (i.e. when considering “green-field conditions”). Indeed, the value obtained in the present analysis is very close to that (0.244) obtained by Kontoe et al. (2013) when using the Imperial College Finite Element Program (ICFEP), as well as to that (of about 0.240) obtained by Loukidis et al. (2003) when performing an upper bound analysis.
As pointed out by Kontoe et al. (2013), and observed in the present study, care should be taken to place the left boundary sufficiently far away from the zone of interest, to minimise its interference on the failure mechanism. In effect, it is apparent in Figure 7 that large shear stresses develop close to the left boundary of the model, due to the application of a pseudo-static force to the elements of the mesh, indicating that the layer mechanism starts developing and may interfere with the obtained results when the left boundary is closer to the main area of interest of the analysis. Regarding the bottom boundary, its effect on the obtained results seems more limited. For instance, similar results to those presented here are obtained when placing the bottom boundary only 4 m below the base of the slope (i.e. when considering a 12 m-thick deposit, rather than a 24 m-thick deposit), as also suggested by Kontoe et al. (2013).
Figure 6 – Incremental shear strains at the last step of the pseudo-static analysis of a homogeneous dry slope.
Figure 7 – Plastic points at the last step of the pseudo-static analysis of a homogeneous dry slope.
The third example consists of a 1.0-thick and 20.0 m-long cantilever retaining wall modelled with solid elements and “wishedin-place” in a homogeneous dry 40 m-thick soil deposit overlying a rigid bedrock (Figure 8). The properties of the soil deposit indicated in the previous example were also employed in this analysis. Regarding the wall, its mechanical response was modelled using a simple isotropic linear elastic model, with a Young modulus of E = 28 GPa and a Poisson’s ratio of n = 0.13. Moreover, its unit weight was considered 24.5 kN/m3. The soil-structure contact was considered perfectly rough, meaning that no interface was used to simulate it. A construction phase sequence similar to that suggested in the literature (Kontoe et al. 2013) was followed, consisting of: (1) generation of the stress state in a level ground deposit using a K0-procedure; (2) simulation of the “wished-in-place” construction of retaining wall and of the excavation required to shape a 8 m-deep cutting; and (3) application of a horizontal pseudo-static acceleration of −0.40 g.
Figure 8 – Cantilever retaining wall model phases and conditions.
Figure 9 depicts the incremental displacements, |Δu|, at failure (i.e. last step reached during the analysis). Note that, since large shear stresses develop at the passive side of the wall (particularly close to the ground surface), this plot was preferred over the deviatoric shear strain plot, allowing for a better visualisation of the failure mechanism developed during the analysis. Indeed, it is apparent that a failure mechanism crossing the wall is developed during the analysis.
Figure 9 – Incremental displacement at the last step of the pseudo-static analysis of a cantilever retaining wall.
Note, nevertheless, that care should always be taken to inspect the possible interaction between the physically expected failure mechanism and the layer mechanism (which, as pointed out before, while theoretically expected, is not expected to occur in the field). In effect, even though the wall mechanism seems to prevail, Figure 10 suggests that the wall mechanism slightly interacts with the layer mechanism. Note that similar observations were reported in the literature (Kontoe et al. 2013).
Figure 10 – Plastic points at the last step of the pseudo-static analysis of a cantilever retaining wall.
The analysis was, therefore, re-run using a wider mesh. In particular, the left boundary of the model was placed farther away from the structure (about 100 m away, rather than 50 m). The adopted mesh was, as much as possible, similar to that used in the previous analysis, with elements with a size of about 0.2 m in the vicinity of the wall. The newly obtained results are depicted in Figure 11, once again, in terms of plastic points at the last step of the analysis. It can be observed that, in this case, the plastic stress points developed at the base of the model do not contact with those developed around the wall, suggesting that, in this case, the two developed mechanisms do not interact with each other. Note, nevertheless, that, for this example, the value of the phase multiplier, SMstage, reached in the former analysis (0.887) is practically identical to that obtained in the latter analysis (of about 0.883). Using the latter value, and taking into account that kx = −0.40 was used in the pseudo-static analysis, kx,crit can be estimated in 0.353 (Equation 1). Indeed, this value is similar to that (0.374) obtained by Kontoe et al. (2013).
Figure 11 – Plastic points at the last step of the pseudo-static analysis of a cantilever retaining wall using a wider mesh.
The force-based pseudo-static analysis may be a reliable tool for the preliminary assessment of the stability of geotechnical structures against seismic failure. In this article, three simple examples were explored to illustrate the fundamentals of the pseudo-static approach, as well as to compare the results obtained when using PLAXIS 2D with those published in the literature (Loukidis et al.,2003; Kontoe et al., 2013). It was shown that a layer mechanism tends to develop during this type of analysis. While theoretically justified, this mechanism has little physical meaning and, therefore, care should be taken to inspect whether this mechanism interacts with physically meaningful mechanisms (e.g. slope and wall failure mechanisms). If deemed necessary, the mesh should be extended to prevent the interaction of these mechanisms from occurring. Noteworthy, in all three cases analysed in this study, critical horizontal pseudo-static acceleration coefficients similar to those reported in the literature were obtained.
Kontoe, S., Pelecanos, L. and Potts, D. (2012). An important pitfall of pseudo-static finite element analysis. Computers and Geotechnics, 48, 41–50.
Loukidis, D., Bandini, P. and Salgado, R. (2003). Stability of seismically loaded slopes using limit analysis. Géotechnique, 53 (5), 463–479.