Discussion: Foundation design for gravity retaining walls under earthquake

The discussers read the paper by Pender (2018) with great interest. The author addresses the important issue of computing the pseudostatic critical acceleration of gravity/cantilever retaining walls, corresponding to which a plastic mechanism is activated within the soil–structure system and the wall starts to move under the applied earthquake. In fact, the critical acceleration is the key ingredient for the seismic design of these structures, controlling both the maximum internal forces and the final displacement (Conti and Caputo, 2018; Conti et al., 2013).

By referring to dry cohesionless backfill and foundation soils, the author concluded that bearing capacity failure is the controlling mechanism when dealing with the seismic design of gravity/cantilever walls.

The scope of this discussion is twofold: (a) to extend the results presented by the author; (b) to provide additional comments on the seismic design of such structures.

The discussers agree that the bearing capacity failure is likely to be the ‘natural’ failure mechanism for a wall on cohesionless soil. This is consistent with experimental data (Conti et al., 2015; Kloukinas et al., 2015; Koseki et al., 2003) and numerical results (Smith and Cubrinovski, 2011). Viggiani and Conti (2016) reported the results of a parametric theoretical study aimed at identifying the critical failure mechanism for gravity walls on different foundation soils. Figures 10 and 11 show the normalised critical acceleration of the wall, k_c = min(k_y,slid, k_y,qlim), for gravity and cantilevered retaining walls, respectively, highlighting the role played by cohesion. Also, numerical and theoretical results for cantilever walls on fine-grained soils indicate that the sliding mechanism usually controls the dynamic behaviour of the wall in undrained conditions (Conti and Caputo, 2018).

While the concept of an admissible wall displacement has been widely accepted within the performance-based seismic design philosophy, the possibility of admitting wall tilting, related to a temporary attainment of the bearing resistance, is still a controversial issue. Indeed, many provisions and codes of practice still recommend that the wall should be designed ensuring an adequate safety margin with respect to a bearing failure of the foundation and assuming the sliding mechanism as the critical one (Anderson et al., 2008; PIANC, 2000). The rationale behind it is twofold: (a) an excessive wall tilting could induce a sudden collapse of the wall by overturning; (b) no reliable procedures are available to accommodate a mixed sliding–rotational failure mode within the well-established Newmark's approach.

On the one hand, the discussers agree with the author that a rational seismic design of gravity/cantilever walls should contemplate the possible activation of both mechanisms, instead of excluding a priori the expected rotation. As a matter of fact, the temporary mobilisation of the soil shear resistance beneath the foundation would not lead to a fragile failure of the system, provided that an excessive wall tilting is prevented. This is a fundamental difference with respect to a pure overturning mechanism, which is indeed a fragile mechanism by nature.

On the other hand, further research is required to develop reliable (and simple) theoretical models, capable of handling combined tilting and sliding failure modes as, in this case, a direct application of the Newmark's sliding block procedure can lead to a significant under-prediction of the final displacement.

The author thanks the discussers for their interest in the paper. He is pleased to see, in the discussion, and the additional papers referenced, confirmation of his conclusion that the critical horizontal earthquake acceleration is an essential gravity retaining wall design parameter. The author's main intention in writing the paper was to illustrate how understanding of the BSS gives insight into conventional bearing strength calculations. Using the appropriate action path the estimation of the value for the critical horizontal acceleration is straightforward (as demonstrated in Appendix 1). Subsequent displacement of the wall system is an important topic but beyond the scope of the paper.

The references provided by the discussers provide additional confirmation of the author's conclusions by way of centrifuge test results (Conti et al., 2015) and numerical analysis using Flac (Conti and Caputo, 2018). The modelling that the author presented in the paper calculates the moment that will induce rotational failure of the foundation. Nevertheless, this is accompanied by possible horizontal movement as both the foundation moment capacity and horizontal shear capacity are mobilised when the action path engages the BSS.

The discussers also explore the effect of cohesive resistance in the foundation material. The author has done some calculations using undrained shear strength values representative of the residual clays found around Auckland. Again, the finding is that the moment capacity of the foundation is mobilised before the horizontal sliding capacity.