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The subject of the influence of the seismic excitation on limit loads of footings is revisited, with emphasis on the moment load. The kinematic approach of limit analysis is employed using two collapse mechanisms allowing footing rotation and one with pure translational kinematics. Two of the mechanisms have novel elements, not presented in earlier literature. The paper is focused on the resistance of the soil weight to activating a mechanism of failure, which can be best cast in terms of the seismic bearing capacity factor Nsγ . Seismic loads from the superstructure are interpreted as those caused by a three-mass model, each mass with its own seismic coefficient. The notion of generalized loads is used to present the yield locus for the footing in terms of the gravity force, horizontal force, and moment. The non-symmetric components of the load are interpreted as seismically activated. The approach yields a strict upper bound to the magnitude of the load vector causing failure. Of the three failure mechanisms considered none yields the best (least) solutions for all combinations of loads. In general, the two mechanisms with footing rotation perform better for large moments, whereas the translational mechanism yields better results when moments are small. However, even in the absence of a moment load, the rotational mechanism can yield better estimates of the limit load when the seismic coefficient is relatively large. 

Three-dimensional stability of roofs in deep flat-ceiling cavities is analyzed. The stability number, factor of safety, and required supporting stress are used as measures of roof stability. Despite the simplicity of the flat roof geometry, the three-dimensional stability analysis presents some complexities owed to the shape of the failure surface geometry in the collapse mechanism. The failure mode assumes a rock block moving downward into the cavity, and the study aims to recognize the most critical shape of the failing block. Three specific block shapes are described in some detail, but more have been analyzed. Blocks defined by a special case of a 4th order conical surface (quartic) on a rectangular base, and a 2nd order elliptic surface (quadric) are found to be the most critical in the stability analysis. The kinematic approach of limit analysis was used, with the rock strength governed by the Hoek-Brown failure criterion. The parametric form of the Hoek-Brown function was employed. Interestingly, an absence of diagonal symmetries in the most critical failure mechanisms was observed in roof collapse of square-ceiling cavities. Computational results in terms of dimensionless measures of stability are presented in charts and tables. 

Engineered cavities in rock formations are often part of an underground transportation infrastructure. Three measures of roof stability in such cavities are discussed: the stability number, the factor of safety, and the support pressure needed to prevent cavity roof failure in weak rock. The stability number is a dimensionless combination of the rock properties and the size of the cavity when roof failure becomes imminent. While there exists substantial experience in application of the stability number and the factor of safety to soil structures, their use to define the safety of rock structures is intricate. This is because the strength envelope for rocks is a non-linear function of the mean stress. The specific function used in the analysis is the Hoek-Brown failure criterion. The kinematic approach of limit analysis is used, and the results are presented in charts. All measures of stability are strongly dependent on the Geological Strength Index, and, to a lesser degree, on other parameters in the Hoek-Brown failure criterion. 

The process credited for high efficacy of embankment pile support systems is the formation of soil arches in the embankment fill. The specific system considered in this presentation is one with cap beams over rows of columns inserted through soft clay to support an embankment fill. Such a system lends itself to a two-dimensional analysis. A formation of distinct arches, as conjectured in many design approaches, was not confirmed. However, based on the calculated stress distribution and the major principal stress trajectories, one can identify “dispersed” arches spanning the distance between two neighboring cap beams. The dispersed arching was not identified in low embankments, whereupon the settlement of the soft soil, a collapse mechanism propagated through the entire embankment height causing differential settlement of the embankment crown. This study was focused on the efficacy of the pile support system and on better understanding of the mechanisms behind the differential settlement at the crown of the embankment. 

Three-dimensional analysis of stability is carried out for slopes with the failure mechanism confined to a narrow space. A curvilinear cone failure surface is modified by removing a slice from its central portion, to make the failure mechanism fit in the narrow space. The resulting surface is no longer smooth, and it is referred to as the ridge mechanism for a distinct ridge in the failure surface. The analysis of narrow slopes based on the ridge mechanism appears to yield lower stability factors than the mechanisms used in geotechnical engineering thus far. Kinematic limit analysis utilized in calculations provides an upper bound to the true stability factor solution, hence the newly proposed mechanism delivers a more accurate solution. 

Three-dimensional failure analyses of slopes are rather elaborate, and for rock slopes, where the rock strength is defined by nonlinear failure envelopes, they are particularly intricate. This is why many earlier approaches used a linear approximation of the strength envelope prior to carrying out the stability analysis. This approximation is avoided in this paper, thanks to using the parametric form of the Hoek-Brown failure criterion. The kinematic approach of limit analysis is used as the method of study. An argument is brought forward that even though rocks tend to fracture at low confining stresses, the ductility of deformation prior to a brittle drop in stress during failure may be sufficient for limit analysis theorems to be applicable. Two measures of rock slope stability are evaluated: the stability number and the factor of safety. Numerical results are presented in the form of charts and tables. Because the limit analysis used allows one to evaluate the rigorous bounds on true solutions, it was possible to demonstrate that the method employed in the paper yields more accurate results than the approaches used formerly in the subject literature. A new and efficient mechanism of failure was devised for very narrow rock slopes. 

A column-supporting system for embankments on soft soils is analyzed using the Finite Element Method. The numerical problem has boundary conditions similar to that of a trap-door, but, contrary to the classical trap-door studies, the focus in this paper is on the load transfer to the supporting columns measured by the system efficacy, and on the critical embankment height that prevents differential settlements on the embankment surface. The stress state simulated in the embankment rapidly evolves in the first stages of the soft soil settlement, measured by millimeters, but the changes to the elastic-plastic stress field developed are marginal in the subsequent stages of settlement simulated up to 10 cm. The displacement field in low embankments with a height comparable to the column spacing is dominated by the failure mechanism with shear bands reaching the embankment crown, causing differential settlements on the embankment surface. In higher embankments, the failure mechanism is confined to the lower portion of the fill, and no differential settlements were detected on the embankment surface. It was suggested that a hypothetical diffused soil arch formed above the failure mechanism, with a shape approximately following the principal stress trajectories. This conjecture was made based on the presence of elevated stress along the symmetry plane of the embankment-column periodic cell. Numbers related to system efficacy and critical height are reported for a limited set of model parameters, but the qualitative outcome of the study is likely applicable to a wider variety of embankments.