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The log-spiral upper bounds presented in this paper correspond to slip surfaces passing through the toe of the slope. Second-order, six-noded, plane-strain triangular elements were employed to construct unstructured meshes.

An elasto-plastic constitutive model following the Mohr-Coulomb failure criterion with the dilatancy angle always set equal to the friction angle was assigned to the soil to enable direct comparison with the limit analysis solutions. Thus, the material model did not have an associated flow rule in the strict sense. The number of elements in the lower and upper bound meshes ranged from to elements and from to elements, respectively.

These differences reduce to half or less when the log-spiral upper bounds are considered instead of the numerical upper bounds; thus, the value of kc is defined with great precision.

The quality of the log-spiral upper bound solutions resides in that this method assumes that all plastic dissipation occurs along an infinitely thin velocity discontinuity with log-spiral shape. The analyses show that the geometry and location of the critical slip surfaces predicted by LE are in very good agreement with those from the log-spiral upper bound, the shear band calculated by FE analysis, and the zone of high power dissipation from the numerical upper bound analysis.

Most FE results were found to lie inside the narrow range defined by the given in Table 1. For direct comparison with limit analysis results, the dilatancy angle was set equal numerical lower bound and the log-spiral upper bound. This was possible because of mesh refinement in the zone where the slip surface was expected to pass through. Table 1. Comparison of horizontal kc Table 2. Comparison of horizontal kc for the slope shown in Figure 3.

Method of analysis kc Method of analysis kc Numerical lower bound 0. The soil profile consists of five soil strata of different shear strength. The thickness of layers 2 and 4 is 0. In this case, the finite element mesh used in the lower bound analysis had triangular elements, 41 extension elements, and stress discontinuities.

The upper bound mesh had triangular elements and velocity discontinuities. As it can be seen in Figure 4b, the collapse mechanism did not involve layers 4 and 5 as their shear resistance is significantly larger than those of overlying soils. The calculated rigorous lower and upper bounds for kc are 0. The calculations were done for zero vertical seismic acceleration.

LE methods estimate a critical seismic coefficient that is in agreement with the numerical upper bound. Example problem: a geometry and soil profile; b zone of high power dissipation from upper bound analysis, FEM shear band and critical slip surfaces from LE methods. Numerical lower and upper bounds to the critical seismic coefficient kc are compared with results from limit equilibrium LE , log-spiral upper bound, and displacement finite element FE methods. For the case of homogeneous slopes, the true value of kc is contained within a very narrow range determined by the numerical lower bound and the log- spiral upper bound solutions.

Values of kc from all LE methods considered in this study are in very good agreement with those from limit analysis and displacement FE analysis. LE methods are capable of predicting the geometry and location of the critical slip surfaces. Numerical limit analysis is a versatile technique that allows the analysis of stability problems with complex loading, geometry and soil profile, providing a result known to be exact with a precision depending on the distance between the bounds.

Sloan, University of Newcastle, Australia. The authors are thankful to Professor Sloan for making his original codes available to them. Numerical limit analysis for slope stability and bearing capacity calculations. Bishop A. The use of the slip circle in the stability analysis of slopes. Chang C-J.

Seismic displacements in slopes by limit analysis. Geotechnical Engineering Div. Hibbitt H. With the free PDF24 Creator you get an interface with which you can freely merge multiple files page-based. MS Word is one of the most used text processing applications and the PDF format is the most used format to publish documents. Click here to learn how to convert Word files to PDF. Read below how it works. A lot of other features are also included that help you to create PDF files.

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