**Introduction**

Hungr *et al. *(1989) described a simple 3D slope stability problem, involving a hemispherical failure surface. The material in the slope has zero friction, therefore it is possible to solve the relevant stability equations explicitly.

**Slope description**

The simple 3D geometry of the slope and failure surface are shown below. The planar slope has a 1:2 grade (26.565 degrees from the horizontal) cut in a homogeneous frictionless cohesive soil, and has a hemispherical failure surface of unit radius. The centre of the hemispherical basal surface is 0.5 units perpendicular from the slope face.

The following image shows the discretised model consisting of an array of columns above the hemispherical failure surface.

**Stability calculations – 3D analysis
**

Hungr *et al. *(1989), using the approach of Baligh and Azzouz (1975) found that a closed form solution for this problem gave a factor of safety of 1.402. We have carried out similar calculations, and confirm their result.

Stability calculations carried out using TSLOPE indicate that a more “exact” solution is provided as the number of columns involved increases. This has also been reported by other workers (Lam and Fredlund, 1993; Huang *et al. *2002).

The TSLOPE results show that Spencer’s method calculates a factor of safety of 1.401 using 113,976 columns (or 700 columns per width). With the same number of columns, the Ordinary method of columns gives a slightly higher factor of safety, 1.442.

The following graph shows that using a modest number of columns, the factor of safety trends towards the closed form result of 1.402.

Other workers have considered this relatively simple problem, and we provide a summary of their results in the following table.

Author | Method of analysis | Number of columns | Factor of safety |
---|---|---|---|

Hungr et al.(1989) | Program CLARA | 1.422 | |

Lam and Fredlund (1993) | Program 3D-SLOPE | 1200 | 1.386 |

Huang and Tsai (2000) | 1300 | 1.388 | |

5300 | 1.399 | ||

Chen et al.(2001) | 1.422 |

Chen *et al. *(2005) have analysed this problem using three dimensional rigid finite elements. They obtained a factor of safety, with a predefined hemispherical slip surface, of 1.430.

**Stability calculations – 2D analysis**

A 2D slope case was constructed through the 3D model, and analysed using Spencer’s method. The result is shown on the following figure. As expected, the 2D factor of safety, 1.0, is less than than the 3D factor of safety.