**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**5

# probability of failure Related Publications

##### 5 Considerations for Effectively Using Probability of Failure as a Means of Slope Design Appraisal for Homogeneous and Heterogeneous Rock Masses

**Authors:**
Neil Bar,
Andrew Heweston

**Abstract:**

Probability of failure (PF) often appears alongside factor of safety (FS) in design acceptance criteria for rock slope, underground excavation and open pit mine designs. However, the design acceptance criteria generally provide no guidance relating to how PF should be calculated for homogeneous and heterogeneous rock masses, or what qualifies a ‘reasonable’ PF assessment for a given slope design. Observational and kinematic methods were widely used in the 1990s until advances in computing permitted the routine use of numerical modelling. In the 2000s and early 2010s, PF in numerical models was generally calculated using the point estimate method. More recently, some limit equilibrium analysis software offer statistical parameter inputs along with Monte-Carlo or Latin-Hypercube sampling methods to automatically calculate PF. Factors including rock type and density, weathering and alteration, intact rock strength, rock mass quality and shear strength, the location and orientation of geologic structure, shear strength of geologic structure and groundwater pore pressure influence the stability of rock slopes. Significant engineering and geological judgment, interpretation and data interpolation is usually applied in determining these factors and amalgamating them into a geotechnical model which can then be analysed. Most factors are estimated ‘approximately’ or with allowances for some variability rather than ‘exactly’. When it comes to numerical modelling, some of these factors are then treated deterministically (i.e. as exact values), while others have probabilistic inputs based on the user’s discretion and understanding of the problem being analysed. This paper discusses the importance of understanding the key aspects of slope design for homogeneous and heterogeneous rock masses and how they can be translated into reasonable PF assessments where the data permits. A case study from a large open pit gold mine in a complex geological setting in Western Australia is presented to illustrate how PF can be calculated using different methods and obtain markedly different results. Ultimately sound engineering judgement and logic is often required to decipher the true meaning and significance (if any) of some PF results.

**Keywords:**
Slope Stability,
Sensitivity Analysis,
probability of failure,
point estimate method,
Monte-Carlo simulations

##### 4 Evaluation of the Mechanical Behavior of a Retaining Wall Structure on a Weathered Soil through Probabilistic Methods

**Authors:**
P. V. S. Mascarenhas,
B. C. P. Albuquerque,
D. J. F. Campos,
L. L. Almeida,
V. R. Domingues,
L. C. S. M. Ozelim

**Abstract:**

Retaining slope structures are increasingly considered in geotechnical engineering projects due to extensive urban cities growth. These kinds of engineering constructions may present instabilities over the time and may require reinforcement or even rebuilding of the structure. In this context, statistical analysis is an important tool for decision making regarding retaining structures. This study approaches the failure probability of the construction of a retaining wall over the debris of an old and collapsed one. The new solution’s extension length will be of approximately 350 m and will be located over the margins of the Lake Paranoá, Brasilia, in the capital of Brazil. The building process must also account for the utilization of the ruins as a caisson. A series of *in situ* and laboratory experiments defined local soil strength parameters. A Standard Penetration Test (SPT) defined the *in situ* soil stratigraphy. Also, the parameters obtained were verified using soil data from a collection of masters and doctoral works from the University of Brasília, which is similar to the local soil. Initial studies show that the concrete wall is the proper solution for this case, taking into account the technical, economic and deterministic analysis. On the other hand, in order to better analyze the statistical significance of the factor-of-safety factors obtained, a Monte Carlo analysis was performed for the concrete wall and two more initial solutions. A comparison between the statistical and risk results generated for the different solutions indicated that a Gabion solution would better fit the financial and technical feasibility of the project.

**Keywords:**
Statistical Analysis,
retaining walls,
probability of failure,
economical analysis

##### 3 Reliability Based Performance Evaluation of Stone Column Improved Soft Ground

**Authors:**
A. GuhaRay,
C. V. S. P. Kiranmayi,
S. Rudraraju

**Abstract:**

The present study considers the effect of variation of different geotechnical random variables in the design of stone column-foundation systems for assessing the bearing capacity and consolidation settlement of highly compressible soil. The soil and stone column properties, spacing, diameter and arrangement of stone columns are considered as the random variables. Probability of failure (*P _{f}*) is computed for a target degree of consolidation and a target safe load by Monte Carlo Simulation (MCS). The study shows that the variation in coefficient of radial consolidation (

*c*) and cohesion of soil (

_{r}*c*) are two most important factors influencing Pf. If the coefficient of variation (COV) of

_{s}*c*

_{r}exceeds 20%,

*P*exceeds 0.001, which is unsafe following the guidelines of US Army Corps of Engineers. The bearing capacity also exceeds its safe value for COV of

_{f}*c*> 30%. It is also observed that as the spacing between the stone column increases, the probability of reaching a target degree of consolidation decreases. Accordingly, design guidelines, considering both consolidation and bearing capacity of improved ground, are proposed for different spacing and diameter of stone columns and geotechnical random variables.

_{s}**Keywords:**
bearing capacity,
probability of failure,
consolidation,
stone columns,
geotechnical random variables

##### 2 Analytical Slope Stability Analysis Based on the Statistical Characterization of Soil Shear Strength

**Authors:**
Bernardo C. P. Albuquerque,
Darym J. F. Campos

**Abstract:**

**Keywords:**
probability of failure,
statistical slope stability analysis,
skew distributions,
functions of random variables

##### 1 Reliability Analysis of Underground Pipelines Using Subset Simulation

**Authors:**
Kong Fah Tee,
Lutfor Rahman Khan,
Hongshuang Li

**Abstract:**

An advanced Monte Carlo simulation method, called Subset Simulation (SS) for the time-dependent reliability prediction for underground pipelines has been presented in this paper. The SS can provide better resolution for low failure probability level with efficient investigating of rare failure events which are commonly encountered in pipeline engineering applications. In SS method, random samples leading to progressive failure are generated efficiently and used for computing probabilistic performance by statistical variables. SS gains its efficiency as small probability event as a product of a sequence of intermediate events with larger conditional probabilities. The efficiency of SS has been demonstrated by numerical studies and attention in this work is devoted to scrutinise the robustness of the SS application in pipe reliability assessment. It is hoped that the development work can promote the use of SS tools for uncertainty propagation in the decision-making process of underground pipelines network reliability prediction.

**Keywords:**
probability of failure,
Underground pipelines,
Reliability and Subset Simulation