Semi-probablistic Design Methodologies for Earth Dam and Foundation Design
Author: Perry Mitchelmore
Introduction
The objective of engineering design for earth fill dams and their foundations are safety, functionality and economy. These three objectives are achieved through a combination of assessing the operating requirements of the structure, designs that incorporate local materials as much as practical, vigilance and rigid quality control during construction, effective operation, maintenance and surveillance practices and emergency and contingency planning. Each consideration contains an element of uncertainty or is targeted to reduce uncertainty in achieving the three objectives.
Earth fill dams have been around for centuries and are common throughout the world. Early dams were used for irrigation and supply of drinking water, flood control and transportation. Originally, earth dams consisted of little more than an earthen embankments. These structures evolved into puddle wall and other core type earth dams in the nineteenth century. Probably the most significant technological development in earth dam design is the use of graded filters in the dam. Graded filters provide both improved safety for dams and improved economy by reducing the material requirements for dams.
In the past half century, dams have been developed for the hydroelectricity and mining industries. These dams use improved technology to become much larger and more sophisticated. As earth dams became larger and more sophisticated, design started to push the frontier of the older empirical design methods, necessitating development of rational and analytical design methods. Many of these methods were developed during the first half of the twentieth century (Terzaghi 1922, Casagrande 1937).
An earth dam and its foundation are geotechnical structures. While the exception of hydrotechnical engineers, structural control of imposed loading is the domain of the geotechnical engineer. Structurally, a dam has to be designed for both internal and external stability. There are four key design considerations;
1. Flood hydrology
2. Internal erosion
3. External erosion
4. Slope stability
Failure to adequately address each of these key design considerations may, and likely will, result in unsatisfactory performance or failure of the structure. Statistics on large embankment dam failures indicate that over nine in ten failures are related to overtopping or piping, as shown in Table 1. Therefore, it is fairly safe to assume the obvious, if we adequately design for 1) and 2) above, we control most of the factors affecting dam failures.
Until recently, geotechnical design methods for safety of earth dams were similar to that for building foundations and other heavy civil structures, the global factor of safety method. In recent decades, Canadian foundation engineers and designers have adopted Load and Resistance Factored Design (LRFD) methods. One of the appeals for the shift in foundation engineering was development of a design method similar to that used by structural engineers for the buildings the foundations will eventually support (Becker 2000).
It is important to note that foundation engineering is largely developed through codes of practice. There is no similar code for dam design. Instead, dam designers rely on best practices and design manuals. While these provide for overall consistent methods, there is some variability on prescribed methods.
While the factor of safety method has served earth dam engineering professionals well during the past half century, there exists a need to keep design practice in line with related professions. This shift has already occurred in hydrotechnical analysis and is occurring in erosion analysis. Any appurtenant structure designed for an earth dam will likely be design by probability methods and freeboard design has adopted probabilistic methods.
Table 1: Statistics on Embankment Failures (Foster, 2000)
Mode of Failure | % of Total Failures |
---|---|
Piping Through the embankment | 31 |
Piping Through the foundation | 15 |
Piping from the embankment to the foundation | 2 |
Overtopping | 46 |
Slope Instability | 5.5 |
Earthquake | 1.5 |
Design Process
The nature of design for a dam differs that of foundations. In a foundation design, variable live and wind loads can have a significant impact on the required structural resistance of the soil or rock. For earth fill dams and their foundations, the dominant loads are earth and pore water pressures, both of which are functions of geology. Therefore, it is the characteristic value and variability of the resisting forces that governs dam design where as both loading and resistance variability’s govern foundation design.
Since most dam locations are determined by the available resource, the insitu geological conditions and local materials will have a significant impact on the earth dam design. The unwritten rule for a new earth dam design is to maximize use of local materials to improve economy. The same rule applies to retrofit and upgrading. As a result, while earth dams may be similar, each is unique. This has more significant implications for assessment and analysis of existing structures, where information is scant, than for new structures.
Generally, there are three critical periods in the life of an earth dam, during or immediately after construction, full supply operation, and rapid draw down of the reservoir. At each critical period, loads as a result of static conditions, hydrological conditions and seismic conditions need to be considered. This is applicable to new construction as well as retrofitting older dams.
Effective design considerations include equally each of the following
1. Loads,
2. Material Properties,
3. Calculation Procedure, and
4. Design Criteria.
Isolated improvements in one of these factors without consideration of the mutual relationship of all factors can result in deterioration of the overall design process (Mortensen, 1983).
Earth dam design can typically evolve through four phases; establishing a geological model of foundation conditions, establishing loading conditions from operational requirements, establishing the geometry of the dam and analysis of the design, both analytically and empirically. Because dams rarely contain critical manufactured products (with the noted exception of gates), considerable judgment is needed to effectively complete each phase in view of the uncertainty associated with environmental loads and natural materials.
The geological model for an earth dams foundation is always a simplification of the actual foundation. To facilitate design, engineers must determine characteristic values for permeability, shear strength, stratification, density and compressibility. As well, effects of anisotropy for each of these parameters must be considered. Because natural deposit characteristics will be variable in space and may be directionally correlated, the engineering parameters will also exhibit these traits.
The characteristic values may be selected based on investigation, judgment, local knowledge or complex testing. Use of statistics is often limited because there is insufficient data for analysis. Common practice is to select pessimistic (i.e., conservative) values based on the deposit type, limited insitu and exsitu test data and judgment.
An earth dam must be designed to achieve both internal and external stability. External stability is achieved by protecting the upstream slope and crest against potentially detrimental effects of wave erosion and the downstream slope and toe against rain and water erosion. Internal stability is achieved by limiting excessive seepage gradients, controlling internal water and earth pressures and use of graded filters between different materials.
Many accepted good practices in earth dam and dam foundation design are not easily assessed analytically. Slope stability is probably the best example of an analytical methodology effectively used for earth dam design. However, we are normally limited to plane strain analysis, often using circular failure assumptions and assumed line(s) of seepage within a structure. Failure modes other than slope stability, such as piping and heave, do not have well developed analytical methods with several critical components based entirely on empirical design tools (i.e., filter design). While the models are limited, so are the parameters used to describe the engineering characteristics for design.
There are many limitations in the present design process. If asked, “How safe is a dam?”, we can only respond that it is safe. We sometimes say it is safe enough, but have no means of defining what we mean. For example, if at an older homogeneous type earth dam, a designer decides to construct an inverted filter at the toe to increase safety against heave, there is no present means to analytically quantify by how much the safety against piping is improved. Likewise, should a HIGH consequence dam be designed to a higher standard than a LOW consequence dam. Absolutely, but cannot say by how much a dam is safer. The calculation procedure is essentially the same for both dams.
Working Stress Design (WSD)
The analytical methods that are used in embankment design rely on a global factor of safety to satisfy stability requirements. The method has many descriptors, Factor of Safety (FoS) method, Allowable Stress Design (ASD), etc. For present purposes, we will refer to the method as the Working Stress Design (WSD) method, consistent with nomenclature in foundation engineering.
In WSD, a single, global factor of safety is used to collectively consider all the uncertainty associated with the design process into a single value with no distinction made as to whether it is applied to material strength and resistance or to load effects. The values of global factor of safety selected for design reflect past experience and the consequences of a failure. Generally accepted values for factor of safety in North America are presented in Table 2.
Table 2: Typical values of Factor of Safety
Parameter | Condition | Value |
---|---|---|
Slope Stability | End of Construction Steady Seepage Rapid Draw Down Earthquake |
1.3 1.5 1.2 to 1.3 1.0 |
Seepage | Exit Gradient Gravel Coarse Sand Fine Sand Exit Gradient for a building |
4 to 5 5 to 6 6 to 7 2 to 3 |
WSD is typically applied as a failure criterion, problems of plastic shear strength rather than elastic deformation and loss of functionality. WSD methods apply to two assessment conditions in a dam, slope stability and seepage considerations.
For slope stability analysis, the factor of safety is defined as the ratio of the resisting moments to the driving moments and is often resolved using the method is slices. The same ratio is used for force equilibrium analysis, except the ratio of vertical and horizontal forces is assessed. The principle loads are soil weight and pore water pressure, although a surcharge load can be present. All slope stability calculation procedures use simplifications in geometry and materials.
For most slope stability analysis models, the soil is considered as either cohesive or cohesionless and the analysis assumes a circular failure shape. As noted, this analysis assumes plane strain conditions, fully mobilized shear strength, and negligible side forces.
For a seepage analysis, two scenarios are typically assessed, piping by heave and piping by internal erosion. The actual erosion mechanism is not well understood, but each of the following contributes in one form or another to internal erosion.
• Permeability or hydraulic conductivity k.
• Hydraulic gradient i.
• Porosity n.
• Critical stress - (shear stress required for flowing water to dislodge soil particles).
• Particle size, expressed as some representative size (e.g., D_{15}).
• Friction angle of the soil.
(USACE 1999)
Piping by heave is assessed by analysis of the exit gradient near the seepage exit point. The factor of safety is defined as the ratio of critical exit gradient to the calculated exit gradient
Under the condition of vertical upward flow, the submerged unit weight should be reduced proportionate to the upward gradient. At the point where the submerged buoyant weight of the soil element is equivalent to the weight of the water, shear strength reduces to zero and the soil become liberated and flows according to the exit gradient. The theoretical exit gradient is established based on flow nets developed on the geological model for design, but the actual exit gradient may vary because of flow concentrations, erosion, geometry, etc. The head in the reservoir that causes the product of the gradient ie and the factor of safety to equal the critical value icr is called the critical head h_{cr}.
To accurately apply the WSD method, nominal or characteristic values of strength, permeability, unit weight, etc. are selected for design. The design values may be the mean value of a series of tests or another value, depending on the judgment and experience of the engineer, complexity of the design and construction considerations. Selection of a nominal load and resistance for design can have a more significant effect on the margin of safety than selection of the factor of safety itself.
Uncertainty can be qualified by sensitivity analysis, but the value of factor of safety tells us very little quantitatively about the level of safety or probability of failure.&nbnbsp; In fact, without a full appreciation of the uncertainty, the factor of safety may provide a misleading sense of safety if uncertainty is high. Management of the uncertainty is essentially a matter of judgment and experience on the part of the engineer, not easily transferred to others. While dam design engineers appreciate the method, it can be challenging to explain the approach to the general public, other engineers and dam owners, all of whom must be satisfied as to the appropriateness of the design.
Load and Resistance Factor Design (LRFD)
LRFD is similar to WSD in the process of selecting nominal or characteristic load and resistance values. As previously noted, the design values may be the mean of a series of tests or another value, depending on the judgment and experience of the engineer, complexity of the design and construction considerations.
The LRFD method is often referred to as Limit States Design (LSD) because it considers separately Ultimate Limit States (ULS), or conditions of plastic equilibrium, and Serviceability Limit States (SLS), or conditions of elastic equilibrium. The former is a structural consideration and incorporates more safety in design than the latter, which is more functional and often based on economic considerations. However, it is worth noting that use of different factors of safety in WSD amount to a form of LSD as well.
In practice, partial factors are used on loads and resistance factors are used on the ultimate resistance that is calculated from unfactored strengths of materials. The LRFD safety criteria is expressed by
where R_{n} and S_{n} represent the nominal resistance and nominal load, respectively and and represent the partial factors for resistance and load.
In foundation engineering, two design methodologies have evolved, the factored strength approach, largely adopted in Europe, and the factored resistance approach, largely adopted in North America. Notable exceptions include the Canadian Foundation Engineering Manual (CFEM 1992) and the Design Code for Construction & Installation of Offshore Structures (CSA 1992).
LRFD and WSD differ in how parameters are factored. In the LRFD method, separate factors are applied to loads and the resistance. These individual factors provide for more flexibility in applying the safety margin in areas where there is more uncertainty. For example, seepage pressures would have a higher partial factor than would earth load, which is less variable.
While the methods differ, the results to the foundation engineers are similar or comparable. This is largely attributable to the calibration methods used to develop partial factors. The resistance factors presently used in foundation practice are developed by calibration with the conventional WSD factor of safety or by semiprobablistic studies of the resistance (Meyerhof 1995, Becker 2000).
LRFD was adopted as an alternative to WSD in Canada in 1975 for structural design of buildings, and is presently in many Canadian structural and foundation codes of practice (NBCC, 1995, CHBDC 2000). It is also used extensively in concrete, wood and steel design codes. The presently accepted partial factors in foundation design are presented in Table 3.
Calibration
A LRFD system for earth dam design must produce, as a minimum, comparable results to the existing system. To achieve this, calibration of the two methods is needed for the particular case of earth dams. It is desirable that results of a calibration are consistent with previous work performed by foundation engineers, but it is important for the work to properly consider unique aspects of dam design.
Table 3: Typical values of Partial Factors from related Design Codes (Cited in Becker 2000)
Parameter |
Eurocode |
Canada |
Ansi A58 (1980) |
|||
CFEM |
NBCC |
CHBDC |
||||
Loads | Dead Load (earth Fill) Hydrostatic Pressure Live Loads Wind Load |
1.1 (0.9) 1.0 1.5 1.5 |
1.25 (0.8) 1.25 (0.8) 1.5 1.5 |
1.25 (0.8) 1.25 (0) 1.5 1.5 |
1.25 (0.8) 1.1 (0.9) 1.4 - 1.7 1.0 - 1.25 |
1.2 - 1.4 (0.9) 0.5 - 1.6 1.3 - 1.6 |
Resistance | Friction (tan ) Cohesion (c) Bearing Capacity Passive Resistance Siding |
1.25 (0.8) 1.4 - 1.6 |
1.25 (0.8) 1.5 |
Resistance Factors of 1.25 - 2.0 at ULS using unfactored strength |
0.5 0.5 0.8 |
Resistance Factors of 1.25 - 2.0 at ULS using unfactored strength |
Anchors | Static Tension (Calc) Static Tension (Tested) |
0.4 0.6 |
0.4 |
Note ( ) – Values in parenthesis are to be applied when the load and resistance acts in the opposite direction
The WSD deterministic method was developed largely based on judgment and practice during the past centuries. By calibrating LRFD with existing WSD practice, we can hopefully preserve these judgments and experience. However, we can also calibrate using probability theory that allows for some optimization. Probability theory can be very complex but can also be accomplished based on certain assumptions.
Probability methods can be represented in several forms. Fully probabilistic methods require the “actual” probability distribution curves be known for each variable in the analysis. The cost of obtaining meaningful data can be prohibitive for all but larger projects. Approximate probabilistic methods, referred to as first order second moment (FOSM) methods, use only the shape of the distribution curves combined with two moments; the mean and coefficient of variability. In probability theory the reliability index, , is the relative measure of the degree of safety. The latter, approximate probabilistic methods have been largely adopted by US Army Corps of Engineers (USACE) and US Bureau of Reclamation (USBR) in risk management policies.
The reliability index, , defined as the number of standard deviations between the expected value (i.e., sometimes the mean) and failure criteria as illustrated in Figure 1.
To understand the concept, we need to consider a probability density function of a random variable, factor of safety as an example. The actual value of factor of safety reported from an analysis is the minimum value determined from the single value input variables. If these variables were altered within a reasonable range and a new factor of safety calculated for each, we would get a number of different factors of safety that could be plotted with the probability of occurrence on the ordinate and the actual value plotted on the abisca. The factor of safety probability distribution generally takes a lognormal shape when plotted in this manner, with a mean value and standard deviation. Normal distributions are easier to work with so it is usually the log of the factor of safety that is analysed.
In addition to the mean and standard deviation, two statistical moments most engineers are familiar with, reliability analysis uses the Coefficient of Variation (V). It is a dimensionless expression for the uncertainty of any parameter and is expressed mathematically by
The expected value, E(X), standard deviation, , and coefficient of variation, V, are independent variables (i.e., if two are known, the third can be calculated). In practice, where data is often scant, it is common to assume a coefficient of variation for a parameter from other sites. Some typical values for V are presented in Table 4.
Becker provides a thorough explanation on derivation of load and resistance factors for LRFD in the Eighteenth Geotechnical Colloquium on Limit States Design for Foundations (Becker 2000). Becker presented the following expression based on reliability theory
which is similar in form to equation (2).
Table 4: Typical Values of Coefficient of Variation (%)
Parameter | USACE, 1999 | Cited in Becker, 2000 | Meyerhof 1995 |
---|---|---|---|
Unit Weight | 4 to 8 | 4 to 16 | 5 to 15 |
Liquid and Plastic Limits | 11 | ||
SPT N Value | 15 to 50 | 30 to 50 | |
Drained Strength of Sand | 3.7 to 9.3 | 5 to 15 | |
Drained Strength of Clay | 7.5 to 10.1 | ||
Undrained Strength of Clay | 30 to 40 | 12 to 85 (37) | 20 to 50 |
Coefficient of Permeability, soils (k) | 90 | ||
Coefficient of Permeability, embankment sands (k) | 30 | ||
Embankment Stability Models | 14 to 32 | ||
Construction Variability | 5 to 15 |
From (4), two expressions are presented, the first expressing resistance factors as a function of and Vr and the other doing the same in terms of Factor of safety
In equations (4), (5) and (6), is a separation coefficient, k_{R} the ratio of the mean resistance to the nominal value for design and is the partial resistance value of interest. The separation coefficient is a function of V_{R}/V_{S}, two factors that are not well defined in geotechnical engineering. For present purposes it is assumed to be 0.75, the same coefficient used by Becker for calibration of the WSD and LRFD for foundation engineering.
The appropriate characteristic value for resistance is defined by kR, which is a function of the coefficient of variation. For resistance values with a normal distribution, 50% of all values will lie within +/-0.7 standard deviations of the mean. It has been suggested that a conservative assessment of resistance is the 75% value, or mean minus 0.7* . For the 90% value, the mean minus 1.3* is applied.
The reliability index, , is related to the probability of failure by the following expression (Rosenbleuth and Esteva, 1972) for
In this manner, target probabilities of failure in earth dam design can be established based on consequences and risk methods. Figure 2 illustrates the estimated risk for different engineering activities (cited in USACE 1999). The acceptable foundation risk is identified as an annual probability of 10^{-2} to 10^{-3} compared to 10^{-4} to 10^{-5} for dams. More recent work suggest these probabilities are acceptable for offshore foundations but that land foundations should have a probability of failure in the range of 10^{-4} (Becker 2000, Meyerhof 1995). Others (MacGregor, 1976) have summarized risk in terms of avoidable and unavoidable risk. For the former, the acceptable level of risk is estimated at between 10^{-3} and 10^{-4} while for the latter, acceptable risk is estimated at 10^{-5}. A population living downstream of a dam is likely to consider that risk unavoidable and likely prefer the more strict level of safety.
Using equation (7), target probability of failures of 10^{-4} and 10^{-5} yield values of 3.56 and 4.10, similar to target values in structural design for buildings (Becker 2000). Probability analysis using 90% reliability is used to determine appropriate partial factors for different coefficients of variation and results are presented in Table 5.
Values of k_{R} of 1.25 and 2.5 were selected as typical for the V_{R} values below and above 30%. In other words, the greater the uncertainty or scatter, the greater the value of the factor used in selecting nominal values. In this manner, selecting a nominal value for insitu permeability would likely require reducing the average by a factor of 2.5 while the same nominal value for angle of internal friction would be reduced by 1.25 from the average.
What is obvious from Table 5 is that as the coefficient of variation increases, the value decreases to reflect the increased uncertainty with respect to the analysis. For the selected parameters, will vary from approximately 0.54 to 0.95.
While the method of analysis is preliminary, the results suggest that for an appropriately factored nominal resistance value using 1.25 and 2.5 for slope stability analysis and seepage analysis, respectively, a partial factor of 0.7 for slope stability and 0.6 for seepage analysis may provide reasonably agreeable partial factors in safety analysis.
Table 5: Representative values for
90% Reliability | |||
Target | |||
V_{R} | k_{R} | 3.6 | 4.1 |
0.1 | 1.25 | 0.95 | 0.92 |
0.2 | 1.25 | 0.73 | 0.68 |
0.3 | 1.25 | 0.56 | 0.50 |
0.4 | 2.50 | 0.85 | 0.73 |
0.5 | 2.50 | 0.63 | 0.54 |
Why Bother?
WSD using the factor of safety method has worked relatively well during the past century in producing dams that are safe, productive and economical. The historical average frequency of failure of large embankment dams is estimated to be 1.2% over the life of the dam. The historical annual probability of failure of large embankment dams is estimated as 4.5 × 10^{–4} per dam per year (Foster 2000). These values are in-line with industry standards where annual rates of probability of stability failure would be in the range of 1-2% for earthworks, earth retaining structures and land-based foundations and about 3% for offshore foundations (Meyerhof 1995).
WSD provides a reasonable approach to design on new structures, but much of the engineering analysis presently occurring is in assessment and upgrading of existing structures. For these structures, the effects of age, deterioration, loading history and operation have a significant impact on safety. As well, there is no control over the quality of construction, selection of materials and frequently scant information on the foundation. In these instances, WSD does not provide a reasonable measure of the actual safety of the structure.
Colleagues in bridge and building foundation engineering have adopted limit states design using LRFD. In time, there will be pressure from other professionals for more uniform design methods.
Geotechnical engineers are already being pressed by structural engineers, probabilistic experts and risk managers to provide more quantification of the level of safety. A full probabilistic analysis could quantify these questions but the methods are beyond what present practice can absorb. The LRFD method provides a convenient and understandable tool to “bridge” to gap between probabilistic experts and classical geotechnical analysis using WSD.
The onus is on present earth dam designers to take a lead in this shift. The current LRFD method used in Canadian design codes (NBCC, 1995, CHBDC 2000) is developed to suit problems dominated by these structures. The process was largely driven by structural engineers and the form of the method suits traditional design methods of structural engineers more so than traditional methods of geotechnical analysis. Earth dams and their foundations have unique requirements that do not necessarily fit with building science and engineering needs.
A comparison between the global factor of safety and reliability index is presented in Figure 3. It is obvious from viewing the figure that assessment of seepage for a dam and its foundation is vastly different from any consideration in foundations.
The shaded values are for seepage analysis using the factor of safety values noted in Table 2. It seems obvious from Figure 3 that the primary concern of design engineering for dams is improved understanding of the variables involved in seepage analysis. However, this is not obvious using the factor of safety method. Likewise, it appears that a factor of safety of 2 instead of the normal 1.5 may be more appropriate for slope stability at dams, provided a consistent probability of failure is the goal.
Figure 3: Comparison between reliability index and factor of safety assuming lognormal distributions (after Meyerhof, 1995)
Conclusions
Present design practice for earth dams and their foundations use a combination of empirical methods and the Working Stress Design (WSD) method for safety assessment. WSD relies significantly on judgment and conservative design assumptions for successful execution. The skill used in design are not easily transferred or articulated and may cause controversy where one designers ideas contradict with that of others. There is a need for a more rational design approach in design, similar to what has occurred in the foundation engineering community.
Over the past three decades, foundation engineering professionals have experimented with, and largely adopted into code, semi-probability methods for design that incorporate, through the LRFD method, uncertainty in a more rational way than the WSD method used in earth dam design.
The concerns and needs of earth dam design engineers are significantly different from foundation engineers. The dominant force in dam design is seepage forces and pore water pressures. Determination of appropriate values for design is very uncertain and large factors of safety are adopted to account for this uncertainty.
A preliminary calibration exercise using reliability theory found that by a combination of adopting a more conservative nominal value for seepage, relatively close partial factors can be employed for both slope stability and seepage analysis. A more thorough calibration exercise is needed that will ensure any new methodology achieves similar results to present practice.
Earth dams and their foundations should be designed with a high probability of failure. An appropriate reliability index of between 3.6 and 4.1 provides a probability of failure on the order of 10^{-5} provided slope stability and seepage resistance values are selected conservatively and partial factors of 0.7 and 0.6 are used. Conservative values for selecting nominal resistance values for slope stability and seepage of 1.25 and 2.5, respectively, were used. These values provided an estimated reliability of 90% for a lognormal distribution.
WSD has served the dam design engineers well producing structures with performance records similar to other structures, 1% to 2% failure rates. An increasing proportion of dam analysis is related to assessment of existing structures where WSD is not as effective. More rational methods would provide geotechnical engineers the tools needed to effectively explain the risk associated with dam design.
The semi-probabilistic LRFD method provides an intermediate step to full probabilistic analysis if dams and their foundations. It is an effective method in itself, but more importantly, it allows geotechnical engineers an opportunity to become familiar with probabilistic principles without having to adopt full probabilistic design methods, something the profession is not yet capable to adopt.
While the building foundation engineer is no more able to answer the question, “How safe is your foundation”, they are further along in the design evolutionary trail to eventually answering that question than are earth dam designers.
References
Becker, D.E. (2000). “18th Canadian Geotechnical Colloquium: Limit States Design for Foundations” Canadian Geotechnical Journal, 33: 956-1007.
Casagrande, A., (1937). “Seepage Through Dams”, Journal of New England Water Works Association, Vol 51, pp. 297-336.
El-Ramy, H., N.R. Morgenstern, and D.M.Cruden. 2002. “Probabilistic Slope Stability Analysis for Practice” Canadian Geotechnical Journal, 39: 665-683.
Foster, M, R. Fell and M. Spannagle (2000) “The Statistics of Embankment Dam Failures and Accidents”, Canadian Geotechnical Journal, 37: 1000-1024.
Foster, M, R. Fell and M. Spannagle (2000) “A Method for Assessing the Relative Liklihood of failure of Embankment Dams by Piping”, Canadian Geotechnical Journal, 37: 1024-1065.
Meyerhof, G.G. 1995. “Development of Geotechnical Limit State Design” Canadian Geotechnical Journal, 32: 128-136.
Mortensen, K. (1983). “Is Limit State Design a Judgement Killer”, Bulletin No. 35, Danish Geotechnical Institute, Denmark.
Terzaghi, K, Peck, R.B,, and Mesri, G. (1996). Soil Mechanics in Engineering Practice, John Wiley & sons, Inc., New York, NY.
U.S. Army Corps of Engineers. 1997. Introduction to Probability and Reliability Methods for Use in Geotechnical Engineering. ETL 1110-2-547, Department of Army, Washington, DC, USA.
U.S. Army Corps of Engineers. 1999. An Overview of Probabilistic Analysis for Geotechnical Engineering Problems, Appendix A. ETL 1110-2-556, Department of Army, Washington, DC, USA.
U.S. Army Corps of Engineers. 1999. Evaluating the Reliability of Existing Levees, Appendix B. ETL 1110-2-556, Department of Army, Washington, DC, USA.
U.S. Army Corps of Engineers. 1952. Soil Mechanics Design, Part CXIX, Chapter 1 – Seepage Control. EM 1110-2-1901, Department of Army, Washington, DC, USA.
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