Naresh C. Samtani, PE, PhD has contributed his second in a series of articles on LRFD for geotechnical engineering titled "LRFD for Substructures - Concept of Failure and Reliability Index". The first aricle was entitled "LRFD for Bridge Substructure Design - A Note on Limit States and Interaction between Structural and Geotechnical Specialists". Click through to read this interesting article.

LRFD for Substructures - Concept of Failure and Reliability Index

By Naresh C. Samtani, PE, PhD
President, NCS Consultants, LLC
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In a previous article (NCS, 2007), the concept of limit state in the framework of Load and Resistance Factor Design (LRFD) as adopted by American Association of State Highway and Transportation Officials (AASHTO) was discussed.  A limit state was defined as a condition beyond which the bridge or component ceases to satisfy the provisions for which it was designed.  Once a limit state is exceeded, a failure is considered to occur from the perspective of the criterion or criteria for which the design was performed. 

The word “failure” connotes an unacceptable difference between expected and observed performance (Leonards, 1982).  The simplest manifestation of a failure may be in the form of cracking of an engineered component or structure during its design life.  Other manifestations may include loss of intended serviceability, excessive maintenance costs, economic losses, collapse and/or loss of life.  Since the design for the case of absolutely no failure is theoretically impossible within the context of stochastic processes and design for a very small probability of failure will be very expensive, a certain acceptable probability of failure, P_f, needs to be defined.  Thus, from practical considerations, an acceptable risk level should be determined for each limit state, i.e., the probability that a failure can occur.  In the AASHTO-LRFD framework (AASHTO, 2007), this is achieved by defining a reliability index, β (also referred to as safety index)  The use of the term “reliability index,” has an important psychological advantage, i.e., it avoids the negative connotation of the word “failure.”  This article discusses the concept of failure and reliability index.  Readers interested in a primer on LRFD are referred to NCS (2006).

1.0  Factor of Safety and Probability of Failure

In the traditional allowable stress design (ASD) approach (AASHTO, 2002) safety is achieved with a single factor of safety (FS) applied to the resistance to obtain an allowable stress (or load).  Since the FS is an all-inclusive entity, it is difficult to identify the portion of the FS value that applies to the structural or geotechnical design.  For example, in the design of drilled shafts a FS=2.5 is commonly used.  In this case, is the FS equally distributed on the structural and geotechnical designs as 1.25 in an additive manner or is it distributed in some other combination?  It is difficult to answer this question with certainty.   This uncertainty occurs because each component (structural and geotechnical) of the substructure design has different load and resistance statistics, i.e., mean value, coefficient of variation, and distribution, e.g., normal or non-normal.  Generally, structural loads are normally distributed while geotechnical resistances are log-normally distributed.

One has to be careful in recognizing that some loads in substructure design may be induced by the geomaterials themselves, e.g., downdrag, which is added to structural loads.  Downdrag load, depending on the method used to determine it, may be distributed in a non-normal pattern resulting in an overall non-normal distribution of the total load on the substructure element.  Due to these considerations, it is difficult to quantify the probability of failure, P_f, for a substructure element in the ASD framework.  The AASHTO-LRFD provides a solution to this problem by explicitly identifying a reliability index, β, which in essence is an alternative quantification of the probability of failure, P_f.

2.0  Reliability Index, β, and Probability of Failure, P_f

A structural component or the entire structure becomes more reliable as the probability of failure decreases.  Thus, reliability may be expressed in terms of the probability of failure, e.g., 1 in 100 (i.e., 1 failure in 100 events), 1 in 1,000, etc.     In statistical terms, reliability may also be thought of as the inverse of the coefficient of variation (COV).  The COV is defined as  the standard deviation (σ) divided by the mean value (μ).  Using this definition the reliability increases as the standard deviation decreases, i.e., as the COV decreases.  This simple concept is generally useful only when there is one normally distributed variable.  In practice, there is a need for a more general definition of reliability that can express the probability of failure in terms of the coefficients of variation of various parameters that may or may not be normally distributed.  One option is to express the reliability in terms of a reliability index, β, which expresses the probability of failure, P_f, as a function of the statistics of the loads, Q, resistances, R, and a limit state function, g.  Failure may be defined as g < 0 for the case of g = φR - γQ where φ is the resistance factor and γ is the load factor.   In other words, the probability of failure, P_f, represents the probability for the condition of failure at which the factored resistance, φR, will be less than factored loads, γQ.   Once failure is defined, the rate of failure per numbers of simulation (physical or numerical) is determined and expressed in terms of reliability index, β.

The First-Order Second-Moment (FOSM) method based on the first two moments, i.e., mean (μ) and variance (V=σ²), of the data for a random variable can be used to develop analytical closed form solutions for relationships between β and P_f.  Such solutions are widely published in the open literature, e.g., Whitman (1984), Withiam, et al. (1998), Haldar and Mahadevan (2000).  Figure 2.1 shows the relationship between β and P_f based on the FOSM approach  for a normally distributed load and resistance and a log-normally distributed load and resistance.  In some publications (e.g., Paikowsky et al., 2004) the “normal” distribution relationship in Figure 2.1 is referred to as an “exact” relationship.  The log-normal relationship in Figure 2.1 is approximate and valid for 2 < β < 6 (Rosenblueth and Esteva, 1972). 

 
Figure 2.1.      Relationships between β and P_f for the case of a single load and single resistance.

Both normal and log-normal relationships yield approximately the same β (≈ 5 to 5.7) at the practical upper limit of the probability of failure between 1 in 5 million to 1 in 100 million; such values of β may be applicable to extremely critical works such as nuclear facilities.   The two relationships continue to be reasonably similar for β > 2.5 while they diverge significantly for β < 2.5.  As discussed in Section 4.0, the target reliability index, β_T, for substructure design is usually between 2.3 and 3.5.  For this range of β-values, the two relationships are practically similar, see Figure 2.2. 

2.1 Probability of Survival, P_s

Instead of the reliability index one may choose to express P_f as the probability of survival, P_s.  The value of P_s in percent is obtained from (100)(1- P_f) where P_f is expressed as a decimal.  In addition to P_f, the value of P_s is also shown in Figure 2.2.  Since the values of P_s are well above 90% in the zone of interest for subsurface design as shown in Figure 2.2, the probability of survival can be used just like the reliability index to avoid the negative connotation of the word “failure.”   

Figure 2.2.      Relationships between β, P_f and P_s in the range of interest for substructures for the case of a single load and single resistance. 

2.2 Cases of Multiple Loads and Resistances with Non-normal Distributions

For cases where multiple loads and multiple resistances with varying non-normal distributions are evaluated, relationships such as those shown in Figure 2.1 may not be possible without significant simplifying assumptions in the FOSM approach.  In such cases, numerical simulation techniques may be used to determine the probability of failure and the corresponding reliability index.  The problem may then be evaluated without simplifying assumptions thus leading to more realistic results.   Many numerical simulation techniques to perform reliability analysis are available and published, e.g., see Law and Kelton (1991), Haldar and Mahadevan (2000).   The Monte-Carlo simulation technique works well with the overall LRFD approach since it has the capability to evaluate random events; this technique is briefly discussed next.

3.0  Determining the Reliability Index by Monte-Carlo Simulation

Monte-Carlo simulation is a scheme employing random numbers and is used for solving deterministic problems where time does not play a substantive role.  Thus, Monte-Carlo simulations are generally static and not dynamic (Law and Kelton, 1991).  Random values of Q and R are generated according to basic statistical parameters. i.e., mean, coefficient of variation and an assigned distribution.  The random values are then combined to form a limit state function, g, according to a pre-determined combination such as g = φR - γQ where φ is the resistance factor and g is the load factor.  Based on the definition of failure, e.g., g < 0, the number of “failing” simulations is counted and the probability of failure determined as follows:

The load and resistance factors used with the Q and R values to determine the value of g can be varied until an owner-specified acceptable value of P_f  is obtained, e.g., 1 failure in 5,000 simulations.  Generally, for substructure design the value of resistance factor, φ, is varied for a given load factor.  Once a value of P_f acceptable to the facility owner is determined, the chart in Figure 2.1 or Figure 2.2 may be used to determine the approximate reliability index, β. 

3.1 Use of β value versus P_f value

The value of β determined based on the Monte-Carlo simulation technique is approximate because the relationships shown in Figure 2.1 or Figure 2.2 are strictly valid only for the case of a single load and a single resistance.  In practice, multiple loads and a resistance are considered in evaluation of g (limit state) values and the actual distribution of g values may be intermediate between normal and log-normal patterns.  Consider a scenario where a numerical simulation using Monte-Carlo technique gave a P_f of 1 in 5,000 based on a normally distributed dead load, a normally distributed live load, and a log-normally distributed resistance.  For this scenario, the g values will be neither normally distributed nor log-normally distributed but will be intermediate between the two distributions.  For P_f of 1 in 5,000, based on Figure 2.2, a β of 3.54 is obtained if g values are assumed to be normally distributed while a β of 3.4 is obtained if g values are assumed to be log-normally distributed.   Since the actual distribution is intermediate between the normal and log-normal distributions the β value is between 3.4 and 3.54.  Assigning and reporting a single value for such a scenario can be misleading.  This is because most designers conventionally use the assumption of normal distribution to evaluate the meaning of β in terms of P_f.  Referring to Figure 2.2, for the case of normal distribution, a β = 3.54 corresponds to P_f of 1 in 5,000 while a β = 3.4 corresponds to P_f of 1 in 3,000.  Such a discrepancy, i.e., 1 in 5,000 versus 1 in 3,000 may be significant for critical facilities.  Therefore, rather than using β value one can simply report P_f as 1 in 5,000.  Alternatively, one can express P_f of 1 in 5,000 as 0.02%.  Use of P_f in this manner can then enable a direct comparison between different numerical simulations using the Monte-Carlo technique.  As a comparison, in this example, the P_s would be 99.98%.

Since the LRFD literature commonly uses reliability index, β, the remainder of this article continues to use the concept of reliability index for the sole purpose of discussions within the AASHTO-LRFD framework.  Nevertheless, based on the above discussions, the reader should be cognizant of the shortcomings of using a β value in lieu of P_f (or P_s).

3.2 Comments on Use of Monte-Carlo Simulation in LRFD

The Monte-Carlo simulation technique is particularly useful when multiple load sources and resistances need to be evaluated.  For example, in a drilled shaft analysis the loads may be comprised primarily of dead loads and live loads while resistances may be from side resistance and base resistance, each of which is mobilized at a different rate as a function of the vertical movement of the shaft.  As noted earlier, the loads may be normally distributed while resistances may not be normally distributed at the component or system level.  Such problems are not analytically tractable and the designer has to resort to numerical approximations.  This is exactly what the designer should bear in mind, i.e., the Monte-Carlo simulation provides an approximation.  Given that the simulation process is based on random numbers, it is not possible to get exactly the same answer for every simulation.  If one does get exactly the same answer, then either it is a coincidence, or else there is something wrong with the random number generator, i.e., the random numbers are not truly random and/or an adequate number of simulations was not evaluated. 

The large volume of numbers produced by Monte-Carlo simulation often creates a tendency to place greater confidence in the results of the simulation than is justified.  The key to Monte-Carlo simulation in the context of LRFD is to critically evaluate the basic statistical parameters for the input variables.  If any of the input variables is not a valid or justifiable representation of the data, the Monte-Carlo simulation data, no matter how impressive they appear, will provide little useful information about the actual limit state under consideration.

Considering the above discussion, the designer should not get carried away in trying to determine the “exact” value of reliability index since that is simply not possible with a numerical simulation technique.  Realistically, the best that the designer can expect is to determine the reliability index within ±0.10.  Thus, for example, if the target β value is 3.0, then the results for β = 3.0 ± 0.10 should be considered to be satisfactory.  No further refinement is warranted because of the inherent variability of the Monte-Carlo simulation technique based on random numbers as well as the difficulty in defining P_f in terms of β value as discussed in Section 3.1.

4.0  Selection of Target Reliability Index, β_T

Identification of the target reliability index, β_T, is an important step for every owner because the cost of the structure is affected by this value.  Even though in Figure 2.2, the relationship between P_f and β appears to be linear in the range of interest, it is actually non-linear because the Y-axis is plotted on a logarithmic scale.  The relationship between the cost of a structure and the β value will increase non-linearly with the exact nature of the non-linear relationship being a function of whether the concept of reliability index is applied at the component level or system level.  The decision to apply the reliability index to the component or the system level is not an easy one and is a function of many factors including, but not limited to, redundancy at the component level, variability of site stratigraphy, construction techniques, and human errors.  Discussion of the various factors that can affect the reliability index is outside the scope of this article. 

Accurate data on actual failures are difficult to compile because systematic databases of failure records do not exist (Leonards, 1982).  This difficulty is further compounded by the general hesitancy of agencies and owners to share data on failures.  Several regulatory agencies for critical facilities such as dams and nuclear plants have adopted the frequency-consequence chart (also known in the published literature as “F-N” chart) as a convenient graphical tool for characterizing exceedance probability of risks against their associated consequence, e.g., society's tolerance for loss of life and property.  Figure 4.1 shows a typical F-N chart.  Both the vertical and horizontal axes are plotted on a logarithmic scale.  In geotechnical engineering, the guidance derived from F-N charts is usually qualitative (Baecher and Christian, 2003).  Baecher (1987) presents the information in Figure 4.1 as an example of empirical probabilities of failure for civil engineering facilities.  The envelopes marked “accepted” and “marginally accepted” reflect risks inferred from the civil works shown in the figure (Baecher and Christian, 2003).  Figure 4.1 shows the expected trend that as the possibility of loss of life increases, the facility is designed for lower probabilities of failure. 

 
Figure 4.1.      Empirical rates of failures for civil engineering facilities (Modified after Baecher, 1987).

Based on a review of published information similar to that presented in Figure 4.1, Phoon, et al. (2003) indicate that the theoretical probability of failure is one (1) order of magnitude smaller than the actual rate of failure.  Using this adjustment and Figure 4.1, Phoon, et al. (2003) indicate that the currently accepted theoretical probability of failure for foundations is between 0.01% (1 in 10,000) and 0.1% (1 in 1,000), which corresponds to β values of 3.1 and 3.7, respectively, as shown in Figure 4.1.  Indeed, the AASHTO-LRFD approach is based on this range of β-values as noted below:

• For strength limit states, the resistance factors for structural design of substructure components have been derived to produce a β ≈ 3.5 (or P_f  ≈ 1 in 5,000 using normal distribution curve in Figure 2.2).
• For strength limit states, the resistance factors for geotechnical design of substructure components have been derived to produce a β ≈ 3.0 (or P_f  ≈ 1 in 1,000 using the normal distribution curve in Figure 2.2).  This is primarily based on comparison with past geotechnical design practice and the redundancy that is usually present in foundation design.

The above values of β are based on the strength limit state only (other limit states have not yet been calibrated based on probability).  Furthermore, these values are at the component level and do not take into account the redundancy of the components in the total system.  For example, if one pile in a group of 20 at a given substructure element fails, does that mean that the substructure element or the bridge it supports would fail?   This may be unlikely because traditionally all piles in the group are designed to carry the load assumed to be applied to the most heavily loaded pile and thus some redundancy will usually be present in substructure design. 

4.1 Consideration of Redundancy in Substructure Design

AASHTO (2007) provides resistance factors based on the redundancy within the substructure element.  While redundancy within the substructure element is considered, at the current time AASHTO guidance does not include any consideration for redundancy at the system level, e.g., redundancy in the bridge structure for which the substructure element is being designed.  When the stability of a system as a whole is considered, the reliability index of the system is often much larger than that at the component level.   A discussion on system reliability is outside the scope of this article and interested readers are referred to Haldar and Mahadevan (2000) for more information on this topic.

Figure 4.2 provides some guidance to assess the redundancy of a piles or shafts in a deep foundation system.  With respect to deep foundations, Paikowsky, et al. (2004) proposed the following guidelines to assess the β-value based on the minimum number of piles or shafts in a group:

• For 5 or more piles or shafts in a group, use β = 2.3 (or P_f  ≈ 1 in 100 using normal distribution curve in Figure 2.2).
• For less than 5 piles or shafts in a group, use β = 3.0 (or P_f  ≈ 1 in 1,000 using normal distribution curve in Figure 2.2).

For the case of a single shaft foundation supporting an entire bridge pier, Allen et al. (2005) recommend the use β ≈ 3.54 (or P_f  ≈ 1 in 5,000 using normal distribution curve in Figure 2.2). The β values cited above are at the component level.  As noted earlier, the β value of the whole system may be considerably higher than that for the component level where redundancy is present.  For example, as noted earlier, a value of β = 3.54 is commonly used for strength limit state evaluation of structural components such as steel girders and prestressed concrete girders.  In comparison at the system level, i.e., for the whole girder bridge, Allen et al. (2005) note that one can have β > 5.5 which is equivalent to P_f ≈ 1 in 50 million based on Figure 2.1.  This is because of the effect of redundancy at the component level is compounded into a larger redundancy at the system level. 

Figure 4.2.      Guidelines for assessing redundancy of deep foundation elements (Paikowsky, et al., 2004).

In general, the resistance factors in AASHTO (2007) for deep foundations have been calibrated for β = 3.0 (or P_f  ≈ 1 in 1,000 assuming normal distribution of limit state condition).  For non-redundant systems, e.g., single shafts, AASHTO (2007) recommends reducing the resistance factor by 20% to account for the recommended β value of 3.54.  AASHTO (2007) provides similar additional recommendations on modification of resistance factor for driven pile foundations based on the number of piles as well as site variability.  The final selection of a target reliability index, β_T, for a given limit state should take into account the importance of the structure, reliability at both the component and system level as well as the redundancy in the system.  The facility owners should carefully evaluate these issues while selecting the target reliability index, β_T.  As part of this process, the facility owners should retain the services of knowledgeable structural and geotechnical specialists and ensure a close interaction between these specialists as discussed in NCS (2007).

5.0  Comparing Factor of Safety and Reliability Index

The factors of safety commonly used in ASD range from 1.5 to 3.0.  Considering that the reliability indices shown in Figure 2.1 or Figure 4.1 appear to be in the same range as values of commonly used factors of safety, it is often tempting to compare the two directly.  This temptation should be avoided because the two values are based on very different assumptions.  For example, the concept of factor of safety is subjective while the concept of reliability index is based on the theory of probability where the data are statistically determined.  Thus, any similarity between values of FS and β is purely coincidental.

6.0  References

AASHTO (2007). AASHTO LRFD Bridge Design Specifications, 4th Edition.  American Association of State Highway and Transportation Officials, Washington, D.C.

AASHTO (2002). Standard Specifications for Highway Bridges, 17th Edition.  American Association of State Highway and Transportation Officials, Washington, D.C.

Allen, T. M., Nowak, A. And Bathurst, R, J. (2005). Calibration to Determine Load and Resistance Factors for Geotechnical and Structural Design.  Transportation Research Circular E-C079, Transportation Research Board of the National Academies, Washington, D.C., 83 p.

Baecher, G. B. (1987). “Geotechnical Risk Analysis User’s Guide,” Report No. FHWA/RD-87-011, Federal Highway Administration, McLean, VA.

Baecher, G. B. and Christian, J. T. (2003). Reliability and Statistics in Geotechnical Engineering.  John Wiley & Sons, Inc., 605 p.

Haldar, A. and Mahadevan, S. (2000). Probability, Reliability and Statistical Methods in Engineering Design.  John Wiley & Sons, Inc., 304 p.

Law, A. M. and Kelton, W. D. (1991). Simulation Modeling and Analysis, 2nd Edition, McGraw-Hill, Inc., 759 p.

Leonards, G. A. (1982). “Investigation of Failures,” – Sixteenth Terzaghi Lecture, Journal of Geotechnical Engineering Division, ASCE, 108 (GT2), pp. 185-246.

NCS (2006). “A Primer on LRFD,” www.ncsconsultants.com.

NCS (2007). “LRFD for Substructures – Limit States and Interaction between Structural and Geotechnical Specialists,” Article # 01-0507-R0, available on GeoPrac.net or in PDF format at www.ncsconsultants.com.

Paikowsky, S. G., Birigsson, B., McVay, M., Nguyen, T., Kuo, V., Baecher G., Ayyub, B., Stenersen, K, O’Malley, K., Chernauskas, L., and O’Neill, M. (2004). Load and Resistance Factor Design (LRFD) for Deep Foundations. NCHRP Report 507, Transportation Research Board of the National Academies, Washington, D.C., 126 p.

Phoon, K-K, Kulhawy, F. H., Grigoriu, M. D. (2003). “Development of a Reliability-Based Design Framework for Transmission Line Structure Foundations,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 129, No. 9, pp. 798-806.

Rosenblueth, E. and Esteva, L. (1972). Reliability Basis for Some Mexican Codes. ACI Publication SP-31, American Concrete Institute, Detroit, MI.

Whitman, R. V. (1984). “Evaluating Calculated Risk in Geotechnical Engineering,” – Seventeenth Terzaghi Lecture, Journal of Geotechnical Engineering Division, ASCE, 110 (GT2), pp. 145-188.

Withiam, J., Votyko, E., Barker R., Duncan, J. M., Kelly, B., Musser, S. and Elias, V. (1998). Load and Resistance Factor Design (LRFD) for Highway Bridge Substructures.  Report No. FHWA HI-98-032, Federal Highway Administration,

7.0  Acknowledgements

The author wishes to acknowledge the following individuals (in alphabetical order of first names) for their effort in reviewing this article and providing comments:

• Dennis R. Mertz, PhD, PE, Professor, Department of Civil and Environmental Engineering, University of Delaware, Newark, DE.
• Edward A. Nowatzki, PhD, PE, Principal Engineer, NCS Consultants, LLC, and Professor Emeritus, Department of Civil Engineering and Engineering Mechanics, Tucson, AZ.
• Jerry A. DiMaggio, PE, Principal Bridge Engineer (Geotechnical), Federal Highway Administration, Office of Bridge Technology, Washington, D.C.

8.0  Comments and Discussions

This article is part of a series of articles in which elements of the author’s continuing work on the subject of LRFD will be reported.  The author welcomes review comments and discussions and can be reached at This e-mail address is being protected from spam bots, you need JavaScript enabled to view it or This e-mail address is being protected from spam bots, you need JavaScript enabled to view it .  Depending on the comments, any given article may be revised and re-issued or a separate article containing the discussions may be issued.  Revised articles that are re-issued will be identified with a revised article number as reflected by the last digit of the article number, e.g., if an article with number ##-####-R0 is revised one time, then it will have the article number ##-####-R1.  Articles containing discussions  will be identified by the suffix D instead of R in the article number.  The readers are therefore encouraged to check for the latest versions of articles by contacting the author or visiting www.ncsconsultants.com. 

9.0  Disclaimer

This article is part of a series of articles published at www.ncsconsultants.com.  Each article presents general discussions on a discrete topic for educational purposes only and is not intended to render engineering or other professional services.  The reader is encouraged to read all previous articles pertaining to the topic of this article, including revisions, in order to obtain a better understanding of the contents of this article.  Use of this or other articles in whole or in part for design, construction or any other application is at the sole risk of the user.  If such engineering or professional services are required, the assistance of an appropriate professional should be sought.  The author or NCS Consultants, LLC, shall not be responsible or liable in any way for any errors, omissions, or damages arising out of the use of the information provided in this article.

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