4.0 Selection of Target Reliability Index, βT
Identification of the target reliability index, βT, 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 Pf 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 â€œF-Nâ€ 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 â€œacceptedâ€ and â€œmarginally acceptedâ€ 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 Pf ≈ 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 Pf ≈ 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 Pf ≈ 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 Pf ≈ 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 Pf ≈ 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 Pf ≈ 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 Pf ≈ 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, βT, 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, βT. 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).