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30 member is one for which failure will directly affect the ele- ficient of variation of 0.4 and target reliability values of 2.0, ment carried by it (i.e., the column) with limited or no ability 2.5, and 3.0 presented in Figure 14, suggest very little sensi- of other foundations supporting the same element to mitigate tivity of the resistance factors to the DL to LL ratio. A simi- the effect of the failure of the member. Referring to Figure 13, lar trend was observed using DL to LL ratio of 10. The large one can intuitively see that, as three points define a plane, a dead-to-live-load ratios represent conditions of bridge con- failure of any deep foundation element in such a configura- struction, typically associated with very long bridge spans. tion cannot be mitigated by the others. Though details of the The relatively small influence of the dead-to-live-load ratio foundation scheme are important--see, e.g., Foundation on the calculated resistance factors suggests that (1) the use Design Standards in the World (Japanese Geotechnical Soci- of a DL to LL ratio of 2 or 2.5 as a typical value is reason- ety, 1998)--one can distinguish between a 5-member scheme able, and (2) the obtained factors are, by and large, applica- (clearly redundant) and a 3- or fewer member scheme (non- ble for long span bridges. redundant) for the purpose of establishing a target reliability. The evaluation of the resistance factors in the present study was originally carried out by using reliability indices 2.8.2 Parameter Study--The Limited Meaning of 2.0, 2.5, and 3.0 associated with pf = 2.28%, 0.62%, and of the Resistance Factor Value 0.14%, respectively. This approach provided a reasonable The use of FORM requires an iterative process and hence range of values to investigate before the final target reliabil- a parametric study more easily obtained by using the FOSM ity values were set. relationships, assuming the results of both are within a close range (to be demonstrated in section 3.2.2). Figure 15 pre- Recommended Concept and Targets sents such relations using Equation 10, the chosen load dis- tribution parameters (Equations 25 and 26), DL to LL ratio of Based on the review of the state of the art, the survey of 2.5 and a target reliability = 2.33 (see section The common practice, and the evaluation of the above values, the obtained relationship shows that a perfect prediction ( = 1, following reliability indices and probability of failure were COV = 0) would result with a resistance factor of ( = 0.80. developed and are recommended in conjunction with meth- With a prediction method for which the bias is one but the ods for capacity evaluation of single piles (see Figure 13): distribution is greater (COV > 0), the resistance factor would sharply decrease so that for COV = 0.4 the resistance factor 1. For redundant piles, defined as 5 or more piles per pile would reduce to = 0.44. The influence of the bias of the cap, the recommended probability of failure is pf = 1%, method (, or mean ratio of measured over predicted) on the corresponding to a target reliability index of = 2.33. resistance factor is equally important. As seen in the figure, 2. For nonredundant piles, defined as 4 or fewer piles per an under predictive method ( > 1) has a "built in" safety and pile cap, the recommended probability of failure is pf = hence a higher resistance factor is used in order to achieve 0.1%, corresponding to a reliability index of = 3.00. the same target reliability as would be obtained by using a method which predicts, on average, more accurately ( 1). For example, for methods having the same distribution 2.8 INVESTIGATION OF (COV = 0.4), an underpredictive method with a bias of = THE RESISTANCE FACTORS 1.5 would result in a resistance factor = 0.67, whereas a 2.8.1 Initial Resistance Factors Calculations method with a bias = 1.0 would result in = 0.44. Although The factors were evaluated using FORM (First Order Reli- 0.8 ability Method) with dead load (DL) to live load (LL) ratios ranging from 1 to 4. The results for a bias of one and a coef- General Case Bias = 1 COV = 0.4 Resistance Factor, 0.6 = 2.33 = 2.0 = 3.00 Pf = 0.1% Pf = 1.0% Pf= 2.28% = 2.5 0.4 pf= 0.62% = 3.0 Pf= 0.14% Logically Redundant Non-Redundant Non - Redundant 0 1 2 3 4 5 DL/LL - Dead to Live Load Ratio Figure 13. Redundant vs. non-redundant pile support and the current research recommendations of target Figure 14. Calculated resistance factors for a general reliability. case showing the influence of the dead-to-live-load ratio.