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43 15 0.18 14 13 0.16 12 log-normal distribution 0.14 11 mlnx = -0.260 10 lnx = 0.502 0.12 Number of Pile-Cases 9 Relative Frequency 8 0.1 normal distribution 7 0.08 6 mx = 0.868 x = 0.416 5 0.06 4 0.04 3 2 0.02 1 0 0 0 0.5 1 1.5 2 2.5 3 KSX = Ratio of Static Load Test Results over the Pile Capacity Prediction using the -API/Nordlund/Thurman design method Figure 25. Histogram and frequency distributions of Ksx for 80 cases of concrete piles in mixed soil. recommendations based on these estimates are presented in loads. Each tested shaft is categorized as either "good" or section 3.4.3. "defective." If no more than c of the n tested shafts are "defective," the set of shafts is accepted. The test parameter, c, is usually a small number. 3.3.4 Testing Drilled Shafts for Major Defects Suppose that the set of N actual shafts includes m shafts 3.3.4.1 Overview with major defects. The fraction defective is denoted, p = m/N. Among samples of n tested shafts, the frequency distri- Drilled shafts require in-field casting and are subject to bution of the number of defective tested shafts, c, is of the defects (especially when unlined in cohesionless soils). Accep- hypergeometric form, tance sampling is used to assess whether an adequate major- ity of a set of shafts is free of major defects. CnN-- m m c Cc fc (c | n, N , m) = (35) CnN 3.3.4.2 Statistical Background in which fc (c | n, N, m) is the frequency distribution, c is the A sample of n from N shafts is tested to identify major number of defective test results within the sample, m is the q defects. Major defects are defined as any defect that signifi- number of defectives in the entire set of N shafts, and C k is cantly compromises the ability of the shaft to carry the assigned the number of combinations of k out of q things.

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44 10 log-normal distribution 0.14 9 mlnx = -0.333 lnx = 0.485 8 0.12 7 0.1 Number of Pile-Cases 6 Relative Frequency normal distribution 0.08 5 4 mx = 0.796 0.06 3 x = 0.348 0.04 2 0.02 1 0 0 0 0.5 1 1.5 2 2.5 3 KSX = Ratio of Static Load Test Results over the Pile Capacity Prediction using the -API/Nordlund/Thurman design method Figure 26. Histogram and frequency distributions of Ksx for 66 cases of pipe and H pile types in mixed soil. For n/N less than about 10%, this frequency distribution that no more than two are defective, (c = 2). This is a very can be reasonably approximated by the more easily calcu- large number of tests, but as can be seen from the nomo- lated binomial distribution, graph, decreasing the tolerable percent defective from the owner's perspective or reducing either the owner's or con- n! tractor's risk, only increases the number of shafts, n, that fc (c | p, n) = n-c p c (1 - p) c! (n - c)! (36) must be tested. This calculation assumes that n/N is less than about 10%, in which p = m /N is the fraction defective (Figure 44). but the conclusion that large sample sizes, n, are required also holds for the case of a larger sampling fraction. Per- forming an iterative solution on the hypergeometric model 3.3.4.3 Sample Calculation for the same case as above, but assuming a finite N = 100, yields a sample size of about 80. Presume that the maximum fraction of shafts with a major defect that the owner is willing to tolerate in a large set of N shafts is 5% and that the owner's risk of incorrectly accept- 3.3.4.4 Conclusion ing a set of shafts with greater than 5% defects is set at = 0.10. Let the contractor's risk of rejecting a set of N shafts The conclusion to be drawn from these simple calculations with no more than, say, 1% defects be set at = 0.10. is that, in order to statistically ensure very low rates of major From the nomograph in Figure 44 (see insert), the assur- defects within a set of drilled shafts, a very high proportion of ance sampling plan is to test n = 110 of the shafts and require the shafts must be tested. Thus, it seems reasonable practically

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45 Dynamic Analysis Construction Design WEAP Dynamic No Dynamic Measurements Measurements Drivability Resistance/ Capacity Dynamic WEAP Equations Pile Stress GRL Analysis EOD Default ENR Equation Gates Equation FHWA Mod. Gates WEAP Analysis GTR 1.602 0.910 1.787 0.848 0.940 0.472 1.656 1.199 WEAP / Dynamic No. = 384 No. = 384 No. = 384 No. = 99 Measurements EOD BOR (last) Load Factor 1.073 0.573 0.833 0.403 No. = 135 No. = 159 < 16 BP10cm 16 BP10cm < 16 BP10cm 16 BP10cm 1.306 0.643 0.929 0.688 0.876 0.419 0.809 0.290 No. = 62 No. = 73 No. = 32 No. = 127 Signal Matching Field Evaluation (CAPWAP) Energy Approach 1.368 0.620 0.894 0367 No. = 377 No. = 371 EOD BOR (last) EOD BOR (last) 1.626 0.797 1.158 0.393 1.084 0.431 0.785 0.290 No. = 125 No. = 162 No. = 128 No. = 153 < 16 BP10cm 16 BP10cm < 16 BP10cm 16 BP10cm < 16 BP10cm 16 BP10cm < 16 BP10cm 16 BP10cm 1.843 0.831 1.460 0.734 1.176 0.530 1.153 0.354 1.227 0.474 0.972 0.359 0.830 0.352 0.775 0.274 No. 54 No. = 71 No. = 32 No. = 130 No. = 56 No. = 72 No. = 29 No. = 124 AR < 350 AR 350 AR < 350 AR 350 AR < 350 AR 350 AR < 350 AR 350 2.589 1.929 1.116 1.308 1.431 1.422 0.764 0.954 2.385 0.698 0.362 0.796 0.727 0.888 0.318 0.396 No. = 37 No. = 22 No. = 22 No. = 10 No. = 39 No. = 23 No. = 19 No. = 10 AR < 350 AR 350 AR < 350 AR 350 AR < 350 AR 350 AR < 350 AR 350 1.717 1.181 1.178 1.110 1.054 0.926 0.736 0.851 0.841 0.468 0.379 0.303 0.459 0.320 0.249 0.305 No. = 37 No. = 34 No. = 83 No. = 47 No. = 39 No. = 34 No. = 82 No. = 42 *All values represent the ratio of the static capacity based on Davisson's failure criterion over the dynamic methods prediction, mean 1 S.D. Figure 27. Statistical parameters of a normal distribution for the various dynamic analyses (applied to PD/LT2000 database) grouped by the controlling parameters.

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46 TABLE 17 The performance of the dynamic methods: statistical summary and resistance factors Resistance Factors for a given No. of Standard Method Time of Driving Mean COV Reliability Index, Cases Deviation 2.0 2.5 3.0 General 377 1.368 0.620 0.453 0.68 0.54 0.43 EOD 125 1.626 0.797 0.490 0.75 0.59 0.46 Dynamic Measurements CAPWAP EOD - AR < 350 & 37 2.589 2.385 0.921 0.52 0.35 0.23 Bl. Ct. < 16 BP10cm BOR 162 1.158 0.393 0.339 0.73 0.61 0.51 General 371 0.894 0.367 0.411 0.48 0.39 0.32 EOD 128 1.084 0.431 0.398 0.60 0.49 0.40 Energy Approach EOD - AR < 350 & 39 1.431 0.727 0.508 0.63 0.49 0.39 Bl. Ct. < 16 BP10cm BOR 153 0.785 0.290 0.369 0.46 0.38 0.32 ENR General 384 1.602 1.458 0.910 0.33 0.22 0.15 Gates General 384 1.787 0.848 0.475 0.85 0.67 0.53 Equations Dynamic General 384 0.940 0.472 0.502 0.42 0.33 0.26 FHWA modified EOD 135 1.073 0.573 0.534 0.45 0.35 0.27 Gates EOD 62 1.306 0.643 0.492 0.60 0.47 0.37 Bl. Ct. < 16BP10cm WEAP EOD 99 1.656 1.199 0.724 0.48 0.34 0.25 Notes: EOD = End of Driving; BOR = Beginning of Restrike; AR = Area Ratio; Bl. Ct. = Blow Count; ENR = Engineering News Record Equation; BP10cm = Blows per 10cm; COV = Coefficient of Variation; Mean = ratio of the static load test results (Davisson's Criterion) to the predicted capacity = KSX = =bias 60 0.15 55 55 0.14 log-normal distribution 0.14 50 50 0.13 mlnx = -0.187 0.13 lnx = 0.379 45 0.12 45 0.12 0.11 mx = 0.894 40 40 0.11 log-normal 0.1 Number of Pile-Cases 0.1 Relative Frequency distribution 35 35 Number of Pile-Cases 0.09 normal distribution Relative Frequency mlnx = 0.233 0.09 30 lnx = 0.387 0.08 30 0.08 mx = 1.368 0.07 25 x = 0.367 0.07 25 normal distribution 0.06 0.06 20 20 0.05 0.05 x = 0.620 15 0.04 15 0.04 0.03 0.03 10 10 0.02 0.02 5 5 0.01 0.01 0 0 0 0 0 0.5 1 1.5 2 2.5 >3 0 0.5 1 1.5 2 2.5 3 Ratio of Static Load Test Results over the Pile Ratio of Static Load Test Results over the Pile Capacity Capacity Prediction using the CAPWAP method Prediction using the Energy Approach method Figure 28. Histogram and frequency distributions for all Figure 29. Histogram and frequency distributions for all (377) CAPWAP pile-cases in PD/LT2000. (371) Energy Approach pile-cases in PD/LT2000.