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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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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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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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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.