The quality of in situ data is key to calculating resistance factor of bored piles. However, it is difficult to summarize accuracy data due to various uncertainties in engineering. This paper employs the Bayesian method and mathematical statistics theory to put forward an estimation method for updating in situ data. A testing database (33 tests in noncohesive soils and 53 tests in cohesive soils) of bored piles is summarized. The model factor of bored piles is quantified as the ratio of the measured capacity to the calculated capacity. The proposed method is used to classify summarized data into three categories, which are “good data,” “general data,” and “bad data.” The “bad data” are discarded because of bad contribution to calculation, and Bayesian theory is incorporated into updating the model factor statistics. Three methods are used to calculate the reliability index and resistance factor of bored piles, and the results show that the reliability index and resistance factor are sensitive to the quality of data. Finally, the available values of resistance factors are proposed based on resistance factor design for bridge design specification, which can offer references to revision relevant specifications. The proposed method can be used to update other geotechnical data.

Bored piles, especially large-diameter piles, are commonly employed to support high-rise buildings and bridges in China and other countries because of their ability to sustain large load [

The resistance factor is calculated incorporating in reliability analysis methods based on pile load test data [

Parameter uncertainty and model uncertainty are two troubles for pile foundation designers. Numerous investigations are conducted to study parameter uncertainty, which shows that parameter uncertainty contains random error of monitor, system error, statistical uncertainty, and so on. Model uncertainty is mainly caused by simplified calculation model. European design specification of geotechnical engineering (EN997-1) clearly suggests that a model revised factor should be incorporated when the pile capacity is calculated using the simplification model [

This paper puts forward a Bayesian estimation method to update the in situ data of bored piles, and three reliability index calculation methods are incorporated into calculating the reliability index of pile capacity using the processing data. Then American LRFD for Bridge Design Specification is used to estimate the resistance factor of bored piles.

Bayesian principle is a tool to update the probability distribution using new information. Assuming that the prior distribution of a random variable (

Assume that

Assume that

If

The model factor is frequently represented as the ratio of the measured capacity to calculated capacity [_{m} is the measured pile capacity; and _{p} is the calculated pile capacity. Numerous investigations show that model factor is a random variable and obeys log-normal distribution [

To improve accuracy of collected data from engineering, this paper employs the biased factor of _{i} is the _{R} is the mean of

Based on equation (

If

If

If

“Good data” are considered to be more reliable and should be treated as the prior information in estimation of the population statistics. However, the sample size of the “good data” is not sufficient to represent the total. This paper employs Bayesian updating technique to evaluate the probability characteristics of the resistance bias factor for bored piles. The “general data” are treated as prior information, and the “good data” are treated as likelihood information. Then, the updating model factor statistics can be obtained using equations (

This paper summarizes various bored pile capacity data shown in Tables

Load testing data of pile capacity in noncohesive soil.

Case number | _{s} (mm) | _{B} (mm) | _{p} (kN) | _{m} (kN) | ||||
---|---|---|---|---|---|---|---|---|

1 | 430 | 430 | 8 | 0.42 | 1321 | 1375 | 0.06 | 0.99 |

2 | 600 | 750 | 9 | 0.76 | 3783 | 4600 | 0.74 | 0.94 |

3 | 750 | 600 | 11 | 1.2 | 2705 | 3000 | 0.40 | 0.97 |

4 | 360 | 350 | 7.8 | 35 | 1084 | 1050 | 7.98 | 0.61 |

5 | 400 | 400 | 9.5 | 5.6 | 1202 | 1357 | 1.40 | 0.89 |

6 | 400 | 400 | 9.5 | 10.1 | 1202 | 1380 | 1.45 | 0.92 |

7 | 400 | 400 | 8 | 6 | 1131 | 1050 | 0.80 | 0.92 |

8 | 400 | 400 | 9.5 | 5 | 1202 | 1225 | 1.97 | 0.84 |

9 | 400 | 400 | 3 | 11 | 1178 | 840 | 3.70 | 0.72 |

10 | 520 | 520 | 16.5 | 12.7 | 6754 | 5600 | 3.91 | 0.86 |

11 | 430 | 430 | 11.5 | 3.41 | 1287 | 1250 | 0.75 | 0.94 |

12 | 450 | 450 | 9 | 0.97 | 1145 | 1500 | 2.08 | 0.80 |

13 | 400 | 400 | 10 | 50 | 940 | 970 | 3.88 | 0.74 |

14 | 400 | 400 | 7 | 70 | 679 | 540 | 2.89 | 0.73 |

15 | 500 | 500 | 7.8 | 75 | 1226 | 600 | 10.93 | 0.46 |

16 | 500 | 500 | 10 | 77 | 1583 | 1175 | 5.29 | 0.59 |

17 | 400 | 400 | 11 | 41 | 605 | 800 | 5.92 | 0.72 |

18 | 400 | 400 | 9.2 | 57 | 472 | 480 | 7.87 | 0.58 |

19 | 500 | 500 | 9.5 | 60 | 793 | 625 | 8.13 | 0.44 |

20 | 500 | 500 | 11.8 | 62 | 1060 | 1025 | 10.46 | 0.51 |

21 | 500 | 500 | 12.3 | 60 | 916 | 1160 | 4.29 | 0.70 |

22 | 500 | 500 | 14.5 | 80 | 1132 | 1500 | 3.90 | 0.75 |

23 | 305 | 305 | 13 | 40.3 | 2028 | 2010 | 8.44 | 0.50 |

24 | 305 | 305 | 13 | 6.4 | 3051 | 6000 | 4.80 | 0.60 |

25 | 305 | 305 | 13 | 97 | 2028 | 1200 | 1.30 | 0.53 |

26 | 305 | 305 | 13 | 18.3 | 3051 | 4650 | 1.13 | 0.97 |

27 | 520 | 760 | 12.2 | 50 | 2563 | 1690 | 2.97 | 0.83 |

28 | 520 | 760 | 12.2 | 93 | 2478 | 1620 | 3.36 | 0.84 |

29 | 520 | 760 | 12 | 26.5 | 2111 | 1650 | 5.36 | 0.80 |

30 | 520 | 760 | 12 | 35 | 2451 | 2100 | 4.60 | 0.70 |

31 | 520 | 760 | 15 | 23.3 | 2974 | 2680 | 8.80 | 0.85 |

32 | 520 | 760 | 6 | 3.5 | 3644 | 3650 | 0.06 | 0.99 |

33 | 520 | 760 | 8 | 5 | 3739 | 5600 | 0.74 | 0.94 |

Load testing data of pile capacity in cohesive soil.

Case number | _{s} (mm) | _{B} (mm) | _{p} (kN) | _{m} (kN) | ||||
---|---|---|---|---|---|---|---|---|

34 | 600 | 600 | 9 | 0.59 | 4807 | 4800 | 0.48 | 0.96 |

35 | 600 | 600 | 11.5 | 2.7 | 6447 | 4100 | 2.87 | 0.82 |

36 | 750 | 750 | 21.8 | 2.34 | 6521 | 7100 | 3.55 | 0.71 |

37 | 350 | 350 | 17.3 | 0.18 | 1398 | 760 | 0.05 | 0.99 |

38 | 610 | 610 | 6.5 | 21.19 | 2062 | 2300 | 3.40 | 0.80 |

39 | 600 | 600 | 6.5 | 13.5 | 3280 | 3100 | 4.96 | 0.84 |

40 | 600 | 800 | 24 | 12 | 2545 | 3360 | 12.43 | 0.64 |

41 | 610 | 610 | 9 | 22.9 | 3040 | 2650 | 5.30 | 0.72 |

42 | 610 | 610 | 7 | 1.65 | 2235 | 1800 | 1.08 | 0.90 |

43 | 750 | 750 | 13 | 37 | 4179 | 3700 | 3.05 | 0.80 |

44 | 450 | 450 | 9 | 1.79 | 3658 | 2930 | 1.76 | 0.88 |

45 | 350 | 350 | 5 | 1.7 | 1196 | 1700 | 2.55 | 0.76 |

46 | 500 | 500 | 6 | 1.7 | 2333 | 1900 | 2.85 | 0.76 |

47 | 600 | 600 | 6 | 85.3 | 707 | 520 | 2.65 | 0.62 |

48 | 450 | 450 | 6 | 20.3 | 1225 | 1175 | 0.82 | 0.94 |

49 | 300 | 300 | 6 | 14.1 | 975 | 1080 | 1.19 | 0.86 |

50 | 600 | 600 | 9.6 | 4.4 | 3223 | 3500 | 2.92 | 0.80 |

51 | 400 | 400 | 8.7 | 2.9 | 1221 | 1240 | 0.82 | 0.93 |

52 | 350 | 350 | 8.7 | 9 | 834 | 825 | 1.76 | 0.84 |

53 | 410 | 410 | 11 | 0.95 | 1444 | 1450 | 1.45 | 0.87 |

54 | 615 | 615 | 12 | 36 | 2375 | 3000 | 4.83 | 0.75 |

55 | 615 | 615 | 12 | 35 | 1948 | 2450 | 3.92 | 0.74 |

56 | 610 | 610 | 7 | 25.4 | 1184 | 1400 | 2.43 | 0.72 |

57 | 610 | 610 | 1.5 | 17.8 | 462 | 510 | 1.00 | 0.93 |

58 | 500 | 500 | 7.8 | 5.58 | 2392 | 3600 | 2.67 | 0.84 |

59 | 430 | 430 | 6.5 | 4.49 | 907 | 1150 | 3.36 | 0.74 |

60 | 450 | 600 | 15.5 | 4.19 | 1818 | 2310 | 2.98 | 0.82 |

61 | 750 | 750 | 10.2 | 1.89 | 5869 | 8500 | 3.94 | 0.79 |

62 | 450 | 450 | 8 | 45.53 | 1292 | 1230 | 8.40 | 0.50 |

63 | 450 | 450 | 8 | 22.7 | 1292 | 1820 | 1.66 | 0.85 |

64 | 450 | 450 | 8 | 3.93 | 2110 | 2580 | 1.52 | 0.89 |

65 | 450 | 450 | 8 | 4.83 | 2110 | 2670 | 2.14 | 0.86 |

66 | 450 | 450 | 8 | 4.0 | 2110 | 2790 | 2.39 | 0.84 |

67 | 450 | 450 | 8 | 3.35 | 2110 | 2900 | 2.20 | 0.85 |

68 | 450 | 450 | 8 | 2.85 | 2110 | 4200 | 3.86 | 0.76 |

69 | 450 | 450 | 8 | 3.2 | 1602 | 2800 | 1.74 | 0.87 |

70 | 450 | 750 | 4.5 | 2.96 | 8906 | 9600 | 4.22 | 0.76 |

71 | 430 | 430 | 7 | 1.19 | 1156 | 1660 | 1.67 | 0.86 |

72 | 550 | 550 | 6 | 6.57 | 2916 | 4800 | 1.46 | 0.90 |

73 | 910 | 910 | 12 | 1.3 | 5088 | 7050 | 1.07 | 0.92 |

74 | 910 | 910 | 9 | 1.8 | 5088 | 5900 | 0.99 | 0.93 |

75 | 910 | 910 | 9 | 1.16 | 7648 | 9700 | 4.58 | 0.72 |

76 | 910 | 910 | 9 | 0.68 | 4721 | 7000 | 1.78 | 0.86 |

77 | 530 | 430 | 8 | 3.38 | 1955 | 1880 | 3.63 | 0.75 |

78 | 600 | 600 | 14.9 | 2.13 | 4230 | 5430 | 3.48 | 0.79 |

79 | 600 | 600 | 14.6 | 3.26 | 4196 | 3000 | 1.31 | 0.92 |

80 | 750 | 750 | 15.4 | 2.35 | 6153 | 6250 | 1.93 | 0.89 |

81 | 600 | 600 | 14.7 | 1.74 | 4207 | 4450 | 1.95 | 0.87 |

82 | 750 | 750 | 15.7 | 1.95 | 10172 | 13000 | 5.37 | 0.75 |

83 | 500 | 500 | 7.2 | 2.2 | 3294 | 3300 | 1.64 | 0.87 |

84 | 750 | 750 | 7.2 | 1.6 | 6266 | 4810 | 0.88 | 0.93 |

85 | 500 | 500 | 7.2 | 9.5 | 1901 | 2200 | 2.79 | 0.81 |

86 | 750 | 750 | 7.2 | 7.2 | 3515 | 5300 | 4.03 | 0.77 |

_{s} is the pile shaft diameter; _{B} is the pile bottom diameter;

Dithinde et al. [

Normalized load-settlement curves of load test data.

The scatter diagram of testing pile capacity and calculation pile capacity.

The classified results are shown in Tables

Classified results of load test data in noncohesive soil.

Good data | General data | |||||||
---|---|---|---|---|---|---|---|---|

Case number | Case number | Case number | ||||||

1 | 1.04 | 0.02 | 14 | 0.80 | 0.22 | 9 | 0.71 | 0.30 |

2 | 1.22 | 0.20 | 18 | 1.02 | 0.00 | 12 | 1.31 | 0.29 |

3 | 1.11 | 0.09 | 19 | 0.79 | 0.22 | 16 | 0.74 | 0.27 |

4 | 0.97 | 0.05 | 20 | 0.97 | 0.05 | 17 | 1.32 | 0.30 |

5 | 1.13 | 0.11 | 21 | 1.27 | 0.24 | 22 | 1.33 | 0.30 |

6 | 1.15 | 0.13 | 23 | 0.99 | 0.03 | 27 | 0.66 | 0.35 |

7 | 0.93 | 0.09 | 29 | 0.78 | 0.23 | 28 | 0.65 | 0.36 |

8 | 1.02 | 0.00 | 30 | 0.86 | 0.16 | 33 | 1.50 | 0.47 |

10 | 0.83 | 0.18 | 31 | 0.90 | 0.11 | — | — | — |

11 | 0.97 | 0.04 | 32 | 1.00 | 0.00 | — | — | — |

13 | 1.03 | 0.01 | — | — | — | — | — | — |

Classified results of load test data in cohesive soil.

Good data | General data | |||||||
---|---|---|---|---|---|---|---|---|

Case number | Case number | Case number | ||||||

34 | 1.00 | 0.13 | 60 | 1.27 | 0.11 | 35 | 0.64 | 0.45 |

36 | 1.09 | 0.05 | 62 | 0.95 | 0.17 | 42 | 0.81 | 0.30 |

38 | 1.12 | 0.03 | 63 | 1.41 | 0.23 | 44 | 0.80 | 0.30 |

39 | 0.95 | 0.18 | 64 | 1.22 | 0.06 | 46 | 0.81 | 0.29 |

40 | 1.32 | 0.15 | 65 | 1.27 | 0.10 | 47 | 0.74 | 0.36 |

41 | 0.87 | 0.24 | 66 | 1.32 | 0.15 | 58 | 1.51 | 0.31 |

43 | 0.89 | 0.23 | 67 | 1.37 | 0.20 | 61 | 1.45 | 0.26 |

45 | 1.42 | 0.24 | 70 | 1.08 | 0.06 | 72 | 1.65 | 0.43 |

48 | 0.96 | 0.17 | 71 | 1.44 | 0.25 | 76 | 1.48 | 0.29 |

49 | 1.11 | 0.04 | 73 | 1.39 | 0.21 | 79 | 0.71 | 0.38 |

50 | 1.09 | 0.06 | 74 | 1.16 | 0.01 | 84 | 0.77 | 0.33 |

51 | 1.02 | 0.12 | 75 | 1.27 | 0.10 | 86 | 1.51 | 0.31 |

52 | 0.99 | 0.14 | 77 | 0.96 | 0.16 | — | — | — |

53 | 1.00 | 0.13 | 78 | 1.28 | 0.12 | — | — | — |

54 | 1.26 | 0.10 | 80 | 1.02 | 0.12 | — | — | — |

55 | 1.26 | 0.09 | 81 | 1.06 | 0.08 | — | — | — |

56 | 1.18 | 0.03 | 82 | 1.29 | 0.11 | — | — | — |

57 | 1.10 | 0.04 | 83 | 1.00 | 0.13 | — | — | — |

59 | 1.27 | 0.10 | 85 | 1.16 | 0.01 | — | — | — |

Log-normal distribution is used as the distribution of model factor, and the model factor statistics are presented in terms of the mean and coefficient of variation (COV). Based on equations (

The updating model factor for pile capacity.

Model factor | ||||||||
---|---|---|---|---|---|---|---|---|

Soil type | All data | Good data | General data | Updating data | ||||

Mean | COV | Mean | COV | Mean | COV | Mean | COV | |

Noncohesive soil | 1.000 | 0.217 | 0.990 | 0.138 | 1.028 | 0.357 | 0.987 | 0.129 |

Cohesive soil | 1.134 | 0.210 | 1.153 | 0.140 | 1.073 | 0. 372 | 1.134 | 0.132 |

According to reliability theory, the limit state equation of bored pile capacity is [

If the three parameters in equation (

The limit-state function is linear at a point on the failure surface; its performance function is [

All the parameters in equation (

Monte Carlo simulation method is an accuracy method to calculate reliability index, which is employed for comparison with the accuracy of other calculation methods. Its performance function is

The calculation can be carried out using MATLAB software; the times of simulation are 10 million, described as

The values of

Figures

Reliability index in noncohesive soil. (a) First-order reliability method. (b) Design point method. (c) Monte Carlo simulation method.

Reliability index in cohesive soil. (a) First-order reliability method. (b) Design point method. (c) Monte Carlo simulation method.

The formula of load and resistance factor design method is [_{n} is standard value of resistance (kN); _{i} is standard value of load (kN); _{i} is load factor.

Reliability analysis is the bias of resistance factor calculation. Load and resistance factor design method proposes the calculation formula shown in equation (

If the reliability index is calculated using equation

The resistance factor is described as

If the reliability index is calculated using equation (

The resistance factor is described as

2.0, 2.5, and 3.0 are selected as the target reliability index. Based on equations (

Calculation results of resistance factor.

Target reliability, _{T} | Noncohesive soil | Cohesive soil | ||||
---|---|---|---|---|---|---|

All data | ||||||

2.0 | 0.440 | 0.452 | 0.450 | 0.461 | 0.476 | 0.475 |

2.5 | 0.375 | 0.389 | 0.389 | 0.387 | 0.396 | 0.394 |

3.0 | 0.336 | 0.354 | 0.353 | 0.344 | 0.362 | 0.360 |

General data | ||||||

2.0 | 0.437 | 0.446 | 0.445 | 0.443 | 0.450 | 0.450 |

2.5 | 0.371 | 0.382 | 0.380 | 0.375 | 0.385 | 0.384 |

3.0 | 0.333 | 0.350 | 0.350 | 0.335 | 0.347 | 0.345 |

Good data | ||||||

2.0 | 0.440 | 0.446 | 0.445 | 0.467 | 0.475 | 0.474 |

2.5 | 0.376 | 0.385 | 0.385 | 0.391 | 0.399 | 0.399 |

3.0 | 0.337 | 0.348 | 0.347 | 0.346 | 0.356 | 0.354 |

Updating data | ||||||

2.0 | 0.440 | 0.449 | 0.448 | 0.463 | 0.472 | 0.471 |

2.5 | 0.376 | 0.385 | 0.384 | 0.390 | 0.400 | 0.400 |

3.0 | 0.337 | 0.348 | 0.348 | 0.345 | 0.358 | 0.356 |

The quality of data has distinct contribution to resistance factor of bored piles. The accuracies of design point method and Monte Carlo simulation method are satisfactory, which can be considered as the criterion to verify the accuracy of proposed method. The results based on two methods are larger than the results based on first-order reliability method, and the difference are 6.9% and 18.3%, respectively. Meanwhile, the difference between the two methods is near 0. The accuracies based on “good data” and “updating data” are better than the accuracies based on “general data” and “all data.”

In summary, Table

Recommended values of resistance factors.

_{T} | Noncohesive soil | Cohesive soil |
---|---|---|

2.0 | 0.456 | 0.473 |

2.5 | 0.386 | 0.400 |

3.0 | 0.347 | 0.356 |

From this study, some conclusions are presented:

The proposed method incorporating probability theory and Bayesian method can not only classify the in situ data but also overcome the deficiency caused by small sample for accuracy data.

Data classification has significant contribution to reliability index and resistance factor. The results according to “good data” and “updating data” are larger than the results according to “general data” and “all data.” Meanwhile the difference of results using two types of data is near 0. Therefore, “good data” and “updating data” can be used as the basis of resistance factor calculation.

Reliability index and resistance factor are sensitive to the type of soil, and the calculation results in cohesive soil are larger than the results in noncohesive soil.

The recommended values are proposed only according to the calculation results and American LRFD for Bridge Design Specification. Its application in engineering fact needs to be further studied. However, the proposed method can be used to update other geotechnical data.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors express their gratitude to the National Natural Science Foundation of China (no. 51978247) and Key Science and Technology Projects of Henan Province (no. 202102310242).