Reliability Analysis of Shaft Resistance of Axially Loaded Bored Piles

Reliability Analysis of Shaft Resistance of Axially

Loaded Bored Piles

Jae Hyun PARK, Chul Soo PARK, and Moonkyung CHUNG

Geotechnical Engineering Research Institute, Korea Institute of Civil Engineering and Building Technology, Republic of Korea

Abstract. As part of Load and Resistance Factor Design (LRFD) implementation effort in Korea, reliability levels for side resistance of bored piles socketed in weak rock were evaluated by using bi-directional pile load test results. Bi-directional pile load tests are widely used to predict the load–settlement behavior of large-diameter bored piles. A database of 24 bidirectional load test data sets and soil property tests was compiled from case studies collected from Korea. The resistance bias factors for the three static design equations[O’Neill and Reese (1999), Horvath and Kenney (1979), and Williams et al. (1980)] were calculated by comparing the representative measured bearing capacities defined by Davisson’s criterion with the predicted values. Reliability analysis was conducted following the First-Order Reliability Method (FORM) using the statistics of bias factors. The reliability levels of the three shaft resistance equations evaluated from FORM method for safety factors in the range of 2.0-5.0 were in the ranges of 1.15-2.79 for O’Neill & Reese equation, 0.97-2.62 for Horvath and Kenney equation, and 1.73-3.38 for Williams et al. equation, respectively.

Keywords. Reliability index, shaft resistance, bored pile, FORM

1. Introduction

The use of drilled shafts is increasing due to its excellence, economic efficiency, and constructability in vertical and lateral behaviors according to the trend of enlargement and complex of structures, and especially, there are increasing applications as the foundation of long span sea bridge. In this study, a reliability analysis was conducted on the shaft resistance of axially loaded drilled shafts where bi-directional pile load test were carried out in order to develop LRFD design method. The pile head settlement, generally, in bi-directional load tests are calculated by the summation of the side shear load̽ displacement curve resulting from the upward movement of the top of the O-cell and the end bearing load ̽ displacement curve obtained from the downward movement of the bottom of the O-cell (LOADTEST, 2000).

An analysis was conducted on the distribution of shaft resistance regarding three representative bearing capacity equations by collecting and analyzing data from 24 bi-directional pile load tests and geotechnical investigations on drilled shafts of 16 sites in Korea. Evaluation on the reliability of each

bearing capacity equation was conducted by first order reliability method (FORM).

2. Shaft Resistance Bias Factors

The selected test data represent general practice of large-size drilled shaft in Korea. The diameter of the test piles was in the range of 800 mm to 3000 mm. Nine of the test piles were less than 30m long and 15 were equal to or more than 30m long.

The specification of the drilled shaft for the bi-directional load test and the results of the geotechnical site investigation collected from this study is shown in Table 1. The test piles were embedded into weak rock within the range of 1.0-10.5m. All of these test piles were driven into mixed soil layers of sand, gravel, silt, and clay, which is common soil profile for deep foundations used in Korea.

Unconfined compressive strength was computed and applied to estimate ultimate unit shaft resistance by taking a weighted average of the measured value at each depth regarding total weak rock layer. The maximum equivalent top-loaded test loads ranged from 20 MPa to 80MPa,

© 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-300

and the ratio of maximum equivalent top-loaded test load over allowable pile load was in the range of 2.0~2.9. Information about tests is summarized in Table 1.

Table 1. Selected pile load tests information no. Pile length (m) Pile diameter (m) Rock socketed depth(m) UCS (MPa) 1 27.1 0.8 2.0 20 2 27.1 0.8 1.5 52 3 26.1 1.5 2.0 20 4 26.1 1.5 1.0 52 5 28.7 0.8 1.4 20 6 28.7 0.8 1.4 52 7 30.5 2.0 6.7 27 8 20.0 1.0 3.5 27 9 22.0 1.0 2.5 52 10 9.0 0.8 1.5 10 11 34.0 1.5 5.5 27 12 30.5 1.5 4.0 27 13 33.5 2.0 5.5 27 14 50.0 2.0 10.0 10 15 50.0 2.0 10.0 18 16 50.0 2.0 5.0 10 17 50.0 2.0 10.0 18 18 39.0 2.5 6.7 11 19 36.2 2.0 6.5 18 20 33.4 1.5 6.2 15 21 36.4 2.0 6.5 53 22 55.4 3.0 10.5 17 23 51.2 2.5 4.7 46 24 52.3 2.5 5.1 97

2.1. Measured Shaft Resistances

With respect to the results from the bi-directional load test, the equivalent top-loaded load-settlement curve was established by an approximate solution (LOADTEST, 2000) using the pattern of developed side shear stress. The equivalent top-loaded curve of bi-directional pile load test is known to have relatively smaller settlements than the measured value for the result of conventional top-down static pile load test (Seol and Jeong, 2009).

Load test data were compiled to determine the measured values of the unit side resistance. Unit side resistance was determined in two steps: (a) the average side resistance derived from the load test results, and (b) the unit side resistance in each soil section from the strain gauge data. Stain gauge measurements were used to calculate the load transfer distribution along the shafts. And the unit skin friction-displacement curve (f-w curve) (f-were prepared for each stratum

(weathered rock, soft rock). Within the same rock layer, by taking a weighted average of the maximum unit skin friction measured for each depth according to the measured length, one representative measured unit skin friction was calculated. It was because when skin friction is calculated by segmenting the layers by measured depth within the same rock layer and its uncertainty is evaluated, the reliability may appear high since the variation is calculated smaller than when skin friction is calculated regarding the overall depth and its uncertainty is evaluated. The maximum measured unit skin friction applied the friction value when yield load of the skin occurred.

The Davisson criterion (1972) was chosen to determine the nominal shaft resistance from the load test. The selection of this criterion was based on the previous study performed by KICT (2008), in which four criteria [Davisson(1972), FHWA 0.05D (O’Neill and Reese, 1999), ASCE 20-96 (ASCE, 1997), and DeBeer(1970)] were used to estimate the nominal resistance of drilled shaft based on the static pile load test data. The Davisson criterion (1972) defines the failure load of a pile as the load corresponding to the settlement that exceeds the elastic compression of the pile, by and offset, equal to 3.81ੈ plus a factor equal to the diameter of the pile divided by 60 for the diameter of the pile equal to or more than 600 ੈ . The equivalent top-loaded load-settlement curves of the 24 tests were evaluated by the Davisson criterion, and the results are summarized in Table 2.

Table 2. Measured and predicted shaft resistances no. Measured (kPa) O&R* (kPa) H&K** (kPa) Williams*** (kPa) 1 716 924 973 709 2 2124 1222 1569 1006 3 628 924 973 709 4 2172 1222 1569 1006 5 872 924 973 709 6 2131 1222 1569 1006 7 609 1073 1130 791 8 841 1073 1130 791 9 1903 1222 1569 1006 10 479 653 688 549 11 463 1073 1130 791 12 636 1073 1130 791 13 614 1073 1130 791 14 322 653 688 549 15 961 879 926 683 16 387 653 688 549 17 1158 879 926 683

18 2232 699 736 577 19 1089 872 918 679 20 1606 808 851 642 21 239 473 499 434 22 1543 858 904 671 23 3349 441 464 411 24 1323 642 676 542

* O’Neill and Reese (1999) ** Horvath and Kenney (1979) *** Williams et al. (1980)

2.2. Predicted Shaft Resistances

The distribution of pile resistance is evaluated with statistical characteristics of resistance bias factors, the ratio of measured bearing capacity over-predicted bearing capacity. In order to calculate the predicated bearing capacity, three static bearing capacity equations [O’Neill and Reese(1999), Horvath and Kenney(1979), and Williams et. al.(1980)] were used to estimate the static shaft resistance of the 24 test piles. The O’Neill and Reese (1999) equation is expressed as follows.

0 5

0 65 .

s a u a

f . p ( q / p )

(1)

where fs = unit shaft resistance (kPa), pa =

atmospheric pressure (Ą101 kPa), qu =

unconfined compressive strength (kPa).

The Horvath and Kenney(1979) equation is expressed as follows.

0 5

6 88 .

s u

f . ( q )

unconfined compressive strength (kPa).

The Williams et al. (1980) equation is expressed as follows.

0 367

4 406 .

s u

f . ( q )

unconfined compressive strength (tf/m2).

Static unit shaft resistances of the 24 test piles were calculated by these three methods and the computed values of resistance are called “predicted capacities” and are listed in Table 2. The measured capacities and predicted capacities are plotted together in Figure 1. As shown in Figure 1, there does not seem to be a good

correlation between the measured capacity and the predicted capacity by static analysis methods. In general, Williams et al. (1980) provides the most conservative prediction of unit shaft resistance among them.

2.3. Shaft Resistance Bias Factors Statistics The resistance bias factor (RM/P) is defined as the ratio of the measured ultimate bearing capacity from a load test over the predicted ultimate bearing capacity by a static bearing capacity formula.

Figure 1. Measured and predicted unit shaft resistances The resistance bias factor accounts for all of the uncertainties from various sources of errors such as model uncertainty, SPT blow count error, spatial variability of the soil parameters, load test error, soil sampling and testing error, and so on. Resistance bias factors for all selected test piles are computed for the three static design methods described above. The bias factor statistics are influenced by the size of the data set and the variation in the bias factors. Extremely outlying data points may not be representative of the resistance due to the large error in either the measured capacity or the predicted capacity. Therefore, it is reasonable to remove the far-outlying data points from the bias factor statistics. The bias factor values outside the boundaries defined by the mean plus or minus two times the standard deviation are discarded.

The statistics of the filtered resistance bias factors are computed for reliability analysis and presented in Table 3. Horvath and Kenney

0 500 1,000 1,500 2,000 2,500 3,000 0 500 1,000 1,500 2,000 2,500 3,000 M e asu red u n it sh af t res ist a n ce , fma x (k P a )

Predicted unit shaft resistance, fs(kPa) O&R H&K Williams

(1979) equation appears to predict the bearing capacity the most closely to the measured capacity. As expected from the comparison of measured capacity and predicted capacity in Figure 1, Williams et al. (1980) equation, in general, underestimates the shaft resistance the most conservatively. However, the conservatism in the design method cannot be assured due to the large variation in the bias factors. Distributions of RM/P for the three bearing capacity methods were also examined and a lognormal distribution was found to most closely represent the bias factor (RM/P) distributions for three methods as shown in Figure 2.

Table 3. Shaft resistance bias factor statistics

Equation Resistance bias factor (RM/P)

Mean(P) COV* Discarded data

O&R 1.18 0.59 No. 23

H&K 1.07 0.59 No. 23

Williams 1.62 0.59 No.23

* COV = coefficient of variation

(a) O’Neill & Reese

(b) Horvath & Kenney

(c) Williams et al.

Figure 2. Distributions of shaft resistance bias factors

3. Reliability Analysis 3.1. Load Factors

There are two sets of information that are required for reliability analysis of pile foundation design: load information in terms of statistics of load bias factors and resistance information in terms of statistics of resistance bias factors. The statistics of RM/P used in this study are shown in Table 3. The load bias factor (QM/P) is defined as the ratio of the observed actual load over the nominal load. This study employed the statistics of load bias factors used for the American Association of State Highway and Transportation Officials (AASHTO) LRFD bridge design specifications (2007). It was based on the assumption that the load statistics in South Korea are not significantly different from those in the United States. This assumption is subject to verification in future research. AASHTO LRFD bridge design specification Strength Case I is considered as the critical loading case for pile bearing capacity, and the following load statistics were used in the reliability analysis: dead load bias factor mean (QDM/P) = 1.05, dead load bias factor coefficient of variation (COVQD) = 0.10, live load bias factor mean (QLM/P) = 1.15, and live load bias factor coefficient of variation (COVQL) = 0.20. Distribution of both dead and live load bias factors was assumed to be lognormal. Hansell and Viest (1971) reported that the ratio of dead load over live load (QD/QL) could be expressed by the empirical formula QD/QL = (1 + IM)(0.0132L), where IM is the dynamic load impact factor (= 0.33 for LRFD

0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 P ro b a b ilit y D en sit y F u n ct io n N u m b er o f P il e-C a ses Bias Factor (RM/P) Normal Log-normal 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 P ro b a b ilit y D en sit y F u n ct io n N u m b er o f P il e C a ses Bias Factor (RM/P) Normal Log-normal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 P ro b a b ilit y D en sit y F u n ct io n N u m b er o f P il e-C a ses Bias Factor (RM/P) Normal Log-normal

loads) and L is the bridge span length in feet. From a survey of bridge span lengths in South Korea, it was found that the span lengths in the range of 30 m ± (or 98 feet ±) were most common (MLTM 2007). Thus, QD/QL of 1.72 was chosen for reliability analysis.

3.2. Reliability Index Calculation

The first order reliability method (FORM) (Rackwitz and Fiessler, 1978) was used to evaluate the reliability of the three shaft resistance equations. Two random variables, the load (Q) and the resistance (R), are considered and they are assumed to be statistically independent and lognormally distributed. The limit state function in this case is defined as: g (R, Q) = ln (R) – ln (Q) = ln (R/Q).

In the FORM analysis, the limit state function is linearized at a point on the failure surface. The limit state function g (R, Q) can be expressed in the following format.

1 R QD QL FS ( QD / QL ) g ln( ) ( QD / QL ) O O O u u  u  (4)

A FORTRAN based computer program (Kwak et al., 2007) was developed following the iteration algorithm of the Rackwitz & Fiessler’s FORM (1978) to facilitate the computing processes of the reliability indices.

The Korean Design Standards for Foundation Structures requires a minimum safety factor of 2.0 for a pile design, and in general, a safety factor equal to or more than 3.0 has been used for static bearing capacity equation in Korean practice. Accordingly, reliability analysis was performed for factors of safety (FS) in the range of 2.0-5.0. The computed reliability indices are shown in Table 4 for all 24 data.

The reliability indices of the three shaft resistance equations evaluated from FORM method for the FS in the range of 2.0-5.0 were calculated as follows: (a) 1.15-2.79 for O’Neill and Reese (1999) equation, (b) 0.97-2.62 for Horvath and Kenney (1979) equation, and (c) 1.73-3.38 for Williams et al. (1980) equation. The reliability indices of the Williams et al. (1980) equation are relatively larger. The differences of reliability indices between Williams et al. and Horvath and Kenney (1979)

equations are about 0.75. The O’Neill and Reese (1999) equation and Horvath and Kenney (1979) equation showed similar level of reliability indices.

The reliability indices evaluated by the FORM methods are shown in Figure 3. An increase in the reliability indexes for the Williams et al. equation occurred due to the high resistance bias factor (RM/P) underlying similar COV comparing with the other two equations.

In Figure 3, the equivalent factors of safety corresponding to reliability index of 2.0 which is suggested as the required minimum safety level for deep foundations are in the range of 3.0-3.5 for O’Neill and Reese (1999) equation and Horvath and Kenney (1979) equation, 2.0-2.5 for Williams et al. (1980) equation, respectively. Table 4.Calculated reliability indices

FS O&R H&K Williams

2.0 1.145 0.971 1.726 2.5 1.546 1.373 2.129 3.0 1.874 1.701 2.459 3.5 2.151 1.979 2.738 4.0 2.391 2.220 2.980 4.5 2.603 2.432 3.193 5.0 2.793 2.622 3.383

Figure 3. Reliability indices for shaft resistance

4. Conclusions

This study presents the following findings and conclusions.

1. Reliability analyses were conducted by first order reliability method on the side resistance of axially loaded drilled shafts 0.0 1.0 2.0 3.0 4.0 2.0 3.0 4.0 5.0 R elia b ilit y In d ex (E ) Factor of Safety O&R H&K Williams

in which bi-directional pile load test were carried out in order to develop LRFD design for piles in Korea.

2. The predicted shaft resistance was calculated by three static design methods and there is no good correlation between the measured resistances and predicted values.

3. The reliability levels of the three shaft resistance equations evaluated from FORM method for the FS in the range of 2.0-5.0 were in the ranges of 1.15-2.79 for O’Neill and Reese equation, 0.97-2.62 for Horvath and Kenney equation, and 1.73-3.38 for Williams et al. equation, respectively.

4. Williams et al. equation provides the most conservative prediction of unit shaft resistance among them. Horvath and Kenney equation appears to predict the bearing capacity the most closely to the measured capacity. The large variation in the resistance bias factors undermines the reliability of the three static equations.

Acknowledgment

This work was supported by the Super Long Span Bridge Project funded by the Ministry of Land, Infrastructure, and Transport, Republic of Korea.

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