Acta Geotechnica DOI 10.1007/s11440-014-0354-8 RESEARCH PAPER A procedure for incorporating setup into load and resistance factor design of driven piles Kam W. Ng • Sri Sritharan Received: 23 February 2014 / Accepted: 9 September 2014 Ó Springer-Verlag Berlin Heidelberg 2015 Abstract In a recent study, the time-dependent increase in axial load resistance of steel H-piles driven into cohesive soils, due to setup, was systematically quantified using measured field data. A method to estimate the setup based on measurable soil properties was subsequently established. These studies highlighted that the uncertainties of the measurements of soil properties and thus the semi-empirical approach to estimate setup are significantly different from those of the methodology used for measuring the pile resistance during retaps at any time after the end of driving. Recognizing that the two sets of uncertainties should be addressed concurrently, this paper presents a procedure for determining the factored resistance of a pile with due consideration to setup in accordance with the load and resistance factor design that targets a specific reliability index. Using the first-order second-moment method, the suggested procedure not only provides a simplified approach to incorporate any form of setup in design, but it also produces comparable results to the computationally intensive firstorder reliability method. Incorporating setup in design and construction control is further shown to reduce foundation costs and minimize retap requirements on piles, ultimately reducing the construction costs of pile foundations. K. W. Ng (&) Department of Civil and Architectural Engineering, University of Wyoming, Dept. 3295, EN3050, 1000 E. University Ave., Laramie, WY 82071-2000, USA e-mail: kng1@uwyo.edu S. Sritharan Department of Civil, Construction and Environmental Engineering, Iowa State University, 376 Town Engineering Building, Ames, IA 50011-3232, USA e-mail: sri@iastate.edu Keywords Foundations  Load and resistance factor design  Piles  Pile setup 1 Introduction The setup in foundation design typically refers to the increase in vertical load resistance (or capacity) of driven piles embedded in soils over time, which results from the healing of remolded soils surrounding the pile, dissipation of pore water pressure induced by pile driving (i.e., consolidation), and/or increase of lateral soil stresses [22, 25]. A systematic field experimental program involving five steel H-piles driven in cohesive soils conducted by the authors concluded that the pile resistance can increase by as high as 55 % within 7 days due to the dissipation of pore water pressure induced by pile driving [18]. The presence of setup was evident by the significant increase in pile driving resistance measured in terms of hammer blow counts required after the end of driving (EOD), which was later confirmed by the subsequent static load test (SLT) measurements. Several researchers have reported field evidence of pile setup in literature [9, 11, 14, 15]. If pile setup can be satisfactorily incorporated into load and resistance factor design (LRFD) framework, it will reduce the foundation cost by requiring: (1) shorter pile length, less number of piles, and/or smaller dimensions for the pile cap; and (2) smaller hammers possibly with less consumption of fuel and reduced labor during construction. A systematic approach to routinely account for setup during pile driving with due consideration to time has not existed until recently. Pile setup is only accounted for in special cases for which setup is accounted for using dynamic strike tests and/or SLTs, which incur additional construction costs. Given the development of a pile setup 123 Acta Geotechnica procedure using soil parameters [18, 19], this paper focuses on incorporating setup into the LRFD framework. This method is targeted for eventual incorporation in design specifications, such as the American Association of State Highway and Transportation Officials (AASHTO), which do not have recommendations to incorporate setup into the LRFD approach. Ensuring simplicity in design and construction methods as well as to allow for incorporation of setup using any predictive methods, the paper proposes a systematic, closed-form approach to determine separate resistance factors for pile resistance at end of driving (REOD) and pile setup resistance (Rsetup). These separate resistance factors will account for varying uncertainties associated with two different resistance components, enabling pile design using Eq. (1) as similarly suggested by Yang and Liang [27]: X cQ  uR ¼ uEOD REOD þ usetup Rsetup ð1Þ where c is load factor, Q is applied vertical load, R is the total nominal pile resistance, u is the resistance factor for total resistance R, uEOD is the resistance factor for REOD, and usetup is the resistance factor for Rsetup. Each of the pile resistance components has its own uncertainties, such as those resulting from the in situ measurement of soil properties and the empirical approach. To satisfactorily incorporate a pile setup estimate into LRFD, the impact of the different uncertainties should be recognized separately, yet accounted for simultaneously to achieve the desired reliability in design. To comply LRFD framework and provide designers with an efficient method involving little computational requirement, the proposed procedure uses the first-order second-moment (FOSM) framework to determine two resistance factors: uEOD and usetup. The application of proposed procedure was demonstrated using a database of steel H-piles embedded in cohesive soils. For verification purposes, the outcomes are compared with the resistance factors determined based on the first-order reliability method (FORM) by Yang and Liang [27]. 2 Background 2.1 Pile setup resistance estimation AASHTO suggests the use of dynamic restrike tests and/or completing SLTs at different times to account for pile setup, which may be appropriate for infrequent use in special projects. Both suggestions are time-consuming and not feasible in practice [3]. Alternatively, the pile setup can be estimated using empirical methods available in literature (e.g., [11, 24, 26]). Among the empirical methods, Skov and Denver’s method [24] has been widely used to estimate pile setup resistances [27], which is summarized in Eq. (2): 123    t Rsetup ¼ Ro A log to ð2Þ where Ro is a reference pile resistance, A is an empirical setup factor, and to is a reference time by which Ro is determined. This method also requires performing a dynamic restrike at to = 1 day to determine the Ro value and multiple restrikes over a period of time to determine the A value. Based on 250-mm square concrete piles and Yoldia clay collected in Germany, Skov and Denver [24] recommended an A value of 0.6 if pile restrikes are not feasible. Due to variability of soil and pile types, Bullock et al. [7] and Yang and Liang [27] concluded that a more suitable A value may be in the range between 0.1 and 1.0. The large variability of A values exhibits no correlations with any soil properties. Consequently, the general application of Eq. (2) becomes limited because A cannot be estimated easily based on soil properties. To improve upon the pile setup estimation, Ng et al. [18] conducted an extensive load test program to quantify the pile setup and develop a semi-empirical method to estimate the setup for low displacement piles in a cohesive soil profile. The experimental results revealed that setup significantly influenced shaft resistance but not the end bearing. Recognizing relative small contribution of end bearing to total resistance and to maintain simplicity in design applications, pile setup in terms of total pile resistance was considered. Based on extensive field evaluations and quantifying the setup using the effect of pore water pressure dissipation, Ng et al. [19] concluded that setup can be satisfactorily estimated using: (1) pile resistance at EOD obtained from either a bearing graph (ultimate pile resistance versus hammer blow count) generated using Wave Equation Analysis Program (WEAP) or CAse Pile Wave Analysis Program (CAPWAP) method; (2) soil properties; and (3) pile geometry, as described below: ! "  # f c ch t þ fr log10 ð3Þ Rsetup ¼ REOD tEOD Na rp2 where REOD is the pile resistance at EOD, Rsetup is the net increase in pile resistance immediately after EOD, fc and fr are empirical factors (13.78 and 0.149, respectively, for WEAP), Ch is the horizontal coefficient of consolidation determined from CPT pore water pressure dissipation tests and strain path method, Na is the weighted average SPT N value accounting for different cohesive soil thicknesses within a soil profile, rp is the equivalent pile radius based on pile cross-sectional area, t is the time of pile setup elapsed after the EOD, and tEOD is the time at EOD (assumes 1 min). It is important to note that Eq. (3) was developed based on load tests performed within 36 days as summarized in Ng et al. [18]. Thus, its application for quantifying long-term setup shall be used with caution. Acta Geotechnica computational effort, which is attributed to an iterative procedure that simultaneously adjusts the load and resistance components until a target reliability index is achieved. This procedure requires a purpose-built program, which may not be readily available to establish usetup reflecting local soil conditions. Measured Pile Resistance, Rm (kN) 8000 6000 Ratio Mean R m /R EOD R m /R t 1.643 0.996 Std. Dev. 0.370 0.194 COV N 0.225 0.195 39 39 4000 Huang [10] Lukas et al. [14] Long et al. [13] Fellenius [8] ISU-PILOT ISU Field Tests 2000 3 Uncertainties of pile resistance 3.1 Evaluation based on resistance ratio 0 0 2000 4000 6000 8000 Estimated Pile Resistance Considering Setup, Rt (kN) Fig. 1 Comparison between the total measured and the total estimated resistances of steel H-piles using Eq. (3) and WEAP Detailed derivation of the pile setup empirical Eq. (3) can be found in Ng et al. [19]. The validation for steel H-piles is shown in Fig. 1, while similar validation for other pile types was documented in Ng et al. [19]. 2.2 Incorporation of setup in pile design Recognizing the different degree of uncertainties associated with nominal values of REOD and Rsetup, Komurka et al. [13] proposed applying separate safety factors for these components within the allowable strength design (ASD) framework. Although the same concept can be extended to the LRFD framework, establishing a suitable resistance factor for Rsetup is not straightforward. If Rsetup is added to REOD as a single resistance component, and is not considered separately using the proposed Eq. (1) during a conventional resistance factor calculation, an unrealistically high resistance factor will be yielded, such as the resistance factor of 1.31 determined by Abu-Farsakh et al. [1] for precast, prestressed concrete piles driven in Louisiana soil and verified using CAPWAP at the EOD condition. Using a database of 37 tests collected by Paikowsky et al. [21] on various pile types in clay, which comprised of five steel H-piles, five closed-end pipe piles, five timber piles, seven prestressed concrete piles, nine concrete piles, and six other pile types, Yang and Liang [27] selected Skov and Denver’s [24] empirical Eq. (2) to estimate Rsetup. The results of their statistical analysis confirmed the different degree of uncertainties for Ro and Rsetup, indicating with coefficients of variation (COVs) of 0.339 for Ro and 0.475 for Rsetup shown in Table 3. Yang and Liang [27] used an invariant FORM to determine the resistance factors for Rsetup, recommending a conservative usetup of 0.30 for a bridge span of\60 m based on a target reliability index (b) of 2.33. Despite its accuracy, the FORM requires intensive The database shown in Table 1 was used to determine the uncertainties of the two resistance components. Table 1 comprises five recently completed full-scale pile tests within the State of Iowa, USA (data sets 1–5; see Ng et al. [16] for more details), ten data sets from the PIle LOad Test (PILOT) database of the Iowa Department of Transportation (data sets 6–15; refer Roling et al. [23] for more information), and four well-documented tests (data sets 16–19) reported by others in literature [9, 11, 14, 15]. Soil condition for data sets 1–15 was mostly glacial till, while soil conditions for data sets 16–19 are described in Table 1, Table 1 also summarizes the elapsed time (t) of pile restrikes or SLT following EOD. The pile driving resistance (hammer blow count) at EOD, needed for estimating initial pile resistance using WEAP (Re-EOD), was available for all data sets. The force and velocity records at EOD, measured using pile driving analyzer (PDA) for CAPWAP signal matching, were available only for data sets 1–5, 16, 18, and 19. Additionally, the driving resistances at the beginning of restrikes (BOR) at different times (Re-restrike), required for estimating pile resistance using WEAP analysis, are available for data sets 1–5, and 18. The WEAP pile resistance at time t (Re-t) can also be established, which is the sum of the estimated EOD resistance (Re-EOD) and setup resistance (Re-setup) calculated using Eq. (3). As suggested by Yang and Liang [27], a measurement-based pile resistance at EOD (Rm-EOD) and/or a restrike time (Rm-t) was taken as the resistance value calculated from CAPWAP analysis using PDA data. A measurement-based pile resistance at the time of SLT (Rm-t) was determined based on Davisson’s criterion [8]. Consequently, it follows that the difference between Rm-t and Rm-EOD is assumed to be the measured pile setup resistance (Rm-setup). CAPWAP method was chosen to determine the pile resistances at EOD (Rm-EOD), because of the following reasons: 1. 2. The pile resistances obtained from CAPWAP are based on field measurements of pile force and velocity records using PDA; Since it is practically infeasible to measure pile resistances at EOD using SLT, measurement-based 123 Acta Geotechnica Table 1 Five pile resistance ratios (RR) based on a database of steel H-piles Data set References {Soil Condition} (Location) 1 ISU2 {GT} (Mills, IA, USA) 2 3 4 5 6 7 8 9 10 11 12 13 ISU3 {GT} (Polk, IA, USA) ISU4 {GT} (Jasper, IA, USA) ISU5 {GT} (Clarke, IA, USA) ISU6 {GT} (Buchanan, IA, USA) PILOT {GT} (Decatur, IA, USA) PILOT {GT} (Linn, IA, USA) PILOT {GT} (Linn, IA, USA) PILOT {GT} (Linn, IA, USA) PILOT {GT} (Johnson, IA, USA) PILOT {GT} (Hamilton, IA, USA) PILOT {GT} (Kossuth, IA, USA) PILOT {GT} (Jasper, IA, USA) RR1 Restrike, (Rm-t/Re-restrike) RR2 Typical RR, (Rm-t/Re-EOD) RR3 RR at time t, (Rm-t/Re-t) RR4 EOD, (Rm-EOD/Re-EOD) RR5 Setup, (Rm-setup/Re-setup) 9d 0.90a 1.62a 0.91a 1.05b 0.74a d 1.14 a 1.84 a 1.00 a 1.21 b 0.76a 1.00 a 1.62 a 0.94 a 1.07 b 0.76a 0.95 a 1.70 a 1.02 a 1.24 b 0.68a 0.84 a 1.54 a 0.90 a 1.05 b 0.69a 1.63 a 1.05 a – 1.34 a 0.85 a 0.58a 0.97 a 0.62 a -0.05c 1.48 a 0.94 a 0.82a 1.50 a 0.97 a 0.93a 1.84 a 1.18 a 1.48a 1.33 a 0.84 a 0.56a 1.70 a 1.15 a 1.45a 1.56 a 0.96 a 0.91a Time elapsed after EOD, t (day) 36 d 16 9 d d 14 3 d 5 d 5 d 5 d 3 d 4 d 5 d 1 d d 14 PILOT {GT} (Poweshiek, IA, USA) 8 15 PILOT {GT} (Poweshiek, IA, USA) 3d 16 Huang [11] {Silty clay over silty sand} (Shanghai, China) 2e 17 18 19 Lukas and Bushell [15] {Fill over clay} (Illinois, USA) Long et al. [14] {Silty clay over sandy till} (Illinois, USA) Fellenius [9] {Mixture of sand, silt and clay over glacial till} (Canada) – – d 31 d 10 – d 26 7 d d 22 e 7 0.77 a 1.11 a – e 13 e 15 e 16 e 18 e 21 e 28 e 32 e 44 1.14a 1.16a 0.75a 1.39b 0.87b 0.52a 2.25 a 1.23 a 1.70 a 1.05 a 1.95 a 1.16 a 1.02 a 0.61 a 2.16 a 0.87 a 1.72 b 1.07 b 1.87 b 1.13 b 0.67b 1.92 b 1.16 b 0.74b 2.11 b 1.27 b 1.02b 2.03 b 1.22 b 0.89b 1.91 b 1.14 b 0.70b 2.24 b 1.32 b 1.16b 2.32 b 1.36 b 1.26b 2.29 b 1.33 b 1.19b 0.93b 0.78b 1.61a 1.12a – 1.38a 0.91 b 1.43 b 0.04c 0.85a 1.19b Measured pile resistance using SLT; b measured pile resistance using CAPWAP; c pile setup was insignificant thus neglected; d time of SLT; e time of restrike; Re = estimated pile resistance using WEAP; Rm = measurement-based pile resistance obtained using CAPWAP or SLT; and GT = glacial till a 3. pile setup using CAPWAP can be effectively used to compare with the estimated value using Eq. (3); and The relatively high-efficiency factor (u/k) for CAPWAP [21]. To investigate various sources of uncertainties for different resistance components, five different resistance ratios (RR), ratios of measurement-based resistance values (Rm) to analysis-based estimated pile resistances (Re), are summarized in Table 1. In this paper, measurement-based pile resistances were obtained from SLT and/or CAPWAP, while estimated pile resistances were determined using WEAP. However, it is important to note that such a comparison shall not be restricted in this arrangement. Each of these ratios is described as follows: 123 1. 2. 3. RR1 is a ratio of measurement-based pile resistance using SLT at time (t) to pile resistance estimated using WEAP during the last restrike event before SLT. RR1 is used to evaluate the uncertainty associated with Rrestrike (i.e., RR1 = Rm-t/Re-restrike); RR2 is a typically used ratio of measurement-based pile resistance at time (t) using either SLT or CAPWAP to estimate WEAP pile resistance at EOD without any setup consideration (i.e., RR2 = Rm-t/ Re-EOD); RR3 defines the ratio of measurement-based SLT or CAPWAP resistances to total estimate resistances, which is the sum of the REOD using WEAP and the Rsetup using Eq. (3) (i.e., RR3 = Rm-t/Re-t); Acta Geotechnica 4. 5. RR4 defines the ratio of measurement-based resistance to estimated resistance at EOD using WEAP (i.e., RR4 = Rm-EOD/Re-EOD); and RR5 defines the ratio of measurement-based pile setup resistance to the setup resistance estimated using Eq. (3) (i.e., RR5 = Rm-setup/Re-setup). The RR4 and RR5 are two essential ratios for calibrating uEOD and usetup, respectively. 3.2 Results of conventional FOSM analysis Table 2 presents the resistance biases (kR) and coefficients of variation (COVR) calculated for all five RRs (i.e., RR1 to RR5) using the database summarized in Table 1. It shows that pile resistances obtained from restrikes (RR1) have the lowest COVR value of 0.14, which reveals the reliability of pile setup quantification using restrike tests, but as noted, they are not always feasible in routine practice. Furthermore, the significant difference in COVR values between EOD (0.157 in RR4) and setup (0.317 in RR5) confirms the different uncertainties with EOD and the setup, justifying the need to develop a separate resistance factor for each component. In compliance with the LRFD limit state (i.e., uR C cQ) and assuming the load (Q) and resistance (R) are mutually independent and follow a lognormal distribution, the resistance factors (u) for RR1 to RR4 were calibrated in accordance with the FOSM method, using Eq. (4) as suggested by Barker et al. [5]. With the focus on the axial pile resistance, the AASHTO [3] ‘‘Strength I’’ load combination was used. The numerical values for the different probabilistic characteristics of dead (QD) and live (QL) loads (c, k, and COV), as documented by Nowak [20] and adopted by Paikowsky et al. [21], were recapitulated in parentheses given in the definition of each parameter as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kR cDL COVDLR ffio u¼ ð4Þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   kDL exp bT ln 1 þ COV2R ðCOVDL Þ cDL ¼ cD QD þ cL QL kDL ¼ kD QD þ kL QL COVDLR ¼ COVDL 1 þ COV2R COVDL ¼ 1 þ COV2D þ COV2L where kR is the resistance bias factor of the RR, COVR is the coefficient of variation of the RR, cD is the dead load factor (1.25), cL is the live load factor (1.75), kD is the dead load bias (1.05), kL is the live load bias (1.15), COVD and COVL are the coefficients of variation of dead load (0.1) and live load (0.2) respectively, and QD/QL is the dead to live load ratio (2.0). As recommended by Paikowsky et al. [21] and adopted by AASHTO [3], the target reliability indices (bT) of 2.33 (corresponding to 1 % probability of failure) for redundant piles and 3.00 (corresponding to 0.1 % probability of failure) for non-redundant piles were used. Table 2 shows the respective resistance factors (u) and efficiency factors (u/k). When the effect of setup was considered based on RR1 using restrikes and RR3 using Eq. (3), a realistic resistance factor of 0.69 for the bT values of 2.33 was yielded. In contrast, for RR2, a comparison was made based on the EOD condition, yielded unrealistically high resistance factors of 1.11 and 0.91 for the bT values of 2.33 and 3.00, respectively. These relatively large kR values were due to indirectly and inaccurately including effect of pile setup in the calculation of the resistance factors. Recognizing the difference in the COV values between the EOD component in RR4 and the setup component in RR5, a single resistance factor determined in RR4 concluded that the conventional LRFD calibrating procedure using Eq. (4) cannot account for the different uncertainties associated with both components. This implies that combining the uncertainties of REOD and Rsetup into one resistance factor fails to satisfy the LRFD philosophy and achieve a consistent and reliable design. Table 2 Comparison of resistance factors obtained using the conventional LRFD framework Resistance ratio (RR) Sample size kR COVR Nominal pile resistance (R) bT = 2.33 bT = 3.00 u u/k u u/k 7 0.959 0.140 Re-restrike 0.69 0.72 0.58 0.61 RR2 = Rm-t/Re-EOD 30 1.723 0.211 Re-EOD 1.11 0.65 0.91 0.53 RR3 = Rm-t/Re-t 30 1.029 0.190 Re-t [Eq. (3)] 0.69 0.67 0.57 0.55 RR4 = Rm-EOD/Re-EOD 8 1.111 0.157 Re-EOD 0.78 0.71 0.65 0.59 RR5 = Rm-setup/Re-setup 28 0.950 0.317 Not applicable for Eq. (4) RR1 = Rm-t/Re-restrike 123 Acta Geotechnica Lognormal Distribution - 95% CI 0.5 99 Resistance Ratio for setup (RR5) Percent 95 90 1.0 2.0 Resistance Ratio for EOD(RR4) Resistance Ratio for setup (RR5) (RR5) Mean -0.09821 Std Dev 0.3109 N 28 AD 0.374 P-Value 0.392 80 70 60 50 40 30 20 Resistance Ratio for EOD (RR4) (RR4) Mean 0.09520 Std Dev 0.1532 N 8 AD 0.255 P-Value 0.620 10 5 for EOD and setup, respectively. Using a total of 17 points given in Table 1 and having both RRs D and S, the correlation coefficient (q) was calculated as 0.071. Compared with the q value of -0.243 computed for the Skov and Denver [24] empirical Eq. (2), which led to the independent relationship between the pile reference resistance (Ro) and setup resistance (Rsetup) as noted by Yang and Liang [27], the much smaller calculated q value of 0.071 concluded that pile resistances for EOD (D) and setup (S) are mutually independent. 1 0.5 1.0 2.0 Fig. 2 Normality test using the Anderson–Darling method for pile setup resistance and initial resistance at EOD 4 FOSM criteria To consider the pile setup resistance estimated using Eq. (3) in pile designs that conforms to the reliability theory used in the LRFD framework, the principle of strength limit state function (g) corresponding to a safety margin is expanded, as in Eq. (5). The first FOSM criterion is satisfied, and this equation is valid, but only if REOD, Rsetup, and applied load (Q) have lognormal distributions as described below:   g ¼ lnðREOD Þ þ ln Rsetup  lnðQÞ ð5Þ To verify that the pile resistances given in Table 1 follow lognormal distributions, a hypothesis test, based on the Anderson–Darling (AD) [4] normality method, was used to assess the goodness of fit of the assumed distributions. Unlike other goodness-of-fit tests such as the Kolmogorov–Smirnov test [12], the AD method was designed to better detect discrepancies in the tail regions of a probability distribution, especially with a relatively small sample size [10]. Figure 2 shows that the AD values of 0.255 and 0.374 are smaller than the critical P values of 0.620 and 0.392 within the 95 % confidence interval (CI) for EOD and setup conditions, respectively. The hypothesis test confirmed the assumed lognormal distributions for both resistances and the limit state function (g). In order to verify the independent relationships between the random variables required for the second FOSM criterion, the correlation between the RR for EOD (let Rm-EOD/Re-EOD be D) and for the setup (let Rm-setup/Re-setup be S) was assessed through the calculation of a correlation coefficient (q) using q¼ covðD; SÞ rD rS ð6Þ where cov(D, S) is covariance between the RR for EOD and setup, and rD and rS are standard deviations of the RR 123 5 Resistance factors for pile setup Following the derivation described in ‘‘Appendix,’’ the resistance factor for pile setup (usetup) based on the FOSM Method can be expressed as:   cDL ksetup  uEOD QDL  usetup ¼  ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kDL exp bT ln½ðCOVRR ÞðjÞ QDL rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kEOD j COVRR QDL ¼ 1 þ QD QL COVRR ¼ 1 þ COV2REOD þ COV2Rsetup Q2D 2 k COV2D þ k2L COV2L Q2L D j¼1þ 2 QD QD 2 kD kL þ k2L 2 kD þ 2 QL QL where ksetup is the resistance bias factor of the setup resistance, COVREOD is the coefficient of variation of the resistance at EOD, and COVRsetup is the coefficient of variation of the setup resistance. This equation reveals that usetup is dependent on several parameters. Considering only the AASHTO [3] ‘‘Strength I’’ load combination, the probabilistic characteristics (c, k, and COV) of the random variables QD and QL are defined in Eq. (4). Considering the database of steel H-piles summarized in Table 1, the probabilistic characteristics (k and COV) of the random variables REOD and Rsetup were selected from RR4 and RR5 of Table 2, respectively. Since the a value is suggested as unity in the ‘‘Appendix,’’ the following analyses primarily focus on the influence of the remaining parameters (i.e., bT, uEOD, and QD/QL) on usetup. Since the uncertainty for the pile resistance at EOD is lower than that for setup (COVEOD = 0.157 less than COVsetup = 0.317) and mean biases of both resistances are close to unity (kEOD = 1.111 and ksetup = 0.950), Fig. 3 shows that the calculated uEOD values are noticeably Acta Geotechnica 0.60 setup=0.327 0.20 0.10 0.00 1.80 2.30 2.80 Target Reliability Index, 3.30 T 0.40 T / =0.42 ( T=2.33) 0.40 / =0.34 ( T=3.00) 0.20 0.10 0.30 0.20 Reduced Safety Margin Redundant Safety Margin = 2.33) 0.70 0.80 0.90 0.10 EOD Fig. 5 Relationship between resistance factors for setup and EOD condition 0.50 0.60 0.50 0.30 0.00 0.60 setup = 2.33 = 3.00 0.10 Resistance Factor for EOD, Efficiency Factor, / setup Resistance Factor for Setup, Note: Based on EOD = 0.78 and 0.65 for T = 2.33 and 3.00, respectively 0.50 0.20 T Fig. 3 Relationship between the resistance factor and the target reliability index 0.60 T setup=0.436 0.30 setup=0.327 setup=0.398 ( 0.40 0.30 EOD=0.783 EOD=0.653 EOD 0.50 0.40 = 3.00) ( = 1.0) T setup 0.60 ( 0.70 0.50 0.00 0.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 = REOD /(QD+QL ) Fig. 4 Relationship between the resistance factor for setup and the ratio of pile resistance at EOD to total load (a) higher than the usetup values at a fixed QD/QL of 2.0. Similar to the trend observed for uEOD, usetup decreases with increasing bT. Based on the data tabulated in Table 1 and the probabilistic characteristics listed in Table 2, usetup values decreased from 0.436 to 0.327 as bT values increased from 2.00 to 3.00. This observation conforms to the LRFD framework suggested by Paikowsky et al. [21], whereby a smaller resistance factor is dependable for a less redundant pile group that has a larger probability of failure, represented with a higher bT. Figure 4 shows the relationship between the usetup and a values. The uncertainties associated with REOD have been accounted for in terms of uEOD, and thus, there is no reason to accommodate any additional safety margins. Any redundant safety margins applied to REOD, such as having an a value greater than 1.0, will reduce the usetup value toward zero. This observation indicates that the resistance contribution from pile setup within the LRFD framework becomes less significant when a pile has a large resistance Resistance Factor for Setup, 0.80 Note: Based on constant QD /QL = 2.0 EOD=0.653 Note: Based on EOD = 0.86, 0.78, 0.75 and 0.65 for T = 2.00, 2.33, 2.50 and 3.00, respectively EOD=0.860 setup 0.90 Resistance Factor for Setup, Resistance Factor for EOD or Setup, EOD or setup 1.00 0.454 0.45 ( T = 2.33) 0.395 0.40 0.371 0.366 0.359 0.35 0.325 ( 0.30 T 0.309 0.306 = 3.00) 0.25 0.20 0.15 0.10 Note: Based on EOD = 0.78 and 0.65 for T = 2.33 and 3.00, respectively 0.05 0.00 0 0.5 1 1.5 2 2.5 3 3.5 4 Ratio of Dead to Live Load, QD /QL Fig. 6 Relationship between resistance factor for setup and dead to live load ratio component contributed from the uEODREOD that sufficiently overcomes the applied factored loads (RcQ). The resistance contribution from pile setup becomes nil specifically at a values equal to 1.80 and 2.16 for the bT values of 2.33 and 3.00, respectively. If the REOD value can be appropriately reduced while satisfying the limit state Eq. (1), the resistance contribution from pile setup can be increased by having a larger usetup value. Therefore, suggesting the a value in Eq. (22) to unity, Eq. (7) is appropriately recommended to determine the usetup value. In order to illustrate the influence of different uEOD values that represent different regional soil conditions and design practices, Fig. 5 illustrates the variation of usetup values as a function of uEOD values based on a constant QD/QL ratio of 2.0. This figure shows that higher usetup values resulted from a higher uEOD. For a non-redundant pile group at a bT value of 3.00, the uEOD and usetup values were determined to be 0.653 and 0.327, respectively. For a redundant pile group at a bT value of 2.33, the uEOD and 123 Acta Geotechnica Table 3 Summary of recommended LRFD resistance factors Empirical method Pile type Pile resistance FOSMa or FORMb Resistance bias (k) Coefficient of variation (COV) Resistance factor, u bT = 2.33 bT = 3.00 REOD FOSM 1.111 0.157 0.78c 0.65c 0.950 0.317 0.36 0.31 FOSM 1.158 0.339 0.58 0.45 0.65 0.50 1.141 0.475 0.27 0.20 0.30 0.00 Equation (3) Steel H-pile Rsetup Skov and Denver [24] All Ro FORM Rsetup FOSM FORM a Based on a suggested a value of one; b based on Yang and Liang’s [27] results; and usetup values were determined to be 0.783 and 0.398, respectively. The aforementioned observations from Figs. 3, 4, and 5 are based on a fixed QD/QL of 2.0, and it is of interest to investigate the influence of this ratio, which is a function of the bridge span, on the usetup values. Based on the AASHTO [2] Specifications, QD/QL ratios of 0.52, 1.06, 1.58, 2.12, 2.64, 3.00, and 3.53 are suggested for span lengths of 9, 18, 27, 36, 45, 50, and 60 m, respectively [27]. Figure 6 illustrates that usetup reduces gradually with increasing QD/QL ratios from 0.52 to about 2.12, and at an even slower rate thereafter. This figure also indicates that an increase in QD/QL ratio, by a factor of 6.8 (i.e., from 0.52 to 3.53), only reduces usetup by a small factor of about 1.2 (i.e., from 0.454 to 0.371). Hence, it can be generally concluded that usetup values are almost insensitive to the QD/QL ratios. Regardless of different QD/QL ratios, usetup values of 0.36 and 0.31 as summarized in Table 3 can be conservatively recommended for bT values of 2.33 and 3.00, respectively. 6 Comparison with FORM Using dynamic and SLT data from 37 piles collected by Paikowsky et al. [21], pile setup resistance factors (usetup) suggested by Yang and Liang [27] based on FORM were compared with the values calculated using the proposed FOSM procedure. Pile resistances were determined using CAPWAP at both EOD and BOR conditions. Skov and Denver’s [24] empirical Eq. (2) was adopted by Yang and Liang [27] to estimate the pile setup resistance with the reference time to set to 1 day and an average A value of 0.50. The k and COV of Ro and Rsetup determined by Yang and Liang [27] are given in Table 3. Accordingly, the random variable Ro was best fitted with a lognormal distribution, while Rsetup was best fitted with a normal distribution. Using these probabilistic characteristics, usetup values computed by Yang and Liang [27] based on FORM were plotted in Fig. 7 with respect to QD/QL ratios ranging 123 c based on a sample size of 8 and use with caution from 0.52 to 4.0. Using the same probabilistic characteristics and the resistance factors for Ro determined based on the FOSM, usetup values determined using the proposed FOSM Eq. (7) are compared in Fig. 7. For redundant pile groups represented by a target reliability index (bT) of 2.33, Fig. 7 (a) shows that the usetup values determined using both methods decrease with increasing QD/QL ratios (i.e., both share a similar trend). The smallest usetup value based on FORM is 0.32, while the smallest usetup value based on the proposed FOSM method is 0.27, which is comparable to the recommended usetup value of 0.30 suggested by Yang and Liang [27]. The usetup values determined using FORM are generally 20 % higher than those calculated using the FOSM method, indicating that usetup values calculated using the simpler and closed-form FOSM method are relatively more conservative than those determined using a more complex and computationally intensive FORM method. The difference usetup value is attributed to: (1) the different reliability theory used in estimating the usetup values; (2) the different uEOD values, as given in Table 3; and (3) the assumed distribution for the probabilistic characteristics of the random variable Rsetup. For non-redundant pile groups (or individual piles) represented by a target reliability index (bT) of 3.00, Fig. 7 (b) shows a similar relationship between usetup values and QD/QL ratios. However, usetup values determined using the FORM are almost nil, while those calculated using the proposed FOSM method are about 0.20. Yang and Liang [27] concluded that a zero usetup value implies that pile resistance will be optimized solely based on Ro, and the benefits of incorporating Rsetup will not be realized due to higher uncertainties associated with the estimation of Rsetup. In fact, pile setup did occur in the 37 pile data series reported by Yang and Liang [27]. Furthermore, extensive pile load tests performed by Ng et al. [17] confirmed that pile setup as high as 55 % occurred when a single pile was embedded in cohesive soil. Hence, it is justifiable to apply the usetup value of 0.20 determined using the proposed FOSM method, so that the benefits of incorporating pile setup in LRFD can be realized. Acta Geotechnica 0.60 0.60 (a) Proposed FOSM Note: Based on FORM by Yang and Liang [26] value of 1.00 0.50 setup value of 1.00 0.40 Resistance Factor for Setup, Resistance Factor for Setup, Note: Based on setup 0.50 (b) Proposed FOSM FORM by Yang and Liang [26] 0.40 0.30 (Recommended setup = 0.30 by Yang & Liang [26] using FORM for T = 2.33) 0.20 0.10 0.30 0.20 0.10 0.00 (Recommended setup = 0.00 by Yang & Liang [26] using FORM for T = 3.00 ) 0.00 -0.10 0 1 2 3 4 Ratio of Dead to Live Load, Q D /QL 0 1 2 3 4 Ratio of Dead to Live Load, Q D /QL Fig. 7 Comparison of usetup based on FOSM and FORM for (a) bT = 2.33, and (b) bT = 3.00 Using the proposed FOSM procedure, Table 3 summarizes the recommended resistance factors for both the Skov and Denver’s [24] method and the setup Eq. (3). Comparing these resistance factors calculated for Skov and Denver’s [24] method and all pile types (i.e., 0.58 and 0.45 for Ro; 0.27 and 0.20 for Rsetup), higher resistance factors were obtained for the setup Eq. (3) and steel H-piles. The higher resistance factors are attributed to (1) the application of a more accurate empirical pile setup Eq. (3) as demonstrated in Ng et al. [19], and (2) the use of the regional database given in Table 1, which contained only steel H-piles, hence reducing the uncertainties caused by various pile types used by Yang and Liang [27]. However, it is important to remind that Eq. (3) was developed based on load tests performed within 36 days, the recommended resistance factors shall be used with caution for a pile setup estimated longer than 36 days. Furthermore, the recommended uEOD for Eq. (3), 0.78 and 0.65 for bT of 2.33 and 3.00, respectively, shall be used with caution as they were determined based on a limited sample size of 8. If more pile load test data are available in future, recalibration of the resistance factors using the proposed procedure is encouraged. while the pile performance in terms of achieving the desired Rtarget is verified using either a dynamic analysis method such as WEAP or SLT. The target nominal pile resistance estimated using a static analysis method (Rtarget,static) is determined by: P cQ Rtarget;static  ð8Þ /static where RcQ is the total factored applied load, and ustatic is the resistance factor of any static analysis method listed in Table 4 recommended by AASHTO [3]. If pile setup would be considered during design using Eq. (3), the limit state Eq. (1) can be written as Eq. (9) by replacing Rsetup with Eq. (3), such that: X cQ  uEOD REOD ! "  # fc Ch t þ usetup REOD þ fr log10 ð9Þ tEOD Na rp2 7 Application framework Rearranging this equation, REOD becomes the target nominal pile driving resistance at EOD (Rtarget-EOD), which may be verified using WEAP during construction, via P cQ ! " Rtarget;EOD   # fc Ch t þ fr log10 uEOD þ usetup tEOD Na rp2 In current practice, driven production piles are designed using a static analysis method to determine a suitable pile length for a given target nominal pile resistance (Rtarget), Note that the denominator is always higher than ustatic, since the uEOD of 0.50 for WEAP (or a higher uEOD of 0.78 based on the data given in Table 1) is larger than any of the ð10Þ 123 Acta Geotechnica Table 4 Summary of AASHTO [3] recommended LRFD resistance factors Method Soil type Resistance factor (u) bT = 2.33 bT = 3.00 WEAP All 0.50 0.40 a-Method Clay 0.35 0.28 b-Method k-Method Clay Clay 0.25 0.40 0.20 0.32 CPT-method Clay 0.50 0.40 Based on 19 data sets of steel H-piles embedded in cohesive soil with pile resistance at EOD estimated using WEAP and pile setup estimations determined using the proposed empirical Eq. (3), conservative resistance factors of 0.36 and 0.31 were recommended for pile setup for redundant and non-redundant pile groups, respectively. The suggested application framework of incorporating pile setup in LRFD facilitates economic advantages in pile designs. Acknowledgments The authors would like to thank the Iowa Highway Research Board for sponsoring the research presented in this paper. static analysis methods (i.e., a-method, b-method, k-method, and CPT-method) listed in Table 4. Having a larger denominator term and the same RcQ, the Rtarget,EOD determined using Eq. (10) considering pile setup will certainly be smaller than the Rtarget,static determined using Eq. (8). If the pile setup is considered and incorporated in design using the proposed LRFD procedure: (1) the target pile driving resistance determined using Eq. (10) will be smaller than that determined using the conventional static analysis method; (2) a shorter pile embedment length will be required to achieve the target pile resistance; (3) the retapping of piles after EOD, for which the assumed target pile driving resistance at EOD based on a static analysis method has not been met, can be reduced since a smaller target pile driving resistance will be required; and (4) the economic advantages of pile setup can be realized while complying with the LRFD framework and ensuring a target reliability level. The aforementioned advantages have been confirmed in a recent study that investigated the impact of incorporating setup on 604 production steel H-piles driven in cohesive soils [17]. This study found that the target driving resistance on average was reduced by 17 % and the number of production piles requiring retapping reduced from 37 % to 15 %. 8 Conclusions A successful incorporation of pile setup in LRFD offers cost-effective foundation solutions in cohesive soils. However, the pile setup is not routinely included in foundation design due to lack of an easily usable setup method in compliance with the LRFD framework. In a recent study, it has been shown that the pile setup can be estimated using soil parameters. To incorporate the setup estimated based on this or other similar procedures into the LRFD framework, a closed-form FOSM method has been presented, so that the uncertainties associated with the pile resistance at EOD and the setup can be accounted for separately. 123 Appendix: Derivation of the resistance factor for setup Satisfying the lognormal distributions and independent relationships of loads and resistances, the reliability index (b) is expressed as a ratio of mean to standard deviation of the limit state function (g), which can be expanded as follows:    EðgÞ EðlnðREOD ÞÞ þ E ln Rsetup  EðlnðQÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q b¼ ¼ rg r2lnðREOD Þ þ r2ln R þ r2lnðQÞ ð setup Þ ð11Þ where REOD is the pile resistance at EOD, Rsetup is the pile setup resistance, E(g) is the expected value of the limit state function g, rg is the standard deviation of the limit state function g, E(ln(REOD)) is the expected value of the natural logarithm of the pile resistance at EOD, and rln(REOD) is the standard deviation of the natural logarithm of the pile resistance at EOD, which can be similarly defined for other random variables. To express Eq. (11) in terms of simple means (i.e., R and Q) and coefficients of variation (COV) for resistances and loads of the normal distributions, the mean and standard deviation of a lognormal distribution for any resistance or load can be transformed using the following general expressions:     EðlnðRÞÞ ¼ ln R  0:5 ln 1 þ COV2R ð12Þ   r2lnðRÞ ¼ ln 1 þ COV2R ð13Þ Using these expressions for the three random variables (REOD, Rsetup, and Q) and substituting them into Eq. (11), the reliability index can be expressed as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0v u   u 1 þ COV2Q REOD þ Rsetup A ln þ ln@t COVRR þ 2COV2REOD COV2Rsetup Q rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b¼ h i ln COVRR þ 2COV2REOD COV2Rsetup 1 þ COV2Q ð14Þ Acta Geotechnica Replacing the simple mean values with their respective bias factors (k), a ratio between average measured and predicted values (i.e., Rm/R or Qm/Q), and neglecting the terms that involve multiplying two squared coefficients of variation (i.e., COV2COV2) since their contribution would be insignificantly small, the expression for b is simplified as: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1   1 þ COV2Q kREOD REOD þ kRsetup Rsetup A ln þ ln@ COVRR kQ Q rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b¼ h i ln ðCOVRR Þ 1 þ COV2Q ð15Þ using the LRFD strength limit state equation (cQ = uR) and replacing uR with uEODREOD ? usetupRsetup, the equation can be rearranged for Rsetup as: Rsetup ¼ cQ  uEOD REOD usetup ð16Þ substituting Eq. (16) into Eq. (15) and isolating the usetup as the variable of interest by rearranging, a preliminary equation for usetup can be established as follows: usetup ¼ ksetup ½cQ  uEOD REOD   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi ðkQ QÞ exp b ln ðCOVRR Þ 1 þ COV2Q vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kEOD REOD u u 1 þ COV2 Q t ðCOVRR Þ ð17Þ considering only the dead (QD) and live (QL) loads, as per the AASHTO [3] ‘‘Strength I’’ load combination, the factored load (cQ) and bias load (kQ Q) are expanded to: cQ ¼ cD QD þ cL QL ð18Þ kQ Q ¼ kD QD þ kL QL ð19Þ As defined in AASHTO [3], cD is the dead load factor of 1.25, cL is the live load factor of 1.75, kD is the dead load bias of 1.05, and kL is the live load bias of 1.15. Furthermore, the coefficient of variation of the load (COVQ), derived by Bloomquist et al. [6], in terms of dead and live load components is given as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u QD 2 u kD COV2D þ k2L COV2L u Q2 L ð20Þ COVQ ¼ u uQ 2 QD t D 2 2 k þ 2 k k þ k D L L QL Q2L D substituting Eqs. [18], [19], and [20] into Eq. (17), the usetup can be revised as: usetup ¼ ksetup ½cD QD þ cL QL  uEOD REOD  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkD QD þ kL QL Þ exp b ln½ðCOVRR ÞðjÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kEOD REOD ðjÞ ðCOVRR Þ ð21Þ Normalizing the above expression with respect to the total load (QD ? QL), and further rearrangement of Eq. (21) in terms of the dead load to live load ratio (i.e., QD/ QL) and representing a as the ratio of pile resistance at EOD to the total load (i.e., a = REOD/[QD ? QL]), the resistance factor of pile setup at a target reliability index (bT) can be expressed as:   c ksetup DL  uEOD QDL  usetup ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kDL exp bT ln½ðCOVRR ÞðjÞ QDL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  kEOD a  j COVRR ð22Þ The parameter a, a ratio of pile resistance at EOD to total load, noted above is analogous to a safety factor applied to the REOD if the traditional allowable stress design (ASD) approach would have been considered. The uncertainties associated with REOD have been accounted for in terms of uEOD in Eq. (22) to comply with the LRFD approach. 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