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Incorporating Setup Effects into the Reliability Analysis of Driven Piles

Incorporating Setup Effects into the Reliability Analysis of Driven Piles applied sciences Article Incorporating Setup Effects into the Reliability Analysis of Driven Piles Xiaoya Bian, Jiawei Chen, Xixuan Bai * and Kunpeng Zheng School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China; wit_bianxy@hust.edu.cn (X.B.); 22004010102@stu.wit.edu.cn (J.C.); 22004010105@stu.wit.edu.cn (K.Z.) * Correspondence: baixx87@wit.edu.cn Abstract: Driven-pile setup is referred to a phenomenon in which the bearing capacity of driven piles increases with time after the end of driving (EOD). The setup effect can significantly improve the bearing capacity (ultimate resistance) of driven piles after initial installation, especially the ultimate shaft resistance. Based on the reliability theory and considering the setup effects of driven piles, this article presents an increase factor (M ) for the ultimate resistance of driven piles to modify setup the reliability index calculation formula. At the same time, the correlation between R and R is setup comprehensively considered in the reliability index calculation. Next, the uncertainty analysis of load and resistance is conducted to determine the ranges of relevant parameters. Meanwhile, the influence of four critical parameters (factor of safety FOS, the ratio of dead load to live load r = Q /Q , M , D L setup the correlation coefficient between R and R , and r ) on reliability index are analyzed. 0 setup R0,Rsetup This parametric study indicates that r has a slight influence on the reliability index. However, the reliability index is significantly influenced by FOS, M , and r . Finally, by comparisons setup R0,Rsetup with the existing results, it is concluded that the formula proposed in this study is reasonable, and more uncertainties are considered to make the calculated reliability index closer to a practical engineering application. The presented formula clearly expresses the incorporation of the pile setup effect into reliability index calculation, and it is conducive to improving the prediction accuracy of the design capacity of driven piles. Therefore, the reliability analysis of driven piles considering Citation: Bian, X.; Chen, J.; Bai, X.; setup effects will present a theoretical basis for the application of driven piles in engineering practice. Zheng, K. Incorporating Setup Effects into the Reliability Analysis of Driven Keywords: driven piles; bearing capacity; setup; reliability; correlation coefficient Piles. Appl. Sci. 2022, 12, 2. https:// doi.org/10.3390/app12010002 Academic Editors: Guoliang Dai, Fayun Liang and Xinjun Zou 1. Introduction Piles that are driven into the soil usually show an increase in bearing capacity (ultimate Received: 20 November 2021 resistance) over time after EOD, which is often referred to as the setup effect of driven Accepted: 16 December 2021 piles. This phenomenon is reported by many geotechnical engineers. Tavenas and Audy [1] Published: 21 December 2021 first put forward the setup effect of driven piles. Samson and Authier [2] illustrated four Publisher’s Note: MDPI stays neutral cases in which the bearing capacity of piles changed significantly over time. Basu et al. [3] with regard to jurisdictional claims in investigated the jacking of piles in clay by finite element method. Komurka et al. [4] published maps and institutional affil- proposed a large number of references related to the topic of pile setup. Ng et al. [5] iations. developed a method for quantifying pile setup by using recent field tests when the steel H-piles were driven into clay. The soil setup phenomenon is mainly composed of three factors: (1) excess pore water pressure dissipation, (2) thixotropic effect, and (3) aging effect [6]. Driven piles have obvious disturbance and remodeling effects on the soil around Copyright: © 2021 by the authors. the pile, which makes the pore water pressure dissipate. Therefore, the effects of setup on Licensee MDPI, Basel, Switzerland. pile resistance depend on the type of soil in which the pile is driven. This article is an open access article As for the bearing capacity of piles, along with some of the most traditional and distributed under the terms and commonly used methods among practitioners [7–11], there are more recent approaches, conditions of the Creative Commons which are based on, for example, the finite element method [12–14]. Meanwhile, some Attribution (CC BY) license (https:// researchers [15–17] indicated that pile setup phenomena should be formally included creativecommons.org/licenses/by/ in the forecast technique of total pile resistance as experience and understanding of the 4.0/). Appl. Sci. 2022, 12, 2. https://doi.org/10.3390/app12010002 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 2 2 of 11 phenomenon grew. For the purpose of predicting the pile’s side resistance at a specific time after EOD and incorporating its influence into the pile design, Bullock et al. [18] presented a conservative method in which the side shear setup was included in pile resistance design. Due to different uncertainties associated with EOD resistance and setup resistance, Komurka et al. [19] proposed an approach to split factors of safety into EOD and setup parts in terms of pile capacity, and this method was especially suitable for load and resistance factor design (LRFD). LRFD is the most important and potential class of reliability-based design approaches, which commonly can quantitatively incorporate more uncertainties into the design process, in particular for uncertainties in loads and resistances [20]. Some research studies [21] were conducted to incorporate the setup effects on the LRFD resistance factor into deep foundation design. Yang and Liang [22] added setup resistance into the LRFD of driven piles. Bian et al. [23] suggested a method for a reliability-based design that takes setup effects into account. Despite the fact that full-scale load tests were undertaken for driven piles with setup effects, and a substantial amount of data was acquired [24], the setup resistance of driven piles was rarely used to the maximum extent due to large uncertainties in the driving process. Therefore, the focus of this study is to establish a model for reliability analysis of driven piles considering setup effects. First, this paper presents a novel reliability index formula for driven piles by in- corporating setup into the reliability evaluation method. Second, the range of relevant parameters is determined by the uncertainty analysis of load and resistance. Next, the influence of four critical parameters (factor of safety FOS; the ratio of dead load to live load r = Q /Q ; the ratio of setup resistance to initial ultimate resistance M ; the correlation D L setup coefficient between R and R , r ) on the reliability index are analyzed. Finally, setup 0 R0,Rsetup through a validation example analysis, it is verified that the method proposed in this study is reasonable, and it is concluded that the method proposed in this study is more accurate in calculating the reliability index and considers more uncertainties. 2. Basic Assessment Methods for Pile Setup Pile setup is the increase in axial bearing capacity of the pile driving into the soil with time. As a result, the ultimate resistance is divided into two components, R and R , as 0 setup shown in the following equation: R = R + R (1) 0 setup where R is the ultimate resistance; R is the initial ultimate resistance; R is the setup setup resistance. Equation (1) emphasizes the importance of appropriately assessing setup resistance for reliability evaluation methods. Therefore, this was a topic that drew the attention of many practitioners and presented empirical relationships for predicting the pile setup. These empirical equations are listed in Table 1. Among existing equations, the logarithmic empirical relationship by Skov and Denver [25] has been widely utilized to predict the pile setup, which is, R = R A log (2) setup 0 where A is a variable that varies depending on the soil type; t is the time since the initial pile driving ended; t is the initial time. 0 Appl. Sci. 2022, 12, 2 3 of 11 Table 1. Empirical equation for predicting setup resistance of driven piles. Reference Equation Comments t = 1.0 and A = 0.6 in clay; t = 0.5 and A = 0.2 0 0 in sand; R is the predicted resistance at time t Skov and Denver [25] R = R 1 + A log t 0 t after driving; R is the measured resistance at time t . Values of a: average = 0.13, lower Long et al. [3] R = 1.1R t bound = 0.05, and upper bound = 0.18. R t EOD EOD is the measured resistance at the EOD. Values of B: lower bound = 1.025, and upper 0.1 Svinkin et al. [26] R = BR t EOD bound = 1.4. R is the ultimate resistance with 100% of h  i t/T Bogard and Matlock [27] setup realized, T is the time required to R = R 0.2 + 0.8 50 t u 1+t/T realize 50% of pile setup. 3. Estimation of the Reliability Index of Driven Piles 3.1. General Reliability Evaluation Method of Driven Piles In engineering practice, there are many factors that affect the bearing capacity of driven piles, including pile geometry size, soil type, spatial randomness, variability, etc. At present, the measurement error of soil physical properties and the influence of the pile forming process on soil properties cannot be accurately analyzed, and the main factors affecting pile bearing capacity can only be reflected in the uncertainty of parameters for calculating bearing capacity [28]. The following limit state equation is established to analyze the reliability of driven piles: g = R Q = 0 (3) The load effect Q, herein, only includes the combination of dead load Q and live load Q ; therefore, the reliability index b can be estimated using the following reliability method [29]: " # 2 2 1+COV +COV l R QD QL R n ln l Q +l Q QD D QL L 1+COV b = (4) h  i 2 2 2 ln 1 + COV 1 + COV + COV R QD QL where l , l , and l are the bias factors for resistance, dead load, and live load, R QD QL respectively; COV , COV , and COV are the coefficients of variation (COVs) for R QD QL resistance, dead load, and live load, respectively. 3.2. Setup Effect in Reliability Evaluation of Driven Piles When the setup effect is incorporated into the driven pile design, the limit state function can be expressed as g = R + R Q Q = 0 (5) 0 setup D L In this work, the increase factor for the ultimate resistance is defined as the proportion of setup resistance to initial ultimate resistance, represented as M by Equation (6). setup setup M = (6) setup Then, Equation (7) is derived using Equation (6) and LRFD method [29]. l + l M FOS(Q + Q ) l R R0 Rsetup setup D L FOS(r + 1) = = l + l M (7) R0 Rsetup setup l Q + l Q l Q + l Q l r + l QD D QL L QD D QL L QD QL Appl. Sci. 2022, 12, 2 4 of 11 where r = Q /Q ; FOS is the factor of safety. D L The computation formula for the reliability index of driven piles considering setup effects can be obtained by substituting Equation (7) into Equation (4) as follows: 2 2 1+COV +COV FOS(r+1) QD QL ln l + l M R0 Rsetup setup 2 2 l r+l QD QL 1+COV +COV R0 Rsetup b = (8) h  i 2 2 2 2 ln 1 + COV + COV 1 + COV + COV R0 Rsetup QD QL As there is an inescapable interplay between R and R , the connection between 0 setup R and R should be taken into account in reliability analysis. When considering 0 setup the correlation between R and R the reliability index of driven piles is expressed setup, as follows: 2 2 1+COV +COV FOS(r+1) QD QL ln l + l M R0 Rsetup setup 2 2 l r+l QD QL 1+COV +2r COV COV +COV R0,Rsetup R0 Rsetup R0 Rsetup b = (9) h  i 2 2 2 2 ln 1 + COV + 2r COV COV + COV 1 + COV + COV R0,Rsetup R0 Rsetup R0 Rsetup QD QL where r is the correlation coefficient between R and R . R0,Rsetup 0 setup The relationship between failure probability and reliability index can be calculated with the following function: P = 1 NOR MD I ST(b) (10) 3.3. Uncertainties of Loads and Resistances The mean (or bias factor), coefficient of variation, distribution type, and other factors are used to describe the uncertainty of random variables. The terms normal and lognormal are frequently used to characterize the load and resistance distributions of engineering constructions [30]. The probabilistic features of loads and resistances for driven piles described in Table 2 were employed for this investigation [21,22,29,31]. Table 2. Probabilistic characteristics of random variables of loads and resistances. Random Standard Coefficient of Bias Factor, l Distribution Reference Variable Deviation, s Variation, COV R 1.158 0.393 0.339 Log-normal Paikowsky et al. [21] 1.141 0.543 0.475 Normal Yang and Liang [22] setup 1.023 0.593 0.580 Log-normal Yang and Liang [31] Q 1.080 0.140 0.130 Log-normal AASHTO [29] Q 1.150 0.207 0.180 Log-normal AASHTO [29] Many researchers reported the r = Q /Q for bridge constructions and speculated D L that it varies with bridge span lengths [32,33]. Meanwhile, Hansell et al. [32] adopted a formula to express the relationship between the ratio of r = Q /Q and the length of the D L bridge span, which is, = (1+ I)(0.0132l) (11) where I is the dynamic load factor, and l is the bridge span length in feet. When the bridge span length l varies from 10 m to 70 m, the value of r = Q /Q virtually spread out from D L 0.576 to 4.0, according to Equation (11). As a result, for this investigation, values ranging from 0.5 to 4.0 for r = Q /Q were chosen. D L The increase factor (M ) is re-expressed by using Equations (2) and (6). setup M = A log (12) setup 0 Appl. Sci. 2022, 12, 2 5 of 11 M estimation is dependent on parameters A and log (t/t ), as shown in Equation (12). setup 0 Yang and Liang [22,31] summarized databases that contained both static and dynamic load test results of driven piles in clay and sand, with the value of A ranging from 0.1 to 1.0. Furthermore, time t following EOD varied between 1 and 100 days in the majority of cases, with log(t/t ) with t = 1 ranging between 0 and 2. As a result of the analysis of A and 0 0 log(t/t ), the increase factor (M ) for this study was determined to be between 0 and 2. setup 4. Reliability Analysis 4.1. The Effect of FOS on Reliability Index Firstly, the effect of FOS on the reliability indices of driven piles considering setup effects is studied. The bias factors (l) and coefficients of variation (COV) of loads and resistances of driven piles are summarized in Table 2, and the value of 1.0 for the increase factor M were also used. Meanwhile, based on the analysis results on the effect of r on setup reliability index, the value of r = 3.69 (65 m span length) was accepted for study [34,35]. Reliability analysis was performed for the FOSs ranging from 1.0 to 5.0. The correlation between R and R was not taken into account in this part. Finally, Figure 1 shows the 0 setup reliability indices corresponding to the factor of safety of driven piles in clay and sand. Figure 1. Reliability indices with FOS for driven piles in clay and sand. The variations in the reliability index with FOS, shown in Figure 1, obviously illustrate that the reliability indices of driven piles increase as FOS increases. This indicates the significant influence of FOS on reliability evaluation results of driven piles. In addition, the rate of increase in the reliability index corresponding to FOS slowly decreases with increasing FOS; a value of 3.0 for FOS is a key point in the transition zone of increase rate in Figure 1. Therefore, the value FOS = 3.0 can be used in later studies. 4.2. The Effect of r on Reliability Index The purpose of this subsection is to investigate the impact of r on the reliability index of driven piles considering setup effects. The l and COV of loads and resistances for driven piles in Table 2 are used, FOS was designed as 3.0, and the value 1.0 for the increase factor (M ) was adopted. In this part, the correlation between R and R was not setup 0 setup considered. Based on these proposed values of critical parameters, the reliability index of driven piles can be calculated using Equation (8). Figure 2 describes the variations in the computed reliability index with r = Q /Q of driven piles. D L Appl. Sci. 2022, 12, 2 6 of 11 Figure 2. Reliability indices with r = Q /Q for driven piles in clay and sand. D L It can be seen from Figure 2 that the reliability indices are insensitive to the variations in r = Q /Q for driven piles, which is consistent with other research in this field [36,37]. D L Notably, this conclusion is beneficial to the selection of ratio r = Q /Q in further studies, D L and it is also reasonable to take r = Q /Q as a constant for the other similar research. D L 4.3. The Effect of M on Reliability Index setup In this subsection, the effect of M on the reliability index of driven piles consid- setup ering setup effects is studied. The l and COV of loads and resistances for driven piles in Table 2 were used, and the value FOS = 3.0 and r = 3.69 were obtained from the analysis of the first two subsections. Reliability analysis was performed for the increase factor M setup ranging from 0 to 2.0. In this part, the correlation between R and R was not taken into 0 setup account. Figure 3 shows the reliability indices corresponding to M of driven piles. setup Figure 3. Reliability indices with M for driven piles in clay and sand. setup The results show that in clay and sand, the reliability indices of driven piles increase with rising M , and the growth rates decrease slowly. Additionally, it also can be seen setup that reliability indices of the driven pile in clay are larger than those in the sand; the difference between them is about 15.5% for a given M . This is because the soil around setup the pile will be disturbed in the process of pile driving, which will lead to the dissipation of pore water pressure and the consolidation of soil. Then, the degree of consolidation of soil is affected by the cohesion of soil, so the corresponding reliability index of different soil will be different. Appl. Sci. 2022, 12, 2 7 of 11 4.4. The Effect of r on the Reliability Index R0,Rsetup In this subsection, the effect of r on the reliability index of driven piles con- R0,Rsetup sidering setup effects is studied. The l and COV of loads and resistances for driven piles in Table 2, FOS = 3.0, M = 1.0, and r = 3.69 were used. The reliability index of driven setup piles was computed using Equation (9) for r , ranging from 1.0 to 1.0. Figure 4 R0,Rsetup depicts the reliability indices corresponding to r of driven piles. R0,Rsetup Figure 4. Reliability indices with r for driven piles in clay and sand. R0,Rsetup The results show that in clay and sand, the reliability indices of driven piles decrease with rising r ; however, the decrease rate of reliability indices with r be- R0,Rsetup R0,Rsetup tween 1.0 and 0 is significantly greater than those with r between 0 and 1.0 for R0,Rsetup clay and sand. 5. Validation Example In order to verify the accuracy of the formula proposed in this paper, it was compared with the formula proposed by Haque et al. [17]. At the same time, the data in the Case Pile Wave Analysis Program (CAPWAP) were compared; the measured and predicted resistance values of 19 test piles are relisted in Table 3 [17]. For calculating the reliability index, the cor- responding load statistical parameters, such as l = 1.335, COV = 0.325 [21], l = 1.080, R R QD l = 1.150, COV = 0.130, COV = 0.180 [29], were considered. Meanwhile, the statisti- QL QD QL cal parameters of setup resistance were calculated at four different intervals of 14 days after EOD (i.e., 30, 45, 60, and 90 days after drive). The values of l are 1.218, 1.092, 1.059, and setup 1.033, respectively, and the values of COV are 0.641, 0.62, 0.64, and 0.66, respectively. setup As for the value of the four critical parameters proposed in this study, the conclusion drawn from the “Reliability Analysis” in Section 4 shows that the value of FOS is 3.0, and the value of r is 3.69. Referring to the parameter A model proposed by Haque et al. [17], the value of A in clay and sand are 0.31 and 0.15, respectively. In this study, the values of M was calculated using Equation (12) at four different intervals of 14 days after EOD setup (i.e., 30, 45, 60 and 90 days after driving), which are 0.551, 0.606, 0.645, 0.699 in clay, and 0.267, 0.293, 0.312, 0.338 in sand, respectively. According to the values of R and R 0 setup shown in Table 3, the value of r can be calculated as 0.312, 0.387, 0.386, and 0.378 at R0,Rsetup four different time intervals. The findings of computing the reliability index using the formulas presented by Haque et al. [17] and presented by this study are summarized in Table 4, and the curve of reliability index is drawn together with time interval, as shown in Figure 5. When the correlation between R and R is considered, the reliability index calculated by 0 setup the formula proposed in this paper is not significantly different from that calculated by the formula proposed in Haque et al. [17], which is usually around 0.3. As a result, this conclusion shows that the formula proposed in this study is feasible. When the correlation Appl. Sci. 2022, 12, 2 8 of 11 between R and R is not considered, the difference between the results calculated in 0 setup this study and those calculated by the formula proposed by Haque et al. [17] is about 0.5. Although the results are slightly higher, the correlation between R and R is considered, 0 setup and more uncertainties in the piling process are investigated, bringing the results closer to engineering application. Table 3. Resistance information of 19 test piles by the Case Pile Wave Analysis Program. Resistance Increased with Respect to 14 Days (kN) Resistance of 14 Day (kN) Project Name R R R R Nos. 30–14 45–14 60–14 90–14 Mea Mea Pre Mea Pre Mea Pre Mea Pre 1 Bayou liberty 356 147 147 227 222 280 276 360 356 2 US 90 LA 222 156 98 196 147 222 182 262 236 3 Calcasieu River TP-2 4310 289 365 445 556 556 694 707 885 4 St. Louis Canal Bridge 178 93 62 120 98 138 120 165 151 5 Morman Slough TP-1 1401 125 151 182 231 227 289 289 369 6 Bayou Bouef (west) 592 182 102 231 156 262 196 311 249 7 Fort Buhlow 409 71 67 111 102 138 129 173 165 8 Caminada Bay TP-3 556 485 356 743 547 925 681 1188 867 9 Caminada Bay TP-5 712 574 302 498 463 618 574 792 734 10 Caminada Bay TP-6 565 338 343 516 529 645 658 823 841 11 Caminada Bay TP-7 222 173 191 267 298 329 369 423 472 12 Bayou Lacasine TP-1 1601 311 111 360 173 396 218 445 276 13 LA-1 TP-2 387 178 173 271 262 334 329 427 418 14 LA-1 TP-4a 770 360 356 556 547 694 681 885 872 15 LA-1 TP-4b 3189 494 614 756 939 943 1170 1205 1495 16 LA-1 TP-5a 787 294 294 449 449 560 560 716 721 17 LA-1 TP-5b 1721 187 254 285 387 356 485 454 618 18 LA-1 TP-6 894 351 347 538 534 672 667 859 854 19 LA-1 TP-10 574 116 111 178 173 222 214 280 276 Note: Mea = measured resistance. Pre = predicted resistance. R , R , R , R = setup resistances at 30, 30–14 45–14 60–14 90–14 45, 60, and 90 days after 14 days, respectively. Nos = Numbers. Table 4. Summary of reliability index. Time Intervals (30, 45, 60, and 90 Days after End of Driving) after the 14 Days from EOD Results of This Paper (not Considering Results of This Paper (Considering Type of Results of Haque et al. (2018) Correlation Coefficient between R and R ) Correlation Coefficient between R and R ) Soil 0 setup 0 setup 30–14 45–14 60–14 90–14 30–14 45–14 60–14 90–14 30–14 45–14 60–14 90–14 Clay 1.976 1.942 1.917 1.899 1.646 1.657 1.652 1.654 1.462 1.466 1.460 1.464 Sand 1.976 1.942 1.917 1.899 1.482 1.495 1.486 1.481 1.299 1.304 1.294 1.291 Compared with Haque et al. [17], this paper proposes a critical parameter (M ), setup which is suitable for various soil types and takes more uncertainties into account, providing a more comprehensive theoretical basis for future research. Figure 5 further demonstrates that the reliability index for the driven pile considering setup effects in clay is much higher than that of the driven pile in sand, which is consistent with the conclusion of Section 4 “Reliability Analysis”. Appl. Sci. 2022, 12, 2 9 of 11 Figure 5. Reliability indices with time intervals for driven piles in clay and sand. 6. Conclusions This paper presented an increase factor for the ultimate resistance for driven piles to modify the reliability index calculation formula. Meanwhile, the study conducted the uncertainty analysis of load and resistance to determine the ranges of relevant parameters. Finally, the impact of four critical parameters on the reliability index were investigated and compared with the existing results. Through parameter analysis, it is concluded that FOS has a significant influence on the reliability index of driven piles considering setup effects. The reliability index is essentially unaffected by r = Q /Q , so it can be used as a constant when calculating D L the reliability index. M was a critical parameter in this study and has a significant setup impact on the reliability index of driven pile considering setup effects. Therefore, the value of M is particularly important in the reliability analysis of driven piles considering setup setup effects and is generally selected according to the type of soil. Meanwhile r R0,Rsetup has a significant influence on the reliability index of driven piles, and when the r R0,Rsetup value is smaller, the corresponding reliability index is higher. Through validation example analysis, the proposed formula in this paper is feasible. Additionally, it is concluded that more uncertainties will be considered when using the formula proposed in this paper to calculate the reliability index of driven pile considering setup effects. To summarize, if the setup effect is not entirely considered, the reliability index obtained is very conservative. Therefore, the reasonable evaluation of setup effects is crucial for the reliability analysis of driven piles. Author Contributions: Conceptualization, X.B. (Xiaoya Bian); methodology, X.B. (Xiaoya Bian) and J.C.; validation, X.B. (Xiaoya Bian), X.B. (Xixuan Bai), J.C. and K.Z.; writing—original draft preparation, X.B. (Xiaoya Bian) and J.C.; writing—review and editing, X.B. (Xiaoya Bian), J.C. and K.Z. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Natural Science Foundation of China (52078396, 51708428). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2022, 12, 2 10 of 11 References 1. Tavenas, F.; Audy, R. 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Reliability of bored pile foundations considering bias in failure criteria. Can. Geotech. J. 2005, 42, 1086–1093. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Incorporating Setup Effects into the Reliability Analysis of Driven Piles

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applied sciences Article Incorporating Setup Effects into the Reliability Analysis of Driven Piles Xiaoya Bian, Jiawei Chen, Xixuan Bai * and Kunpeng Zheng School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China; wit_bianxy@hust.edu.cn (X.B.); 22004010102@stu.wit.edu.cn (J.C.); 22004010105@stu.wit.edu.cn (K.Z.) * Correspondence: baixx87@wit.edu.cn Abstract: Driven-pile setup is referred to a phenomenon in which the bearing capacity of driven piles increases with time after the end of driving (EOD). The setup effect can significantly improve the bearing capacity (ultimate resistance) of driven piles after initial installation, especially the ultimate shaft resistance. Based on the reliability theory and considering the setup effects of driven piles, this article presents an increase factor (M ) for the ultimate resistance of driven piles to modify setup the reliability index calculation formula. At the same time, the correlation between R and R is setup comprehensively considered in the reliability index calculation. Next, the uncertainty analysis of load and resistance is conducted to determine the ranges of relevant parameters. Meanwhile, the influence of four critical parameters (factor of safety FOS, the ratio of dead load to live load r = Q /Q , M , D L setup the correlation coefficient between R and R , and r ) on reliability index are analyzed. 0 setup R0,Rsetup This parametric study indicates that r has a slight influence on the reliability index. However, the reliability index is significantly influenced by FOS, M , and r . Finally, by comparisons setup R0,Rsetup with the existing results, it is concluded that the formula proposed in this study is reasonable, and more uncertainties are considered to make the calculated reliability index closer to a practical engineering application. The presented formula clearly expresses the incorporation of the pile setup effect into reliability index calculation, and it is conducive to improving the prediction accuracy of the design capacity of driven piles. Therefore, the reliability analysis of driven piles considering Citation: Bian, X.; Chen, J.; Bai, X.; setup effects will present a theoretical basis for the application of driven piles in engineering practice. Zheng, K. Incorporating Setup Effects into the Reliability Analysis of Driven Keywords: driven piles; bearing capacity; setup; reliability; correlation coefficient Piles. Appl. Sci. 2022, 12, 2. https:// doi.org/10.3390/app12010002 Academic Editors: Guoliang Dai, Fayun Liang and Xinjun Zou 1. Introduction Piles that are driven into the soil usually show an increase in bearing capacity (ultimate Received: 20 November 2021 resistance) over time after EOD, which is often referred to as the setup effect of driven Accepted: 16 December 2021 piles. This phenomenon is reported by many geotechnical engineers. Tavenas and Audy [1] Published: 21 December 2021 first put forward the setup effect of driven piles. Samson and Authier [2] illustrated four Publisher’s Note: MDPI stays neutral cases in which the bearing capacity of piles changed significantly over time. Basu et al. [3] with regard to jurisdictional claims in investigated the jacking of piles in clay by finite element method. Komurka et al. [4] published maps and institutional affil- proposed a large number of references related to the topic of pile setup. Ng et al. [5] iations. developed a method for quantifying pile setup by using recent field tests when the steel H-piles were driven into clay. The soil setup phenomenon is mainly composed of three factors: (1) excess pore water pressure dissipation, (2) thixotropic effect, and (3) aging effect [6]. Driven piles have obvious disturbance and remodeling effects on the soil around Copyright: © 2021 by the authors. the pile, which makes the pore water pressure dissipate. Therefore, the effects of setup on Licensee MDPI, Basel, Switzerland. pile resistance depend on the type of soil in which the pile is driven. This article is an open access article As for the bearing capacity of piles, along with some of the most traditional and distributed under the terms and commonly used methods among practitioners [7–11], there are more recent approaches, conditions of the Creative Commons which are based on, for example, the finite element method [12–14]. Meanwhile, some Attribution (CC BY) license (https:// researchers [15–17] indicated that pile setup phenomena should be formally included creativecommons.org/licenses/by/ in the forecast technique of total pile resistance as experience and understanding of the 4.0/). Appl. Sci. 2022, 12, 2. https://doi.org/10.3390/app12010002 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 2 2 of 11 phenomenon grew. For the purpose of predicting the pile’s side resistance at a specific time after EOD and incorporating its influence into the pile design, Bullock et al. [18] presented a conservative method in which the side shear setup was included in pile resistance design. Due to different uncertainties associated with EOD resistance and setup resistance, Komurka et al. [19] proposed an approach to split factors of safety into EOD and setup parts in terms of pile capacity, and this method was especially suitable for load and resistance factor design (LRFD). LRFD is the most important and potential class of reliability-based design approaches, which commonly can quantitatively incorporate more uncertainties into the design process, in particular for uncertainties in loads and resistances [20]. Some research studies [21] were conducted to incorporate the setup effects on the LRFD resistance factor into deep foundation design. Yang and Liang [22] added setup resistance into the LRFD of driven piles. Bian et al. [23] suggested a method for a reliability-based design that takes setup effects into account. Despite the fact that full-scale load tests were undertaken for driven piles with setup effects, and a substantial amount of data was acquired [24], the setup resistance of driven piles was rarely used to the maximum extent due to large uncertainties in the driving process. Therefore, the focus of this study is to establish a model for reliability analysis of driven piles considering setup effects. First, this paper presents a novel reliability index formula for driven piles by in- corporating setup into the reliability evaluation method. Second, the range of relevant parameters is determined by the uncertainty analysis of load and resistance. Next, the influence of four critical parameters (factor of safety FOS; the ratio of dead load to live load r = Q /Q ; the ratio of setup resistance to initial ultimate resistance M ; the correlation D L setup coefficient between R and R , r ) on the reliability index are analyzed. Finally, setup 0 R0,Rsetup through a validation example analysis, it is verified that the method proposed in this study is reasonable, and it is concluded that the method proposed in this study is more accurate in calculating the reliability index and considers more uncertainties. 2. Basic Assessment Methods for Pile Setup Pile setup is the increase in axial bearing capacity of the pile driving into the soil with time. As a result, the ultimate resistance is divided into two components, R and R , as 0 setup shown in the following equation: R = R + R (1) 0 setup where R is the ultimate resistance; R is the initial ultimate resistance; R is the setup setup resistance. Equation (1) emphasizes the importance of appropriately assessing setup resistance for reliability evaluation methods. Therefore, this was a topic that drew the attention of many practitioners and presented empirical relationships for predicting the pile setup. These empirical equations are listed in Table 1. Among existing equations, the logarithmic empirical relationship by Skov and Denver [25] has been widely utilized to predict the pile setup, which is, R = R A log (2) setup 0 where A is a variable that varies depending on the soil type; t is the time since the initial pile driving ended; t is the initial time. 0 Appl. Sci. 2022, 12, 2 3 of 11 Table 1. Empirical equation for predicting setup resistance of driven piles. Reference Equation Comments t = 1.0 and A = 0.6 in clay; t = 0.5 and A = 0.2 0 0 in sand; R is the predicted resistance at time t Skov and Denver [25] R = R 1 + A log t 0 t after driving; R is the measured resistance at time t . Values of a: average = 0.13, lower Long et al. [3] R = 1.1R t bound = 0.05, and upper bound = 0.18. R t EOD EOD is the measured resistance at the EOD. Values of B: lower bound = 1.025, and upper 0.1 Svinkin et al. [26] R = BR t EOD bound = 1.4. R is the ultimate resistance with 100% of h  i t/T Bogard and Matlock [27] setup realized, T is the time required to R = R 0.2 + 0.8 50 t u 1+t/T realize 50% of pile setup. 3. Estimation of the Reliability Index of Driven Piles 3.1. General Reliability Evaluation Method of Driven Piles In engineering practice, there are many factors that affect the bearing capacity of driven piles, including pile geometry size, soil type, spatial randomness, variability, etc. At present, the measurement error of soil physical properties and the influence of the pile forming process on soil properties cannot be accurately analyzed, and the main factors affecting pile bearing capacity can only be reflected in the uncertainty of parameters for calculating bearing capacity [28]. The following limit state equation is established to analyze the reliability of driven piles: g = R Q = 0 (3) The load effect Q, herein, only includes the combination of dead load Q and live load Q ; therefore, the reliability index b can be estimated using the following reliability method [29]: " # 2 2 1+COV +COV l R QD QL R n ln l Q +l Q QD D QL L 1+COV b = (4) h  i 2 2 2 ln 1 + COV 1 + COV + COV R QD QL where l , l , and l are the bias factors for resistance, dead load, and live load, R QD QL respectively; COV , COV , and COV are the coefficients of variation (COVs) for R QD QL resistance, dead load, and live load, respectively. 3.2. Setup Effect in Reliability Evaluation of Driven Piles When the setup effect is incorporated into the driven pile design, the limit state function can be expressed as g = R + R Q Q = 0 (5) 0 setup D L In this work, the increase factor for the ultimate resistance is defined as the proportion of setup resistance to initial ultimate resistance, represented as M by Equation (6). setup setup M = (6) setup Then, Equation (7) is derived using Equation (6) and LRFD method [29]. l + l M FOS(Q + Q ) l R R0 Rsetup setup D L FOS(r + 1) = = l + l M (7) R0 Rsetup setup l Q + l Q l Q + l Q l r + l QD D QL L QD D QL L QD QL Appl. Sci. 2022, 12, 2 4 of 11 where r = Q /Q ; FOS is the factor of safety. D L The computation formula for the reliability index of driven piles considering setup effects can be obtained by substituting Equation (7) into Equation (4) as follows: 2 2 1+COV +COV FOS(r+1) QD QL ln l + l M R0 Rsetup setup 2 2 l r+l QD QL 1+COV +COV R0 Rsetup b = (8) h  i 2 2 2 2 ln 1 + COV + COV 1 + COV + COV R0 Rsetup QD QL As there is an inescapable interplay between R and R , the connection between 0 setup R and R should be taken into account in reliability analysis. When considering 0 setup the correlation between R and R the reliability index of driven piles is expressed setup, as follows: 2 2 1+COV +COV FOS(r+1) QD QL ln l + l M R0 Rsetup setup 2 2 l r+l QD QL 1+COV +2r COV COV +COV R0,Rsetup R0 Rsetup R0 Rsetup b = (9) h  i 2 2 2 2 ln 1 + COV + 2r COV COV + COV 1 + COV + COV R0,Rsetup R0 Rsetup R0 Rsetup QD QL where r is the correlation coefficient between R and R . R0,Rsetup 0 setup The relationship between failure probability and reliability index can be calculated with the following function: P = 1 NOR MD I ST(b) (10) 3.3. Uncertainties of Loads and Resistances The mean (or bias factor), coefficient of variation, distribution type, and other factors are used to describe the uncertainty of random variables. The terms normal and lognormal are frequently used to characterize the load and resistance distributions of engineering constructions [30]. The probabilistic features of loads and resistances for driven piles described in Table 2 were employed for this investigation [21,22,29,31]. Table 2. Probabilistic characteristics of random variables of loads and resistances. Random Standard Coefficient of Bias Factor, l Distribution Reference Variable Deviation, s Variation, COV R 1.158 0.393 0.339 Log-normal Paikowsky et al. [21] 1.141 0.543 0.475 Normal Yang and Liang [22] setup 1.023 0.593 0.580 Log-normal Yang and Liang [31] Q 1.080 0.140 0.130 Log-normal AASHTO [29] Q 1.150 0.207 0.180 Log-normal AASHTO [29] Many researchers reported the r = Q /Q for bridge constructions and speculated D L that it varies with bridge span lengths [32,33]. Meanwhile, Hansell et al. [32] adopted a formula to express the relationship between the ratio of r = Q /Q and the length of the D L bridge span, which is, = (1+ I)(0.0132l) (11) where I is the dynamic load factor, and l is the bridge span length in feet. When the bridge span length l varies from 10 m to 70 m, the value of r = Q /Q virtually spread out from D L 0.576 to 4.0, according to Equation (11). As a result, for this investigation, values ranging from 0.5 to 4.0 for r = Q /Q were chosen. D L The increase factor (M ) is re-expressed by using Equations (2) and (6). setup M = A log (12) setup 0 Appl. Sci. 2022, 12, 2 5 of 11 M estimation is dependent on parameters A and log (t/t ), as shown in Equation (12). setup 0 Yang and Liang [22,31] summarized databases that contained both static and dynamic load test results of driven piles in clay and sand, with the value of A ranging from 0.1 to 1.0. Furthermore, time t following EOD varied between 1 and 100 days in the majority of cases, with log(t/t ) with t = 1 ranging between 0 and 2. As a result of the analysis of A and 0 0 log(t/t ), the increase factor (M ) for this study was determined to be between 0 and 2. setup 4. Reliability Analysis 4.1. The Effect of FOS on Reliability Index Firstly, the effect of FOS on the reliability indices of driven piles considering setup effects is studied. The bias factors (l) and coefficients of variation (COV) of loads and resistances of driven piles are summarized in Table 2, and the value of 1.0 for the increase factor M were also used. Meanwhile, based on the analysis results on the effect of r on setup reliability index, the value of r = 3.69 (65 m span length) was accepted for study [34,35]. Reliability analysis was performed for the FOSs ranging from 1.0 to 5.0. The correlation between R and R was not taken into account in this part. Finally, Figure 1 shows the 0 setup reliability indices corresponding to the factor of safety of driven piles in clay and sand. Figure 1. Reliability indices with FOS for driven piles in clay and sand. The variations in the reliability index with FOS, shown in Figure 1, obviously illustrate that the reliability indices of driven piles increase as FOS increases. This indicates the significant influence of FOS on reliability evaluation results of driven piles. In addition, the rate of increase in the reliability index corresponding to FOS slowly decreases with increasing FOS; a value of 3.0 for FOS is a key point in the transition zone of increase rate in Figure 1. Therefore, the value FOS = 3.0 can be used in later studies. 4.2. The Effect of r on Reliability Index The purpose of this subsection is to investigate the impact of r on the reliability index of driven piles considering setup effects. The l and COV of loads and resistances for driven piles in Table 2 are used, FOS was designed as 3.0, and the value 1.0 for the increase factor (M ) was adopted. In this part, the correlation between R and R was not setup 0 setup considered. Based on these proposed values of critical parameters, the reliability index of driven piles can be calculated using Equation (8). Figure 2 describes the variations in the computed reliability index with r = Q /Q of driven piles. D L Appl. Sci. 2022, 12, 2 6 of 11 Figure 2. Reliability indices with r = Q /Q for driven piles in clay and sand. D L It can be seen from Figure 2 that the reliability indices are insensitive to the variations in r = Q /Q for driven piles, which is consistent with other research in this field [36,37]. D L Notably, this conclusion is beneficial to the selection of ratio r = Q /Q in further studies, D L and it is also reasonable to take r = Q /Q as a constant for the other similar research. D L 4.3. The Effect of M on Reliability Index setup In this subsection, the effect of M on the reliability index of driven piles consid- setup ering setup effects is studied. The l and COV of loads and resistances for driven piles in Table 2 were used, and the value FOS = 3.0 and r = 3.69 were obtained from the analysis of the first two subsections. Reliability analysis was performed for the increase factor M setup ranging from 0 to 2.0. In this part, the correlation between R and R was not taken into 0 setup account. Figure 3 shows the reliability indices corresponding to M of driven piles. setup Figure 3. Reliability indices with M for driven piles in clay and sand. setup The results show that in clay and sand, the reliability indices of driven piles increase with rising M , and the growth rates decrease slowly. Additionally, it also can be seen setup that reliability indices of the driven pile in clay are larger than those in the sand; the difference between them is about 15.5% for a given M . This is because the soil around setup the pile will be disturbed in the process of pile driving, which will lead to the dissipation of pore water pressure and the consolidation of soil. Then, the degree of consolidation of soil is affected by the cohesion of soil, so the corresponding reliability index of different soil will be different. Appl. Sci. 2022, 12, 2 7 of 11 4.4. The Effect of r on the Reliability Index R0,Rsetup In this subsection, the effect of r on the reliability index of driven piles con- R0,Rsetup sidering setup effects is studied. The l and COV of loads and resistances for driven piles in Table 2, FOS = 3.0, M = 1.0, and r = 3.69 were used. The reliability index of driven setup piles was computed using Equation (9) for r , ranging from 1.0 to 1.0. Figure 4 R0,Rsetup depicts the reliability indices corresponding to r of driven piles. R0,Rsetup Figure 4. Reliability indices with r for driven piles in clay and sand. R0,Rsetup The results show that in clay and sand, the reliability indices of driven piles decrease with rising r ; however, the decrease rate of reliability indices with r be- R0,Rsetup R0,Rsetup tween 1.0 and 0 is significantly greater than those with r between 0 and 1.0 for R0,Rsetup clay and sand. 5. Validation Example In order to verify the accuracy of the formula proposed in this paper, it was compared with the formula proposed by Haque et al. [17]. At the same time, the data in the Case Pile Wave Analysis Program (CAPWAP) were compared; the measured and predicted resistance values of 19 test piles are relisted in Table 3 [17]. For calculating the reliability index, the cor- responding load statistical parameters, such as l = 1.335, COV = 0.325 [21], l = 1.080, R R QD l = 1.150, COV = 0.130, COV = 0.180 [29], were considered. Meanwhile, the statisti- QL QD QL cal parameters of setup resistance were calculated at four different intervals of 14 days after EOD (i.e., 30, 45, 60, and 90 days after drive). The values of l are 1.218, 1.092, 1.059, and setup 1.033, respectively, and the values of COV are 0.641, 0.62, 0.64, and 0.66, respectively. setup As for the value of the four critical parameters proposed in this study, the conclusion drawn from the “Reliability Analysis” in Section 4 shows that the value of FOS is 3.0, and the value of r is 3.69. Referring to the parameter A model proposed by Haque et al. [17], the value of A in clay and sand are 0.31 and 0.15, respectively. In this study, the values of M was calculated using Equation (12) at four different intervals of 14 days after EOD setup (i.e., 30, 45, 60 and 90 days after driving), which are 0.551, 0.606, 0.645, 0.699 in clay, and 0.267, 0.293, 0.312, 0.338 in sand, respectively. According to the values of R and R 0 setup shown in Table 3, the value of r can be calculated as 0.312, 0.387, 0.386, and 0.378 at R0,Rsetup four different time intervals. The findings of computing the reliability index using the formulas presented by Haque et al. [17] and presented by this study are summarized in Table 4, and the curve of reliability index is drawn together with time interval, as shown in Figure 5. When the correlation between R and R is considered, the reliability index calculated by 0 setup the formula proposed in this paper is not significantly different from that calculated by the formula proposed in Haque et al. [17], which is usually around 0.3. As a result, this conclusion shows that the formula proposed in this study is feasible. When the correlation Appl. Sci. 2022, 12, 2 8 of 11 between R and R is not considered, the difference between the results calculated in 0 setup this study and those calculated by the formula proposed by Haque et al. [17] is about 0.5. Although the results are slightly higher, the correlation between R and R is considered, 0 setup and more uncertainties in the piling process are investigated, bringing the results closer to engineering application. Table 3. Resistance information of 19 test piles by the Case Pile Wave Analysis Program. Resistance Increased with Respect to 14 Days (kN) Resistance of 14 Day (kN) Project Name R R R R Nos. 30–14 45–14 60–14 90–14 Mea Mea Pre Mea Pre Mea Pre Mea Pre 1 Bayou liberty 356 147 147 227 222 280 276 360 356 2 US 90 LA 222 156 98 196 147 222 182 262 236 3 Calcasieu River TP-2 4310 289 365 445 556 556 694 707 885 4 St. Louis Canal Bridge 178 93 62 120 98 138 120 165 151 5 Morman Slough TP-1 1401 125 151 182 231 227 289 289 369 6 Bayou Bouef (west) 592 182 102 231 156 262 196 311 249 7 Fort Buhlow 409 71 67 111 102 138 129 173 165 8 Caminada Bay TP-3 556 485 356 743 547 925 681 1188 867 9 Caminada Bay TP-5 712 574 302 498 463 618 574 792 734 10 Caminada Bay TP-6 565 338 343 516 529 645 658 823 841 11 Caminada Bay TP-7 222 173 191 267 298 329 369 423 472 12 Bayou Lacasine TP-1 1601 311 111 360 173 396 218 445 276 13 LA-1 TP-2 387 178 173 271 262 334 329 427 418 14 LA-1 TP-4a 770 360 356 556 547 694 681 885 872 15 LA-1 TP-4b 3189 494 614 756 939 943 1170 1205 1495 16 LA-1 TP-5a 787 294 294 449 449 560 560 716 721 17 LA-1 TP-5b 1721 187 254 285 387 356 485 454 618 18 LA-1 TP-6 894 351 347 538 534 672 667 859 854 19 LA-1 TP-10 574 116 111 178 173 222 214 280 276 Note: Mea = measured resistance. Pre = predicted resistance. R , R , R , R = setup resistances at 30, 30–14 45–14 60–14 90–14 45, 60, and 90 days after 14 days, respectively. Nos = Numbers. Table 4. Summary of reliability index. Time Intervals (30, 45, 60, and 90 Days after End of Driving) after the 14 Days from EOD Results of This Paper (not Considering Results of This Paper (Considering Type of Results of Haque et al. (2018) Correlation Coefficient between R and R ) Correlation Coefficient between R and R ) Soil 0 setup 0 setup 30–14 45–14 60–14 90–14 30–14 45–14 60–14 90–14 30–14 45–14 60–14 90–14 Clay 1.976 1.942 1.917 1.899 1.646 1.657 1.652 1.654 1.462 1.466 1.460 1.464 Sand 1.976 1.942 1.917 1.899 1.482 1.495 1.486 1.481 1.299 1.304 1.294 1.291 Compared with Haque et al. [17], this paper proposes a critical parameter (M ), setup which is suitable for various soil types and takes more uncertainties into account, providing a more comprehensive theoretical basis for future research. Figure 5 further demonstrates that the reliability index for the driven pile considering setup effects in clay is much higher than that of the driven pile in sand, which is consistent with the conclusion of Section 4 “Reliability Analysis”. Appl. Sci. 2022, 12, 2 9 of 11 Figure 5. Reliability indices with time intervals for driven piles in clay and sand. 6. Conclusions This paper presented an increase factor for the ultimate resistance for driven piles to modify the reliability index calculation formula. Meanwhile, the study conducted the uncertainty analysis of load and resistance to determine the ranges of relevant parameters. Finally, the impact of four critical parameters on the reliability index were investigated and compared with the existing results. Through parameter analysis, it is concluded that FOS has a significant influence on the reliability index of driven piles considering setup effects. The reliability index is essentially unaffected by r = Q /Q , so it can be used as a constant when calculating D L the reliability index. M was a critical parameter in this study and has a significant setup impact on the reliability index of driven pile considering setup effects. Therefore, the value of M is particularly important in the reliability analysis of driven piles considering setup setup effects and is generally selected according to the type of soil. Meanwhile r R0,Rsetup has a significant influence on the reliability index of driven piles, and when the r R0,Rsetup value is smaller, the corresponding reliability index is higher. Through validation example analysis, the proposed formula in this paper is feasible. Additionally, it is concluded that more uncertainties will be considered when using the formula proposed in this paper to calculate the reliability index of driven pile considering setup effects. To summarize, if the setup effect is not entirely considered, the reliability index obtained is very conservative. Therefore, the reasonable evaluation of setup effects is crucial for the reliability analysis of driven piles. Author Contributions: Conceptualization, X.B. (Xiaoya Bian); methodology, X.B. (Xiaoya Bian) and J.C.; validation, X.B. (Xiaoya Bian), X.B. (Xixuan Bai), J.C. and K.Z.; writing—original draft preparation, X.B. (Xiaoya Bian) and J.C.; writing—review and editing, X.B. (Xiaoya Bian), J.C. and K.Z. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Natural Science Foundation of China (52078396, 51708428). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2022, 12, 2 10 of 11 References 1. Tavenas, F.; Audy, R. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Dec 21, 2021

Keywords: driven piles; bearing capacity; setup; reliability; correlation coefficient

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