Computers and Geotechnics 81 (2017) 229–238

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Computers and Geotechnics

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Research Paper

Calibration of resistance factor for design of pile foundations consideringfeasibility robustness

http://dx.doi.org/10.1016/j.compgeo.2016.08.0180266-352X/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Key Laboratory of Road and Traffic Engineering of theMinistry of Education, Tongji University, 4800 Caoan Road, Shanghai 201804, China.

E-mail addresses: dianqing@whu.edu.cn (D.-Q. Li), px09@whu.edu.cn (X. Peng),skhoshn@g.clemson.edu (S. Khoshnevisan), hsein@clemson.edu (C.H. Juang).

Dian-Qing Li a, Xing Peng a, Sara Khoshnevisan b, C. Hsein Juang c,b,⇑a State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, ChinabGlenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USAcKey Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, 4800 Caoan Road, Shanghai 201804, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 April 2016Received in revised form 8 August 2016Accepted 19 August 2016

Keywords:Pile foundationsLoad and resistance factor designUncertaintyTarget reliability indexFeasibility robustnessCalibration

The resistance factor for pile foundations in load and resistance factor design (LRFD) is traditionally cal-ibrated considering target reliability index (bT) and statistics of load and resistance bias factors. However,the resistance bias factor is hard to quantify statistically. Consequently, the design obtained using the cal-ibrated resistance factor can still miss bT if the variation in resistance bias factor has been underesti-mated. In this paper, we propose a new resistance factor calibration approach to address this dilemmaby considering ‘‘feasibility robustness” of design in the calibration process. Herein, the feasibility robust-ness is defined as a probability that the bT requirement can still be satisfied even in the presence of uncer-tainty or variation in the computed bearing capacity. For illustration, LRFD approach for pile foundationscommonly used in Shanghai, China is examined. Emphasis is placed on re-calibration of resistance factorsat various feasibility robustness levels, with due consideration of the variation in the resistance bias fac-tor. A case study is presented to illustrate the use of the re-calibrated resistance factors. The results showthat the feasibility robustness is gained at the expense of cost efficiency; in other words, the two objec-tives are conflicting. To aid in the design decision-making, an optimal feasibility robustness level and cor-responding resistance factors are suggested in the absence of a designer’s preference.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Foundations have traditionally been designed based on theallowable stress design (ASD) approach, which normally employsa single global factor of safety (FS) to cope with all uncertaintiesassociated with load and resistance (e.g., [5,28,3,13]). However,the nominal FS obtained from a deterministic method cannot accu-rately reflect the true level of safety [10,22]. Currently, the load andresistance factor design (LRFD) approach, which is a simpler vari-ant of the reliability-based design method, has been gaining accep-tance. Compared with ASD approach, the LRFD approach that isbased on reliability theory can reasonably consider load and resis-tance uncertainties in the design [28,21]. The LRFD approach gen-erally uses load factors and resistance factor to account for theuncertainty in load and resistance, respectively. In recent years,extensive research (e.g., [34,26,1,19,36,22]) was conducted to cali-brate resistance factor for the design of pile foundation for a given

set of load factors. Generally, the resistance factor is calibrated to aprescribed target reliability index bT considering the statistics ofload and resistance bias factors [38,35].

In LRFD, the resistance bias factor is defined as the ratio of themeasured bearing capacity from a load test to the predicted (orcomputed) bearing capacity by a static bearing capacity model,and is modeled as a random variable reflecting mainly the uncer-tainty in the model that is used to compute the capacity. A properstatistical characterization of resistance bias factor requires collec-tion of reliable static load test data, which is the most importanttask for LRFD calibration [19]. In practice, however, the resistancebias factor statistics are hard to ascertain, particularly when thedata are limited in quality and/or quantity [2]. Thus, uncertaintyis inherent in the derived statistical parameters of the resistancebias factor. Unfortunately, the resistance factor calibrated for LRFDis very sensitive to the uncertainty in the resistance bias factor.Consequently, a design obtained using the calibrated resistancefactor may not achieve bT (i.e., the design is not feasible) if the vari-ation in the resistance bias factor is underestimated.

To address this dilemma, the authors propose a new approachfor resistance factor calibration that considers explicitly the feasi-bility robustness of design [25]. Emphasis of this paper is placed

230 D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238

on re-calibration of resistance factor with due consideration ofvariation in the resistance bias factor. By considering the feasibilityrobustness, design using the re-calibrated resistance factor willalways satisfy the bT requirement to the extent defined by engineereven if uncertainty exists in the computed capacity.

It should be noted that the robustness design concept is notnew; in fact, it was introduced by Taguchi [30] and has been usedwidely in various engineering fields (e.g., [31,6,9,24,4,20,18,27]).Furthermore, examples of geotechnical design with LRFD approachconsidering design robustness have been reported [16,11]. How-ever, this paper represents the first attempt at introducing therobustness concept into the LRFD calibration. The novelty of thispaper is evidenced in the results presented.

This paper is outlined as follows. First, the traditional approachof resistance factor calibration and its possible drawback are pre-sented through a LRFD calibration practice of pile foundations inShanghai, China. Next, the feasibility robustness concept is intro-duced, followed by the development of the new resistance factorcalibration approach considering feasibility robustness. Then, theresistance factors are re-calibrated at various predefined levels offeasibility robustness and illustrated through a bored pile designexample. Finally, a most preferred feasibility robustness level andthe corresponding resistance factors are suggested in the absenceof a designer’s preference.

2. Traditional approach for resistance factor calibration

In this section, the traditional resistance factor calibration pro-cess is reviewed using an example reported by Li et al. [22] thatdescribes Shanghai, China experience. In Li et al. [22], resistancefactors for total load-carrying capacity are calibrated for drivenpiles and bored piles designed by three commonly used methodsin Shanghai, i.e., the static load test-based method (LT method),the design table method (DT method), and the cone penetrationtest-based method (CPT method). The details of these methodsare summarized in Appendix A. Let R, QD, and QL denote totalcapacity, dead load, and live load, respectively. The design equationin Shanghai can be expressed as:

Rn

cR¼ cDQDn þ cLQLn ð1Þ

where Rn, QDn, and QLn are the nominal values for R, QD, and QL,respectively; and cR, cD, and cL are the partial factors for R, QD, andQL, respectively. Note that in some codes, such as AASHTO [1], a par-tial factor / is applied to resistance in a form such that cR = 1//.

According to AASHTO [1], using an assumption of lognormaldistribution function for resistance and loads, reliability index bcan be calculated using first order secondmoment method as (after[33,37]):

b �ln

kRcRðcD þ cLqÞkD þ kLq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ COV2

Q

1þ COV2R

vuut0@

1A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ COV2

R

� �1þ COV2

Q

� �h ir ð2Þ

where kR, kD, and kL are mean bias factors of resistance, dead load,and live load, respectively; q is the live load to dead load ratio; Qis the total load (i.e., Q = QD + QL); COVR and COVQ are the coeffi-cients of variation of the resistance bias factor and load bias factor,respectively. According to Li et al. [22], COVQ can be calculated as:

COVQ ¼ 11þ q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCOV2

D þ q2COV2L

qð3Þ

where COVD and COVL are COVs of dead load bias factor and liveload bias factor, respectively. As noted in Zhang et al. [39], when

an empirical relationship is used to compute the bearing capacity,the computed capacity is subjected to two types of uncertainties,i.e., the within-site variability and the cross-site variability. Thewithin-site variability is mainly caused by the inherent variabilityof soil properties in the zone influencing each pile and by the con-struction errors associated with the site-specific workmanship. Thecross-site variability is mainly caused by the regional variation insoil properties and by the construction errors associated with theworkmanship in a region. In Li et al. [22], both the within-site vari-ability and the cross-site variability of the pile capacity are consid-ered; thus kR and COVR can be further written as:

kR ¼ kR1kR2 ð4ÞCOVR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCOV2

R1 þ COV2R2

qð5Þ

where kR1 and COVR1 are the mean and COV of the bias factoraccounting for within-site variability, respectively; and kR2 andCOVR2 are the mean and COV of the bias factor accounting forcross-site variability, respectively.

In resistance factor calibration, a target reliability index bT ispre-defined. Based on Eq. (2), the value of cR required to achievebT can be obtained as:

cR ¼ kD þ kLqkRðcD þ cLqÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ COV2

R

1þ COV2Q

vuut

� exp bT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ COV2

R

� �1þ COV2

Q

� �h ir� �ð6Þ

Eq. (6) shows that cR is a function of bT, load bias factor statis-tics, and resistance bias factor statistics. The load bias factor statis-tics employed by Li et al. [22] are those used in the national codefor foundation design in China [23]: kD = 1.0, kL = 1.0, COVD = 0.07,and COVL = 0.29. Based on MOC [23], load partial factors cD = 1.0and cL = 1.0 are adopted; additionally, a live load to dead load ratioof q = 0.2 is used [22]. The resistance bias factor statistics (i.e., kRand COVR) can be obtained by conducting statistical analysis oncases with both static load test and prediction results.

The within-site variability can be characterized by comparingcapacities of piles within a site. In Li et al. [22], a load test databaseconsisting of 146 piles from 32 sites and another database com-prising 37 piles from 10 sites were used to characterize thewithin-site variability for driven piles and bored piles, respectively.In these load tests, piles with identical geometry at each site wereloaded until failure occurred. The ultimate bearing capacity wasdetermined with a comprehensive analysis on the load-displacement (Q-s) curve and the corresponding displacement-logarithm of time (s-lgt) curve. The load at the start point of a steepdrop on the Q-s curve and the load beyond which the settlementwill not converge on the s-lgt curve was taken as the ultimate bear-ing capacity [29,37]. Details on these piles are summarized inTables 1 and 2.

According to Zhang et al. [39], the within-site variability refersto the variability in the pile capacity values within a site and thus,the mean of these values is truly reflected by the mean of the mea-sured capacity values, which is based on the proven theory that thesample mean is an unbiased estimate of the population mean.Therefore, the within-site variability of the pile capacity predictionis unbiased [39,22], i.e., kR1 = 1. On the other hand, the value ofCOVR1 is determined by calculating the COV of the measuredcapacities of the piles within a site. Note that values of COVR1 varyfrom site to site. The values of COVR1 of driven piles are in the rangeof 0.031–0.155 with a mean of 0.087 and a COV of 0.36. The valuesof COVR1 of bored piles are in the range of 0.049–0.179 with a meanof 0.093 and a COV of 0.44. In Li et al. [22], as in other traditionalLRFD calibration studies (e.g., [26]), the means of those COVR1

Table 1Summary of driven piles for code calibration (modified after [22]).

Siteno.

Site name Dimension(mm)

No. ofpiles

Measured ultimate bearing capacity (kN) Statistics ofmeasured bearingcapacity

Mean(kN)

COV

1 Shanghai Jiangping HighSchool

250 � 250 6 558; 496; 558; 558; 558; 550 546 0.046

2 Tangzhen CommodityHousing

250 � 250 3 765; 720; 810 765 0.083

3 Shanghai F1 Speedway 250 � 250 3 980; 900; 1000 960 0.0554 Yueda Residential Building 300 � 300 3 462; 594; 600 552 0.1415 Shenyuan Construction Site

#1200 � 200 3 500; 450; 425 458 0.083

6 Shenyuan Construction Site#2

250 � 250 4 750; 780; 720; 636 722 0.086

7 Kangtai Residential Building 200 � 200 4 460; 440; 396; 480 444 0.0818 Shanghai Haide Apartment 250 � 250 3 240; 270; 210 240 0.1259 Donglan Xincun 250 � 250 8 700; 770; 840; 770; 700; 770; 840; 630 753 0.09610 Xincheng Residential

Building250 � 250 5 540; 540; 600; 540; 540 552 0.049

11 Rongnanyuan Extension 250 � 250 11 403; 403; 403; 403; 403; 403; 403; 403; 403; 403; 358 399 0.03412 Qingchi Project Phase I 250 � 250 3 567; 491; 567 542 0.08113 Shanghai Dekui Plant 250 � 250 8 461; 461; 461; 461; 461; 461; 461; 403 454 0.04514 2205 Gonghexin Rd 250 � 250 3 736; 736; 644 705 0.07515 Shanghai Tennis Club 300 � 300 3 720; 600; 720 680 0.10216 Jinda Residential Building 300 � 300 14 786; 672; 784; 896; 672; 896; 672; 784; 784; 784; 1008; 1008; 1008;

1008840 0.155

17 Runjiang Project Phase I 300 � 300 3 448; 538; 538 508 0.10218 Wenhuayuan Phase V 350 � 350 3 1550; 1639; 1806 1665 0.07819 Shanghai Dingxin Apartment 350 � 350 8 1280; 1440; 1280; 1280; 1280; 1280; 1600; 1600 1380 0.10620 ECNU Middle School 350 � 350 3 630; 720; 717 689 0.07421 ECNU Science Park 350 � 350 6 1079; 960; 960; 960; 969; 969 983 0.04822 Feilipu Plant 400 � 400 3 1425; 1425; 1350 1440 0.03123 Site #2 – 3 1250; 1200; 1500 1317 0.12224 Site #4 – 3 2700; 2300; 2600 2533 0.08225 Site #7 – 4 470; 560; 440; 420 473 0.13126 Site #9 – 3 890; 1050; 1100 1013 0.10827 Site #14 – 3 1300; 1520; 1550 1457 0.09428 Site #19 – 4 – – 0.06029 Site #24 – 3 – – 0.08030 Site #30 – 4 – – 0.10331 Site #31 – 4 – – 0.12232 Site #32 – 4 – – 0.097

Table 2Summary of bored piles for code calibration (modified after [22]).

Site no. Site name Diameter (mm) No. of piles Measured ultimate bearingcapacity (kN)

Statistics ofmeasured bearingcapacity

Predicted bearing capacityusing DT method (kN)

Mean (kN) COV

1 Zhongjian Mansion 850 3 8192; 8192; 7168 7850 0.075 97172 Metro Line 10 Parking Lot 1 600 3 4008; 5025; 3685 4239 0.131 39003 Metro Line 10 Parking Lot 2 600 4 3900; 3600; 3600; 3300 3600 0.068 29654 Yuyuan Community 550 6 2420; 2420; 2240; 1960; 2240; 2800 2347 0.119 21405 Gaofufang 700 3 8840; 8840; 7480 8387 0.094 75726 Shanghai Quality Inspection 650 3 1860; 1780; 1670 1770 0.054 22187 Taihongxinyuan 600 5 2700; 2700; 3000; 2700; 2700 2760 0.049 28308 Yueyang Plaza 850 3 10,000; 8000; 11,500 9833 0.179 11,5299 Zhongsheng Commercial Center 900 3 5460; 5460; 4550 5157 0.102 407810 Chunguang Garment Factory 600 4 2400; 2400; 2400; 2700 2475 0.061 3273

D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238 231

values for driven piles and bored piles are adopted in calibration, aslisted in Table 3, and the uncertainties in those COVR1 values areignored.

The cross-site variability can be characterized by comparing themeasured and predicted bearing capacities of piles from differentsites. A bearing capacity ratio is first obtained for each site bydividing the mean of measured bearing capacities at this site bythe computed bearing capacity (e.g., predicted capacity using DTmethod for bored piles listed in Table 2). The mean and the COV

of the bearing capacity ratios are regarded as the estimates of kR2and COVR2. For the piles designed using LT method, the computedcapacity is taken as the mean of measured capacities, thus the val-ues of bearing capacity ratio for all sites are taken as 1 which leadto kR2 = 1 and COVR2 = 0. For the bored piles designed using DTmethod, the bearing capacity ratios for 10 sites in Table 2 are cal-culated (e.g., a ratio of 0.808 for site 1). The mean and COV of the10 ratios are computed and taken as kR2 and COVR2, respectively,i.e., kR2 = 0.996 and COVR2 = 0.184, for the DT method for bored

Table 3Summary of resistance bias factor statistics used in calibration and resultingcalibrated partial factors (modified after [22]).

Driven piles Bored piles

LT method DT method CPT method LT method LT method

kR1 1 1 1 1 1COVR1 0.087 0.087 0.087 0.093 0.093kR2 1 1.025 1.006 1 0.996COVR2 0 0.144 0.093 0 0.184kR 1 1.025 1.006 1 0.996COVR 0.087 0.168 0.127 0.093 0.206cR 1.53 1.93 1.72 1.56 2.26

Fig. 1. Cumulative frequency of the observed COVR1 with fitted lognormal CDF fordifferent types of piles.

232 D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238

piles. Note that the driven piles in Table 1 are not used for charac-terizing the cross-site variability using DT method and CPTmethod. In Li et al. [22], for driven piles designed using DT methodand CPT method, the values of kR2 and COVR2 based on the previousdesign code are reviewed and adopted. The characterization ofcross-site variability for different design methods and differentpiles are also summarized in Table 3.

The statistics of resistance bias factor (i.e., kR and COVR) are thenobtained using Eqs. (4) and (5) based on the characterized within-site variability and cross-site variability. Among the design meth-ods, the LT method has the smallest COV, as it is free from cross-site variability. On the other hand, the DT method has the largestuncertainty. Having obtained the statistics of load bias factor andresistance bias factor, the required cR corresponding to a specifiedtarget reliability index, say, bT = 3.7 [22], can be calculated usingEq. (6). The calibrated values of cR for different design methodsand different piles obtained by Li et al. [22] are listed in Table 3.

In the aforementioned resistance factor calibration process,COVR1 = 0.087 and COVR1 = 0.093, the mean values of the respec-tive COVs, are adopted for the analysis of driven piles and boredpiles, respectively. However, COVR1 actually varies from site to siteas seen in Tables 1 and 2. This variation in COVR1 is not reflected inthe traditional resistance factor calibration; and as such, uncer-tainty in the computed capacity is not fully accounted for whenthe calibrated cR is applied to a future case. To characterize the dis-tribution of COVR1, Fig. 1(a) and (b) illustrates the cumulative fre-quency of observed COVR1 from the adopted database (see Tables 1and 2) for driven piles and bored piles, respectively. Lognormalcumulative distribution function (CDF) curve is also plotted inFig. 1 to fit the data set. Approximations are superimposed onthe CDF curve using the computed values of the mean and COV.Visually, the lognormal CDF matches the cumulative frequency ofCOVR1 well. Therefore, COVR1 is treated as a lognormal randomvariable in this paper.

To investigate the effect of variation in COVR1 on the b of designusing the calibrated resistance factors in Li et al. [22], Monte Carlosimulation (MCS) [14] is performed to generate 5000 random sam-ples of COVR1 and the corresponding values of b are computedusing Eq. (2) for different types of piles and different design meth-ods. Fig. 2 shows histograms of relative frequency of b for differenttypes of piles and different design methods. It is observed that bvalues distribute in wide ranges and many of the designs cannotachieve bT = 3.7 (i.e., b < bT). Furthermore, Fig. 2 also shows thecumulative frequency curves of b. With the aid of these cumulativecurves, the probability of (b < bT) can be readily obtained. The val-ues of this probability for driven piles designed with the LTmethod, DT method, and CPT method are 0.466, 0.511, and 0.485,respectively. For bored piles using LT method and DT method,the values are 0.442 and 0.500, respectively. The implication is thatwhen cR obtained from the traditional LRFD calibration that doesnot consider the variation in COVR1 is used, there is a high chancethat the safety requirement (i.e., bP bT) will not be satisfied (i.e.,the design is not feasible). In other words, the design using this

cR may not meet the safety requirement due to the uncertaintyin the computed capacity. This is a drawback of the traditionalresistance factor calibration. To reduce the chance for the violationof safety requirement, or to improve the feasibility robustness ofthe design using partial factors, a new resistance factor calibrationapproach considering robustness is proposed.

3. Resistance factor calibration considering robustness

3.1. Feasibility robustness

In this paper, ‘‘feasibility robustness” [25] is adopted to measurethe robustness of partial-factor design with respect to uncertainparameters (i.e., COVR1) in the calibration process. Herein, the fea-sibility robustness is the robustness against the uncertainty in theassessed safety level, and is defined as the probability that bT canstill be satisfied even with the variation in COVR1. Symbolically,feasibility robustness is formulated as [15,16]:

P½ðb� bTÞ P 0� P P0 ð7Þ

where P[(b�bT)P 0] is the probability that bT is satisfied; and P0 is apre-defined acceptable level of this probability (i.e., feasibilityrobustness).

Determining P[(b�bT)P 0] of a design given cR requires knowl-edge of the distribution of the corresponding b. As shown in Fig. 2,the values of P[(b�bT)P 0] of those piles designed with the cali-

Fig. 2. Relative and cumulative frequency of b associated with calibrated cR for different types of piles and different design methods (note: bT = 3.7).

D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238 233

brated cR can be evaluated based on the corresponding cumulativefrequency curves obtained from MCS. The results are summarizedin Table 4. For example, for driven piles designed using LT methodwith cR = 1.53, the value of P[(b�3.7)P 0] is 0.534. If the feasibilityrobustness level is pre-selected as P0 = 0.6, the design usingcR = 1.53 cannot meet the feasibility robustness requirement (since0.534 is less than the prescribed level of 0.6). Although MCSprovides a rigorous way to determine the feasibility robustnessP[(b�bT)P 0], it is time consuming. Simulations show that theresulting histogram of b can be approximated well with a normaldistribution, as depicted in Fig. 2. Thus, if the mean and standarddeviation of b, denoted as lb and rb, can be determined, the

feasibility robustness of design can readily be computed with theassumption of normal distribution as [16,32]:

P½ðb� bTÞ P 0� ¼ Ulb � bT

rb

� �P P0 ð8Þ

where U is cumulative standard normal distribution function.Therefore, the evaluation of feasibility robustness is reduced to

a task of determining lb and rb. In this paper, seven point estimatemethod (PEM) developed by Zhao and Ono [40] is adopted to esti-mate lb and rb. Since b is the function of single random variableCOVR1 in this problem, the seven-point PEM first assigns the values

Table 5Calibrated resistance factors (cR) for load-carrying capacity at different feasibilityrobustness levels.

P0 Driven piles Bored piles

LT method DT method CPT method LT method DT method

0.5 1.52 1.95 1.72 1.53 2.300.6 1.55 1.98 1.75 1.57 2.340.7 1.59 2.01 1.79 1.62 2.380.8 1.64 2.05 1.83 1.69 2.440.9 1.72 2.11 1.90 1.81 2.520.99 2.04 2.27 2.10 2.39 2.76

Note: Load factors used in this calibration are: cL = 1.0 and cD = 1.0 (Shanghai code).

Table 4Feasibility robustness of calibrated partial factors in Li et al. [22] obtained from MCSand PEM.

Approach Driven piles Bored piles

LTmethod

DTmethod

CPTmethod

LTmethod

DTmethod

MCS 0.534 0.489 0.515 0.558 0.500PEM 0.541 0.479 0.506 0.578 0.464

Note: The numerical entries in this table are feasibility robustness, P[(b�bT)P 0].

234 D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238

of COVR1 at each estimating point; then bi (i = 1, 2, . . ., 7) isevaluated at each point using Eq. (2). The resulting seven bi valuesare then used to compute the lb and rb, as [40]:

lb ¼X7i¼1

Wibi ð9Þ

r2b ¼

X7i¼1

Wiðbi � lbÞ2 ð10Þ

where Wi (i = 1, 2, . . ., 7) are the weights corresponding to estimat-ing points as per Zhao and Ono [40]. The readers are referred toZhao and Ono [40] for details on PEM.

Using PEM, the feasibility robustness of the design using thecalibrated cR reported by Li et al. [22] are calculated and displayedin Table 4. Comparing with the results computed with MCS, itshows that the two approaches are quite consistent and thePEM-based approach for feasibility robustness (Eqs. (8)–(10)) isjudged adequate in terms of accuracy.

3.2. Calibration of resistance factor with robustness consideration

The aforementioned procedure to evaluate feasibility robust-ness of a design using the existing cR actually is the inverse ofthe task of resistance factor calibration considering robustness.Here, the resistance factor calibration considering robustness is aprocess of determining the value of cR such that the resultingdesign can achieve the pre-defined feasibility robustness level,i.e., P[(b�bT)P 0] = P0.

The resistance factor calibration process considering feasibilityrobustness proposed herein can be implemented using a trial-and-error approach, in which a trial cR is assumed and thenchecked against feasibility robustness requirement using MCS orPEM, followed by revision of the cR, if necessary. Alternatively, itis noted that the feasibility robustness can be expressed as a func-tion of cR, e.g., in a form of P[(b�bT)] = G(cR), by combining Eqs. (2),(8), (9), and (10). Thus, once the target P0 is defined, the corre-sponding cR can be readily determined by solving the equation G(cR) = P0, in lieu of the trial-and-error process.

In the present study, six feasibility robustness levels (i.e.,P0 = 0.5, 0.6, . . ., 0.9, 0.99), are selected for illustration purpose.Among these pre-defined target levels, the lower bound P0 = 0.5is selected to ensure an average level that lb P bT as per Eq. (8);the upper bound P0 = 0.99 represents an extreme design scenario.To be consistent with Li et al. [22], bT = 3.7 is adopted in thecalibration. Using the proposed calibration procedure (i.e., solvingG(cR) = P0), the values of cR compatible with cL = 1.0 and cD = 1.0to achieve pre-defined target robustness levels for different typesof piles and different design methods are determined and theresults are given in Table 5. For example, when the feasibilityrobustness level is set as P0 = 0.5, the values of cR for driven pilesdesigned with LT method, DT method, and CPT method are 1.52,1.95, and 1.72, respectively; for bored piles designed using LTmethod and DT method, the values are 1.53 and 2.30, respectively.

Noted that the somewhat unusual load factors of 1.0 adopted fromChinese and Shanghai codes should only be used in conjunctionwith the resistance factors calibrated to the target reliability indexunder this setting of load factors. As the values of load factors areset to 1.0, the values of cR are essentially the same as the nominalFS (see Eq. (1)). Note that the required cR increases with increasingP0, as expected. To achieve the same P0, DT method requires largercR, as it is associated with greater uncertainties while the requiredcR for LT method is smaller due to the lower uncertainties involvedwith LT method. Thus, although a smaller nominal FS is required inLT method, the reliability level and robustness level are indeedsame as those using DT method that yield a larger nominal FS. Thisfurther illustrates the limitation of the ASD approach, i.e., the nom-inal FS cannot accurately reflect the true level of safety. UsingTable 5, the designer can first select the P0 for a given projectand then use the corresponding cR to perform the design of piles.Such design is guaranteed to have a level of feasibility robustnessof P0 (meaning that the chance for the design to remain feasibleregardless of the variation of COVR1 is P0).

Furthermore, the values of lb and rb at various feasibilityrobustness levels corresponding to the resistance factors listed inTable 5 are calculated using Eqs. (9) and (10), respectively, andthe results are summarized in Table 6. Note that for the same P0,piles designed with LT method have the largest lb, e.g., 4.38 for dri-ven piles and 4.54 for bored piles when P0 is 0.8. This is consistentwith the view that LT method is the most reliable. On the otherhand, piles designed with DT method have the smallest lb, e.g.,3.98 for both driven piles and bored piles when P0 is 0.8. It shouldalso be noted that both the values of lb and rb increase withincreasing P0. The implication is that feasibility robustness is pri-marily affected by lb as per Eq. (8).

4. Design example using the re-calibrated resistance factors

To illustrate use of the calibrated resistance factors consideringfeasibility robustness listed in Table 5, a bored pile is designedusing DT method for a sand profile. The bored pile with diameterD = 0.85 m shown in Fig. 3 is embedded at a depth of L, (i.e., the pilelength is L). The soil layer surrounding bored pile is divided into 5sub-layers, as shown in Fig. 3. The suggested values of unit toeresistance qt and unit side resistance fs for each soil layer are alsogiven in Fig. 3. The bored pile in this problem is intended to carryaxial compressive loads of 2500 kN dead load (QDn) and 500 kN liveload (QLn). Using live load and dead load factors of 1.0 and 1.0,respectively, the design load is 3000 kN.

Using the corresponding cR in Table 5, the values of requiredpile length L at various feasibility robustness levels can be deter-mined by Eqs. (1) and (A1) using DT method, and the design resultsare illustrated in Fig. 4. For example, the required depths forP0 = 0.5 and P0 = 0.99 are L = 45.9 m and L = 53.6 m, respectively.It can be seen that the value of L increases with increasing P0. That

Fig. 4. Design results at various feasibility robustness levels.

Table 6Values of lb and rb at various feasibility robustness levels.

P0 Driven piles Bored piles

LT method DT method CPTmethod

LT method DT method

lb rb lb rb lb rb lb rb lb rb

0.5 3.70 0.67 3.70 0.30 3.70 0.43 3.70 0.83 3.70 0.300.6 3.88 0.70 3.79 0.30 3.80 0.44 3.90 0.89 3.79 0.310.7 4.11 0.74 3.87 0.31 3.95 0.46 4.18 0.93 3.86 0.310.8 4.38 0.79 3.98 0.32 4.10 0.47 4.54 1.01 3.98 0.320.9 4.81 0.87 4.13 0.33 4.36 0.50 5.14 1.13 4.12 0.330.99 6.32 1.13 4.53 0.36 5.04 0.56 7.56 1.65 4.54 0.36

D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238 235

is to say, larger pile length is required to achieve higher feasibilityrobustness levels (i.e. larger P0 values). Note that the L (at a givenD) can be treated as an index of construction cost. The implicationis that design with high robustness against variation in the com-puted capacity can always be achieved at the expense of cost effi-ciency. However, this may not be desired and a tradeoff betweencost efficiency and design robustness must be made. The discus-sion in the section that follows focuses on this tradeoff issue.

5. Further discussion

Table 5 provides the resistance factors calibrated to bT = 3.7 andvarious target levels of feasibility robustness P0. Designers canselect cR corresponding to the desired P0 for a given project. Gener-ally, a design with higher feasibility robustness and relativelylower cost is desired. However, as shown in the above bored pile

Fig. 3. A bored pile d

design example, feasibility robustness and cost efficiency are con-flicting to each other. When there is no strong preference by thedesigner, a tradeoff decision may be based on an optimization ofcR performed with respect to two objectives, design robustnessand cost efficiency.

Note that according to Eq. (1), for a given loading, use of ahigher cR implies the need for a higher nominal bearing capacity,which in turn, points to the need for a larger L and thus, a greatercost. Thus, the cost can be effectively reflected in cR. In this study, adiscrete design space is considered for the optimization of cR, inwhich P0 will be selected from the range of 0.5–0.99 with anincrement of 0.01. The multi-objective optimization is set up asfollows:

esign example.

236 D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238

Find:

Fig. 5. P

An optimal cR compatible with cL = 1.0 andcD = 1.0

Subject to:

G(cR) = P0 2 {0.5, 0.51, 0.52, . . ., 0.99} Objectives: Maximizing the feasibility robustness (in terms

of P0)

Minimizing the construction cost (in terms of cR)

The values of cR corresponding to the 50 potential feasibilityrobustness levels are determined by solving G(cR) = P0, respec-tively. Fig. 5 shows the variation of calibrated cR as a function of

areto front showing tradeoff between feasibility robustness and resi

P0, for different combinations of pile types and design methods.As shown in Fig. 5, cR increases as P0 increases, and the curves havethe similar trend with the relationship between L and P0 shown inFig. 4, as expected. Thus, a tradeoff exists between design robust-ness and cost efficiency; the tradeoff relationship is presented hereas a Pareto front [7,17,12]. The knee point [8] on the Pareto frontconceptually yields the best compromise among conflicting objec-tives. In other words, the knee point may be taken as the most pre-ferred choice. In this study, the minimum distance approach[17,11] is used to determine the knee point, in which a point thathas the minimum distance from the ‘‘utopia point” in a normalizedspace of the objective functions is taken as the knee point.

stance factors for different types of piles and different design methods.

Table 7Calibrated partial factors obtained from knee point of Pareto front.

Piles Design method Knee point Suggested

P0 cR P0 cR

Driven piles LT method 0.86 1.68 0.85 1.67DT method 0.84 2.07 0.85 2.08CPT method 0.85 1.86 0.85 1.86

Bored piles LT method 0.88 1.78 0.85 1.74DT method 0.84 2.46 0.85 2.47

Note: Partial factors for loads used in this calibration are: cL = 1.0 and cD = 1.0(Shanghai code).

D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238 237

Interested readers are referred to Khoshnevisan et al. [17] andGong et al. [11] for additional details on the minimum distanceapproach.

With the minimum distance approach, knee points for differenttypes of piles designed using different methods are readily identi-fied, as also shown in Fig. 5. Consider, for example, driven pilesdesigned using LT method, the knee point identified in Fig. 5(a)has the following parameters P0 = 0.86 and cR = 1.68. As can beobserved in Fig. 5(a), on the left side of knee point, the curve is rel-atively flat, indicating a slight reduction in cR (i.e., cost) would leadto a large decrease in design robustness (as reflected by a markedlydecrease in P0), which is undesirable. On the other side of the kneepoint, the slope is relatively sharp, indicating a slight gain in P0 (i.e.,a slight improvement of feasibility robustness) requires a largeincrease in cR, rendering it cost inefficient. Therefore, the kneepoint on the Pareto front represents the best compromise betweendesign robustness and cost efficiency.

The values of P0 and cR corresponding to those knee pointsshown in Fig. 5 are summarized in Table 7. Note that the valuesof P0 vary from 0.84 to 0.88. For consistency and simplicity, in allscenarios, P0 is set at 0.85 and the cR values are adjusted slightly,and the final results are listed in Table 7, which are recommendedfor typical geotechnical practice. The previous bored pile exampleis redesigned herein using cR = 2.47 selected form Table 7compatible with cL = 1.0 and cD = 1.0. To meet P0 = 0.85, the finalL is determined to be 48.75 m and is shown in Fig. 4.

6. Concluding remarks

This paper presents a new resistance factor (cR) calibrationapproach considering feasibility robustness. The new approach isdemonstrated with a pile design calibration example that describesShanghai, China experience. Herein, the feasibility robustness isdefined as a probability that the safety requirement can still be sat-isfied even if the capacity of the designed system is uncertain (i.e.,exhibiting significant variation) due to the uncertainty in resis-tance bias factors. With due consideration of the variation in theresistance bias factor, the proposed calibration approach aims toobtain cR so that the resulting design is robust against uncertaintyin the computed bearing capacity.

The presented results show that without considering designrobustness, the design using calibrated cR obtained from thetraditional LRFD calibration process can still miss the target reliabil-ity index (bT) because of the underestimation of the variation inresistance bias factor. By considering feasibility robustness in thecalibration process, cR is re-calibrated in this study, which can beselected for future applications based on the desired feasibilityrobustness level. A case study is performed using the re-calibratedcR. It shows that feasibility robustness and cost efficiency are twoconflicting objectives. To aid in decision-making in the design pro-cess, a feasibility robustness level of 0.85 and corresponding valuesof cR, which are obtained through a multi-objective optimization,are suggested in the absence of a designer’s preference.

It should be noted that the calibrated resistance factors areobtained specifically for design of piles in Shanghai, as assump-tions based on local practice are made. Further studies are war-ranted to confirm the general applicability of the proposedcalibration approach.

Acknowledgments

The study on which this paper is based was supported in part byNational Science Foundation through Grant CMMI-1200117(‘‘Transforming Robust Design Concept into a Novel GeotechnicalDesign Tool”). The results and opinions expressed in this paperdo not necessarily reflect the views and policies of the NationalScience Foundation. This work was also supported by the NationalScience Fund for Distinguished Young Scholars (Project No.51225903), and the National Natural Science Foundation of China(Project Nos. 51329901, 51579190, 51528901). The second authorwishes to thank the Glenn Department of Civil Engineering ofClemson University, South Carolina for hosting his two-year visitas an exchange PhD student. The fourth author wishes to acknowl-edge the support of the Shanghai Summit Discipline DevelopmentProgram through Key Laboratory of Road and Traffic Engineering ofthe Ministry of Education at Tongji University, Shanghai, China.

Appendix A. Pile design methods used in Shanghai, China

A.1. Load test-based (LT) method

LT method is considered as the most reliable approach for piledesign. When the static load test is used to determine the designcapacity, at least 3 piles or 1% of the total number of productionpiles should be tested. However, if the number of production pilesis smaller than 50, the number of piles to be tested can be reducedto a minimum of 2. The ultimate pile capacity at the site is chosenas the mean of the measured bearing capacities. For each pile, theultimate bearing capacity was determined by examining the load-displacement (Q-s) curve and the corresponding displacement-logarithm of time (s-lgt) curve. The load at the start point of a steepdrop on the Q-s curve and the load beyond which the settlementwill not converge on the s-lgt curve was taken as the ultimate bear-ing capacity [29,37].

In LT method, the piles are initially designed based on thedesign table (DT) method or the cone penetration test-based(CPT) method. If static load tests show that the bearing capacityof the initially designed piles is not adequate and the design needsto be revised, the bearing capacity of the revised piles should alsobe measured by static load tests. LT method is often used forimportant projects, or when the site condition is complex, newconstruction methods are employed, or new types of piles are used.

A.2. Design table (DT) method

In DT method, the nominal bearing capacity of a pile is deter-mined empirically based on the soil profile as:

Rn ¼ Rt þ Rs ¼ qtAt þ Up

Xnj¼1

f sjzj ðA1Þ

where Rt is toe resistance; Rs is side resistance; qt is unit toe resis-tance; At is cross-section area of pile end; Up is perimeter of pile;fsj is unit side resistance of the jth soil layer, zj = thickness of pileinterfacing with the jth soil layer; and n is the number of soil layerssurrounding the pile.

238 D.-Q. Li et al. / Computers and Geotechnics 81 (2017) 229–238

A.3. Cone penetration test-based (CPT) method

In CPT method, the bearing capacity of a pile is calculated usingthe following equation:

Rn ¼ Rt þ Rs ¼ abqctAt þ Up

Xnj¼1

f sjzj ðA2Þ

where ab is correction factor; and qct is cone tip resistance measuredat the pile toe.

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