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## Journal of the South African Institution of Civil Engineering

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*On-line version* ISSN 2309-8775

*Print version* ISSN 1021-2019

### J. S. Afr. Inst. Civ. Eng. vol.60 n.2 Midrand Jun. 2018

#### http://dx.doi.org/10.17159/2309-8775/2018/v60n2a6

**TECHNICAL PAPER**

**Determination of base and shaft resistance factors for reliability-based design of piles**

**X-Y Bian; X-Y Chen; H-L Lu; J-J Zheng**

**ABSTRACT**

This paper aims to propose a procedure for calculating separately the resistance factors for ultimate base and shaft resistances for the reliability-based design of piles. The proposed procedure can clearly explain the different sources of uncertainties of the bearing capacity, including those from ultimate base and shaft resistances. The study evaluates the convergence of the proposed procedure, and the effects of relevant parameters on resistance factors. Finally, two examples are used for comparison and application of the presented method for determining ultimate base and shaft resistance factors. Convergence analysis proves that the final resistance factors can be rapidly obtained, and maintain good stability using the iteration algorithm included in the proposed procedure. A parametric study indicates that the ratio of dead load to live load and initial values of the ultimate shaft (or base) resistance factor, have a limited effect on the final convergence values of ultimate shaft (and base) resistance factors. However, the target reliability index has significant influence on the ultimate shaft and base resistance factors. The validation example shows that the ultimate shaft and base resistance factors, as calculated in this paper, are conservative compared to the results by Kim *et al *(2011), due to the consideration of more uncertainties. The recommended ultimate shaft and base resistance factors for the reliability-based design of piles can be obtained conveniently using the proposed procedure in the application example.

**Keywords**: pile, reliability, resistance factor, ultimate base resistance, ultimate shaft resistance

**NOTATION**

The following symbols are used in this paper:

bratio of R_{ult,b}to (L_{D}+ L_{L})coefficient of variation for

coefficient of variation for

coefficient of variation for L

_{D }coefficient of variation for

L_{L}

glimit state function

LloadL

_{D}dead loadL

_{L}live load

Pfailure probability_{f}

Rultimate resistance of pile_{ult }R

_{ult,b}ultimate base resistance of pileR

_{ult,s}ultimate shaft resistance of pile

sratio of R_{ult,s }to (L_{D}+L_{L})

βreliability index

β_{T }target reliability index

Yload factor for dead load_{D}Y

_{L}load factor for live load

θlimit value of calculation accuracyλ

_{D}bias factor for L_{D}λ

_{L}bias factor for L_{L}λ

_{ult,b}bias factor for R_{ult,b}A

_{ult,s}bias factor for R_{ult,s}

ρratio of Ld to L_{L}Ø

_{ult,b}resistance factor for R_{ult,b}Ø

_{ult,s}resistance factor for R_{ult,s}initial value of Ø

_{ult,s}

**INTRODUCTION**

Load and resistance factor design (LRFD) is conceptually a more advanced design method than the existing working stress design (WSD). The key improvements of LRFD over the traditional WSD are the ability to provide a more consistent level of reliability and the possibility of accounting for load and resistance uncertainties separately (Foye *et al *2006). Successful implementation of LRFD in geotechnical engineering contributes to an economical and safe design.

Many researchers and practitioners are now recognising the great advantages of LRFD in practice, and more and more relevant research is being incorporated into LRFD for driven piles based on reliability analysis (Zhang *et al *2001; Paikowsky 2004; AASHTO 2007). Many countries and regions, such as the United States, Canada, South Africa, China mainland, Japan, Korea, Singapore, Europe and Hong Kong, are replacing or have already replaced WSD with LRFD for structural design. However, LRFD in geotech-nical engineering has not been fully developed yet (Kim *et al *2011).

Against this background, a rational framework for LRFD development should be established for the replacement of resistance factors calculated based on factors of safety with those calculated based on reliability analysis. The LRFD framework is conducive to maintaining the same levels of load factors for all loads under different conditions. A number of studies have looked at calculating and calibrating resistance factors for geotechnical engineering. Zheng *et al *(2012) presented a Bayesian optimisation approach to determine the resistance factor of piles, and recommended values for the resistance factors of driven piles. Bian *et al *(2015) incorporated the serviceability limit state requirements into LRFD for the ultimate limit states of piles to determine the resistance factors for reliability-based design of piles. Phoon and Kulhawy (2002), and Phoon *et al *(2003) proposed a multiple resistance factor design concept for foundations and studied the uplift resistance factors for uplift side resistance, uplift tip resistance and dead weight of foundation against uplift force. Honjo *et al *(2002) established a procedure for the calculation of partial factors for dead load, seismic load, base resistance and shaft resistance of axially-loaded cast-in situ piles. Kim *et al *(2011) contributed to the development of LRFD for axially-loaded driven piles in sands, the evident feature of which is that the resistance factors for base and shaft resistances were calculated separately to account for their different uncertainty levels. Basu and Salgado (2012) developed resistance factors for drilled shafts for a design method based on soil parameters.

However, methods to determine the ultimate base and shaft resistance factors are not well developed. This paper will present a novel method to calculate the ultimate base and shaft resistance factors for the reliability-based design of piles. First, an iterative algorithm to estimate the ultimate base and shaft resistance factors will be presented using the reliability theory and LRFD criteria. Second, the convergence of the proposed procedure will be analysed. Third, the effects of relevant parameters - ratio of dead load to live load, initial value of base (or shaft) resistance factor, and target reliability index - on resistance factors will be evaluated. Finally, the validation and practical application of the presented method will be shown with two examples to illustrate the feasibility and availability of the presented method.

**LOAD AND RESISTANCE FACTOR DESIGN**

**Design criterion**

Considering an axially-loaded driven pile, the ultimate pile resistance (or bearing capacity) R_{ult} is generally expressed as the summation of ultimate base resistance (or end resistance) R_{ult,b} and ultimate shaft resistance (or shaft friction) R_{ult,s}_{s}(Poulos & Davis 1980):

The key advantage of the LRFD approach is that significant uncertainties (e.g. load and material resistance) can be incorporated quantitatively into the design process. If only dead load L_{D} and live load *Ll *are considered, the LRFD design formula for an axially-loaded driven pile can be written as (AASHTO 2007):

where Ø_{ult,b} and Ø_{ult,s} are the resistance factors for R_{ult,b} and R_{ult,s} respectively; and *γ** _{D}* and

*γ*

_{L}are the specified load factors for dead and live loads respectively.

**Resistance factors**

Suppose resistance R_{ult} and load *L *follow lognormal distribution and they are statistically independent (Ang & Tang 2007; Wang & Kulhawy 2008; Dithinde *et al *2011). It should be pointed out here that the probability distribution for load *L *is certainly suitable to dead load L_{D} and live load L_{l}, namely lognormal distribution. The limit state function (g) in accordance with LRFD framework is established:

The reliability index β, which is used to estimate the reliability of piles and reflect the safety status of piles, can be calculated using the following formula (Federal Highway Administration 2001; Bian *et al *2016):

where λ_{ult,b}, λ_{ult,s}, λ_{D} and λ_{L} are the bias factors for R_{ult,b}, R_{ult,s}, L_{d} and L_{L} respectively; and , , and are the coefficients of variation (COVs) for R_{ult,b}, R_{ult,s}, L_{D}, and L_{L} respectively. Here the bias factor includes the net effect of various sources of errors, such as inherent soil variability, measurement error, and transformation uncertainty.

LRFD is the limit state design (mainly including ultimate limit state and serviceability limit state for piles), and only the ultimate limit state requirements are focused on in this paper. As the critical state of the design formula (Equation 2) is necessary for the study, replacing the inequality sign in Equation 2 with an equality sign gives:

Then R_{ult,b} and R_{ult,s}_{s}can be expressed respectively as:

Substituting R_{ult,b} and R_{ult,s}, expressed respectively by Equations 6 and 7, into Equation 4, and replacing *β*in Equation 4 with target reliability index *(**β** _{T}), *gives the following expressions for Ø

_{ult,b }and Ø

_{ult,s}

Equations 8 and 9 indicate that Ø_{ult,b }and Ø_{ult,s }are functions of many parameters, such as ρ, β_{T} s, *b *and so on. Among these parameters ρ, β_{T}, *s *and *b *are key factors influencing Ø_{ult,b }and Ø_{ult,s }due to great uncertainties for them. Here β_{T} is a certain level of reliability, for which piles designed using the LRFD method will guarantee. In other words, the reliability index of a pile designed using LRFD is greater than or equal to the target reliability index.

**Probabilistic parameters**

Based on the foregoing discussion, there are two sets of information required to estimate resistance factors: load and resistance information (including load factor, bias factor, Cov). A review of literature (AASHTo 2007) suggests that the following probabilistic parameters can be used for L_{D} and L_{L}: λ_{D} = 1.08, = 0.13, λ_{L} = 1.15, = 0.18, y_{D} = 1.25 and y_{L} = 1.75. *ρ*= *L _{D}/L_{L }*is structure-specific and changes with span length (Hansell

*et al*1971; Withiam

*et al*2001). Hansell

*et al*(1971) also proposed an empirical formula, L

_{D}/L

_{L}= (1 + I)(0.0132l), to relate L

_{D}/L

_{L}with span length, where

*L*is the dynamic load factor (taken as 0.33 for LRFD loads), and

*L*is the span length in feet.

*ρ*= L

_{D}/L

_{L}spreads over from 0.576 to 5.184 when

*L*varies from 10 m to 90 m, and

*ρ*= 3.0 is a frequently used value.

Formula *p _{f}*= Φ(-β) expresses the relationship between failure probability

*(p*and

_{f})*β*(see Table 1). The acceptable β

_{τ}is in essence the maximum acceptable failure probability. For example, determining acceptable β

_{T}= 3.0 means the acceptable maximum failure probability is 0.001.

Barker *et al *(1991) reduced the target reliability index for driven piles to a value between 2.0 and 2.5, especially for a group system effect. Paikowsky (2004) suggested an initial target reliability index between 2.0 and 2.5 for a pile group, and 3.0 for a single pile. Paikowsky (2004) also recommended target reliability indices of 2.33 (corresponding to 1% probability of failure) and 3.00 (corresponding to 0.1% probability of failure) for representing redundant and non-redundant pile groups, respectively. As suggested by Barker *et al *(1991) and Paikowsky (2004), five levels (2.0, 2.5, 3.0, 3.5 and 4.0) of target reliability index will be considered in this study and the corresponding resistance factors calculated.

Probabilistic parameters for R_{ult}, *R _{ult,b}* and R

_{ult,s}from literature are summarised in Table 2. Equations 8 and 9 demonstrate that

*b*(= R

_{ult,b}/(L

_{D}+ L

_{L})) and

*s*(= R

_{ult,s}/(L

_{D}+ L

_{L})) are two key parameters for evaluation of resistance factors. However, both

*b*and

*s*are difficult to determine, because

*R*and

_{ult,b}*R*depend largely on site conditions and pile types. For example,

_{ult,s }*R*of friction piles is generally very small and may be ignored with respect to R

_{ult,b}_{ult,s}, which means that , resulting in and

*.*Moreover, for a safe design . For end-bearing piles R

_{ult,s }is generally very small, and may be ignored with respect to

*R*which means that , resulting in . Moreover, for a safe design . For end-bearing friction piles and friction end-bearing piles,

_{ult,b},*b*and

*s*are complex and need further study.

**PROCEDURE TO CALCULATE RESISTANCE FACTORS**

**Procedure flow chart**

Equations 8 and 9 indicate that Ø_{ult,b} and Ø_{ult,s} mainly depend on ρ, *β*_{T},*s,b* and other parameters. Especially, Ø_{ult,b} computed using Equation 8 will be submitted into Equation 9 to compute Ø_{ult,s}, and this Ø_{ult,s }will be resubmitted into Equation 8 to compute Ø_{ult,b }again. This computation process is in fact an iteration process, which contributes to build a procedure for resistance factors calculation, as shown in Figure 1 (Bian *et al *2016). In Figure 1, is the initial value of _{s}where *θ*is the limit value of calculation accuracy.

With reference to Figure 1, the proposed approach to determine resistance factors 0_{ult,b} and 0_{ult,s} for reliability-based design of piles is outlined in the following steps:

■

Input statistical parameters of load and resistance. Bias factors, COVs and load factors LStep 1_{d}and L_{l}can be determined referring to the previous literature. Bias factors and COVs of R_{ult,b}and R_{ult,s}_{s}are estimated using the load test database of piles.■

Input combination parametersStep 2p= L_{D}/L_{L}s= R_{ult,s}/ (L_{D}+L_{L}) andb= R_{ult,b }/ (L_{D}+L_{L}). For ρ, the commonly used values (such as 0.5, 1.0, 2.0, 3.0, 4.0 and 5.0) from the previously mentioned literature are welcome, whilesandbneed to be evaluated depending on site conditions and pile types.■

Input βStep 3_{T}and precision limit value θ. For β_{T}the commonly used values (such as 2.0, 2.5, 3.0, 3.5 and 4.0) will be accepted for further study.θis set as 0.0001.■

Determine initial value (or ). Six values, namely 0, 0.2, 0.4, 0.6, 0.8 and 1.0, will be suggested to study the influence of (or ) on the final results.Step 4■

Calculate ∅Step 5_{ult,b}and ∅_{ult,s}Submit into Equation 8 to compute ∅_{ult,b}, and denote as Submit into Equation 9 to compute ∅_{ult,s}, and denote as . Repeat this process to obtain and■

Examine convergence. If , , and satisfy , and are taken as the final resistance factors , repeatStep 6Step 5.

**Convergence analysis of calculation procedure**

The validity and application conditions of the procedure are investigated in depth in this section. All related computation tasks will be completed using MS Excel. Convergence of the proposed procedure to calculate resistance factors was made using the following parameters: *ρ*= 3.0, imagemaqui = 0.5, *β** _{T}*= 3.0 and

*θ*= 0.0001. Parameter

*s*was set as 0.5, 1.0, 2.0 and 3.0. For each value of

*s, b*was set as 0, 0.5, 1.0, 2.0, 3.0 and integer ≥ 4.0, respectively. Here statistics of L

_{D}and L

_{L}(namely λ

_{D}= 1.08, = 0.13, λ

_{L}, = 1.15, = 0.18, y

_{D}= 1.25 and y

_{L}= 1.75) presented by AASHTO (2007) are accepted to compute the resistance factors Ø

_{ult,b}and Ø

_{ult,s},. Bias factors and COVs of

*R*and R

_{ult,b}_{ult,s}presented by Jardine

*et al*(2005), and summarised in Table 2 by the authors, are also used in this section. Final resistance factors obtained for each combination and the number of iterations required to reach the set level of accuracy (θ = 0.0001) are summarised in Table 3.

In Table 3, NM indicates that the combination (b = 0 and *s *= 0.5) is meaningless for the reliability-based design of piles, while NRC indicates that the iterative process could not reach convergence. Employing the information from Table 3, a bold judgement can be made that the iterative process proposed in this paper reached convergence only when the sum of *s *and *b *was less than 4. This requirement meets the demand for reliability-based design of piles satisfactorily. The LRFD criteria for piles, the most important limit state design method, has been expressed by Equation 2. For reliability-based design of piles, R_{ult}_{t}does not excessively exceed the load effects. Due to this, and given that safe designs are those with R_{ult}/(L_{d} + L_{L}) ≥ 1.0, *b *= R_{ult,b}/(L_{D}+ L_{L}) and *s *= R_{ult,s}/(L_{D}+ L_{L})will be limited to between 0 and 3.0. It is also pointed out that runs with *b *= 0 are done purely for comparison, as *b *= 0 implies a pure friction pile which is not possible.

From Table 3, the following preliminary conclusions can be drawn: for a given *s, *the number of iterative steps increase with increasing *b, *as shown from columns 10 to 13. For example, with *s *= 0.5, the iterative steps are 4, 5, 7 and 20 corresponding to *b *with values of 0.5, 1.0, 2.0 and 3.0 respectively; with *s *= 2.0, the iterative steps are 3, 8 and 15 corresponding to *b *with values of 0, 0.5 and 1.0 respectively.

**PARAMETER ANALYSIS AND DISCUSSION**

In this section, load statistical parameters (including λ_{D} = 1.08, = 0.13, λ_{L} = 1.15, = 0.18, *r _{D}*= 1.25 and

*Y*= 1.75) presented by AASHTO (2007), and resistance statistical parameters (including A

_{L}_{ult,b}= 1.023, λ

_{ult,s}= 1.088, = 0.201 and = 0.287) presented by Jardine

*et al*(2005) are used to compute the resistance factors Ø

_{ult,b}and Ø

_{ult,s}

**Effect of ****ρ****on resistance factors**

To study the effect of *ρ*on Ø_{ult,b} and Ø_{ult,s}, the following parameters were kept constant: = 0.5, *s *= 1.0, *b *= 1.0, *β*_{τ}= 3.0 and *θ*= 0.0001. It is also well reasoned to set *ρ*as 0.5, 1.0, 2.0, 3.0, 4.0 and 5.0 respectively. Using these parameters, Ø_{ult,b} and Ø_{ult,s }were calculated using the proposed procedure and plotted against *ρ*in Figure 2.

Varying *ρ*did not influence convergence, with convergence generally obtained within six iterative steps. It can be seen from Figure 2 that both Ø_{ult,b} and Ø_{ult,s} decrease slightly with increasing *ρ**. *However, the difference between Ø_{ult,b} and Ø_{ult,s }is an approximate constant for all p. Under the given assumptions, Ø_{ult,s }is larger than Ø_{ult,b}, and the average difference between Ø_{ult,b} and Ø_{ult,s} is about 0.025.

**Effect of **** on resistance factors**

To study the effect of on final resistance factors (Ø_{ult,b} and Ø_{ult,s}), the following parameters were kept constant: *ρ*= 3.0, *s *= 1.0, *b *= 1.0, β_{T} = 3.0 and *θ*= 0.0001. Parameter was set as 0, 0.2, 0.4, 0.6, 0.8 and 1.0 respectively, and the two resistance factors were calculated and plotted against in Figure 3.

Varying did not influence convergence significantly, with convergence reached within no more than seven iterative steps. It can be seen from Figure 3 that both Ø_{ult,b} and Ø_{ult,s }versus are approximate horizontal lines, which illustrate that convergence values Ø_{ult,b} and Ø_{ult,s }are both independent of , as Ø_{ult,b} and Ø_{ult,s} are determined as 0.37 and 0.39 respectively. This conclusion provides support to the rationality of the proposed procedure for resistance factor calculation.

**Effect of ****β**_{τ}** on resistance factors**

To study the effect of β_{T} on Ø_{ult,b} and Ø_{ult,s }the following parameters were kept constant: *ρ*= 3.0, = 0.5, *s *= 1.0, *b *= 1.0 and *θ*= 0.0001. β_{T} was set as 2.0, 2.5, 3.0, 3.5 and 4.0. Calculated values of Ø_{ult,b} and Ø_{ult,s} are plotted against β_{T} in Figure 4.

In Figure 4, both Ø_{ult,b} and Ø_{ult,s} decrease sharply with an increase of β_{T}, which shows that both Ø_{ult,b} and Ø_{ult,s} are very sensitive to β_{T} For example, when β_{T} increases from 2.0 to 4.0, Ø_{ult,b} decreases from 0.55 to 0.25, and Ø_{ult,s} decreases from 0.59 to 0.26. Varying β_{T} had a significant effect on iterative steps. For β_{T} = 2.0, the iterative steps were 12; for β_{T} = 2.5, 3.0 and 3.5 respectively, the iterative steps were all nearly 6; and for β_{T} = 4.0, the iterative steps were only 4.

In summary, engineers should very seriously consider a suitable β_{T} to conduct the reliability-based design of pile foundations. Selecting a small β_{T} will leave the piles designed using LRFD methods at risk. Selecting a large β_{T} will lessen the identified resistance factors, the design scheme will be conservative and the cost will be uneconomical. The analysis in this section indicates that the values of β_{T} between 2.5 and 3.0 are suitable. β_{T} between 2.5 and 3.0 indicates the acceptable maximum failure probability between 0.1% and 0.6%, which is low enough for general pile foundation engineering. Besides, the iterative steps are nearly 6 for βτ with values between 2.5 and 3.0, and the computational efficiency is good, too.

**validation and application**

Practical validation and application of the proposed method will be illustrated by the following two examples, respectively.

**Validation example**

According to Kim *et al *(2011), Ø_{ult,b} and Ø_{ult,s} values were calibrated using their proposed method for building and bridge structures, which are compatible with the ASCE/SEI 7-05 (ASCE 2005) load factors and the AASHTO (2007) load factors. For comparison and validation, the corresponding load statistical parameters are considered in this section (Kim *et al *2011): in the case of ASCE/ SEI 7-05, λ_{D} = 1.05, *COV _{L}*= 0.1, λ

_{L}= 1.0 and = 0.25; while in the case of AASHTO, λ

_{D}= 1.05, = 0.1, λ

_{L}= 1.2 and = 0.205. Also,

*y*= 1.25 and

_{D}*Y*= 1.75 are selected for both ASCE/SEI 7-05 and AASHTO cases. The process of estimating the resistance factors for building and bridge structures must satisfy the following conditions:

_{l}1. Utilise

ρ=Lwith four different values: 1.0, 2.0, 3.0 and 4.0_{D}/L_{L}2. Utilise β

_{T}with four different values: 2.0, 2.5, 3.0 and 3.53. Determine both

s=R/(L_{ult,s}_{d}+ L_{l}) andb= R_{ult,b}/(L_{D}+ L_{L}) as 1.04. Select resistance statistical parameters referring to Jardine

et al(2005), namely λ= 1.023, λ_{ult,b}= 1.088, = 0.201 and = 0.287._{ult,s}

Calculated results of Ø_{ult,b} and Ø_{ult,s }using the proposed method in this paper are shown in Table 4. For building structures with the ASCE/ SEI 7-05 (ASCE 2005) load factors, the calculated Ø_{ult,b} and Ø_{ult,s} values vary within the ranges 0.55-0.62 and 0.58-0.66 for β_{T} = 2.0; 0.44-0.50 and 0.47-0.53 for β_{T} = 2.5; 0.36-0.40 and 0.38-0.43 for β_{T} = 3.0; and 0.29-0.32 and 0.31-0.34 for β_{T} = 3.5, respectively, depending on the ratio *ρ*= *L _{D}/L_{L} (*

*ρ*=

*L*range of 1.0-4.0).

_{D}/L_{L }For bridge structures with the AASHTO (2007) load factors, the calculated Ø_{ult,b} and Ø_{ult,s} values vary within ranges 0.55-0.58 and 0.58-0.62 for β_{T} = 2.0; 0.45-0.48 and 0.48-0.51 for β_{T} = 2.5; 0.36-0.39 and 0.39-0.41 for β_{T} = 3.0; and 0.30-0.32 and 0.32-0.34 for β_{T} = 3.5, respectively, depending on the ratio *ρ*= L_{D}/L_{L} (ρ = L_{D}/L_{L} range of 1.0-4.0).

For ease of comparison, the results of Ø_{ult,b} and Ø_{ult,s} from Kim *et al *(2011) are also given in Table 4 (columns 11 to 16). Comparison shows that the results of Ø_{ult,b} and Ø_{ult,s }for building and bridge structures compatible with load factors from ASCE/ SEI 7-05 (ASCE 2005) and AASHTO (2007) computed in this paper, are smaller than the results from Kim *et al *(2011). These differences may be due to many probable reasons, but it should be pointed out here that some main uncertainty factors are not considered in Kim *et al *(2011), such as proportions of shaft (base) resistance to load (namely R_{ult,s}/(L_{D} + L_{L}) and R_{ult,b }/ (L_{D}+ L_{L})), the correlation between Ø_{ult,b} and Ø_{ult,s} and so on.

For example, in Kim *et al *(2011), Ø_{ult,b} and Ø_{ult,s} for building and bridge structures compatible with load factors from AASHTO (2007) vary within ranges of 0.82-0.87 and 0.70-0.75 for β_{T} = 2.5; 0.76-0.79 and 0.63-0.66 for β_{T} = 3.0; and 0.69-0.73 and 0.57-0.60 for β_{T} = 3.5. These resistance factors in Kim *et al *(2011) seem to be very large for the reliability-based design of piles. The resistance factors proposed by AASHTO (2007) for strength limit state for shallow foundations are generally between 0.35-0.60, which perhaps more strongly support the results in this paper.

**Application example**

Luo (2004) compiled a database of pile load tests, including 151 driven pile load tests. From these databases only 128 driven pile load tests with sufficient information (measured ultimate bearing capacity, base resistance and shaft resistance) were analysed. The bias factors and COVs of pile resistances were calculated by authors referring to Luo (2004): λ_{ult,b} = 1.18, λ_{ult,s}= 1.21, = 0.34, and = 0.22, respectively. Computed resistance factors using the proposed procedure are summarised in Table 5 for the following conditions:

1. Set

ρ= L_{d}/L_{l}at four different values: 1.0, 2.0, 3.0 and 4.02. Set β

_{T}at four different values: 2.0, 2.5, 3.0 and 3.53. Both sets

s= R_{ult,s}/(L_{D}+ L_{L}) andb= R_{ult,b}/(L_{D}+ L_{L}) at 1.04. Accepted load statistical parameters presented by AASHTO (2007), namely λ

_{D}= 1.08, = 0.13, λ_{L}= 1.15, = 0.18, y_{D}= 1.25 and y_{L}= 1.75.

Based on the results shown in Table 5, one can see that the calculated Ø_{ult,b} and Ø_{ult,s }values vary within the ranges of 0.57-0.62 and 0.58-0.63 for β_{T} = 2.0; 0.45-0.49 and 0.46-0.51 for β_{T} = 2.5; 0.36-0.39 and 0.37-0.40 for β_{T} = 3.0; and 0.29-0.32 and 0.30-0.32 for β_{T} = 3.5, respectively, depending on the ratio *ρ*= L_{D}/L_{L} (ρ = L_{D}/L_{L} range of 1.0-4.0). The variations of Ø_{ult,b} with the different *ρ*values are very small; this is also the case for Ø_{ult,s}. This further verifies the conclusion obtained in the section above titled "PARAMETER ANALYSIS AND DISCUSSION", that both Ø_{ult,b} and Ø_{ult,s} decrease slightly with increasing *ρ**. *Therefore, the recommended resistance factors can be proposed with different target reliability levels by considering the influence of *ρ*referring to Table 5. For a target reliability level, the mean value of four resistance factor values, corresponding to *ρ*= 1, 2, 3 and 4, is determined as the recommended resistance factor. By this method, based on the calculated resistance factors shown in Table 5, this study presents the recommended resistance factors for 128 driven pile load test cases, coming from Luo (2004) and summarised in Table 6.

Table 6 shows that the recommended resistance factors are significantly different for different β_{T} indices. This phenomenon is actually compatible with the conclusion, as obtained in the section above, titled "PARAMETER ANALYSIS AND DISCUSSION", that β_{T} is an important factor for the determination of resistance factors, as it is essential to choose the appropriate target reliability index for the reliability-based design of piles using the proposed method in this paper. According to the presented values of β_{T} between 2.5 and 3.0 mentioned earlier, the recommended values of the resistance factors are 0.38-0.47 for Ø_{ult,b} and 0.39-0.49 for Ø_{ult,s}.

**CONCLUSIONS**

Uncertainties regarding the bearing capacity of piles actually derives from the ultimate base and shaft resistances, and should be explained separately in the reliability-based design of piles. The way to solve this problem is by developing a method to evaluate and study the ultimate base and shaft resistance factors respectively. This is the major contribution achieved in this paper.

Convergence analysis demonstrates that the presented iteration algorithm to estimate ultimate base and shaft resistance factors converges rapidly and remains stable. The condition of convergence (i.e. *b *and *s *between 0 and 3.0) can meet the demand of the reliability-based design of piles satisfactorily.

In addition, parameter analysis indicates that the ratio of dead to live loads has a limited influence on calculated resistance factors. The overall consideration of dead and live loads in the determination of ultimate base and shaft resistance factors is reasonable. Similarly, the initial seed resistance factor also has little effect on convergence of the final resistance factors. Any initial seed resistance factor could be selected in the reliability-based design of piles. However, the target reliability index significantly influences computed resistance factors, and an appropriate target reliability index is required for the reliability-based design of piles.

In a nutshell, the ultimate base and shaft resistance factors for the reliability-based design of piles can easily be obtained using the proposed procedure with an appropriate target reliability index. The application example has illustrated this point.

**ACKNOWLEDGEMENTS**

The authors would like to express their gratitude to the National Key Research and Development Program of China (2016YFC0800208) and the National Natural Science Foundation of China (51708428, 51408444, 51378404).

**REFERENCES**

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**Correspondence**:

DR XIAO-YA BIAN

Wuhan Institute of Technology School of Civil Engineering and Architecture Wuchang Campus

Wuhan, Hubei, China

T: +86 27 8719 4698

E: bianxy@hust.edu.cn

PR0FXU-Y0NG CHEN

Wuhan Institute of Technology School of Civil Engineering and Architecture Wuchang Campus

Wuhan, Hubei, China

T: +86 27 8719 4698

E: cxy1314kl@126.com

PROF HAI-LIN LU

Wuhan Institute of Technology, School of Civil Engineering and Architecture, Wuchang Campus

Wuhan, Hubei, China

T: +86 27 8719 4698

E: hail_lu@yangtzeu.edu.cn

PROF JUN-JIE ZHENG

Huazhong University of Science and Technology School of Civil Engineering and Mechanic

Wuhan, Hubei, China

T: +86 27 8755 7024

E: zhengjj@hust.edu.cn

DR XIAO-YA BIAN is a lecturer in the School of Civil Engineering and Architecture at the Wuhan Institute of Technology, China. He obtained his PhD in Geotechnical Engineering from the School of Civil Engineering and Mechanics at the Huazhong University of Science and Technology, China. His special interests are risk and reliability of geotechnical engineering, and ground improvement.

PR0FXU-Y0NG CHEN Is an assistant professor In the School of Civil Engineering and Architecture at the Wuhan Institute of Technology, China. He obtained his PhD in Bridge Engineering from the School of Civil Engineering and Mechanics at the Huazhong University of Science and Technology, China. His special interests are bridge reinforcement and non-probabilistic reliability of bridges.

PROF HAI-LIN LU is a professor in the School of Civil Engineering and Architecture at the Wuhan Institute of Technology, China. He obtained his PhD in Structural Engineering from the School of Civil Engineering at the Tianjin University, China. His special interest is earthquake resistance ofengineering structures.

PROF JUN-JIE ZHENG is a professor in the School of Civil Engineering and Mechanics at the Huazhong University of Science and Technology, China. He obtained his PhD in Geotechnical Engineering from the Zhejiang University, China. His special interests are reliability in geotechnical engineering, ground improvement and tunnel engineering.