A procedure for incorporating setup into load and resistance factor design of driven piles

Abstract

In a recent study, the time-dependent increase in axial load resistance of steel H-piles driven into cohesive soils, due to setup, was systematically quantified using measured field data. A method to estimate the setup based on measurable soil properties was subsequently established. These studies highlighted that the uncertainties of the measurements of soil properties and thus the semi-empirical approach to estimate setup are significantly different from those of the methodology used for measuring the pile resistance during retaps at any time after the end of driving. Recognizing that the two sets of uncertainties should be addressed concurrently, this paper presents a procedure for determining the factored resistance of a pile with due consideration to setup in accordance with the load and resistance factor design that targets a specific reliability index. Using the first-order second-moment method, the suggested procedure not only provides a simplified approach to incorporate any form of setup in design, but it also produces comparable results to the computationally intensive first-order reliability method. Incorporating setup in design and construction control is further shown to reduce foundation costs and minimize retap requirements on piles, ultimately reducing the construction costs of pile foundations.

Appendix: Derivation of the resistance factor for setup

Satisfying the lognormal distributions and independent relationships of loads and resistances, the reliability index (β) is expressed as a ratio of mean to standard deviation of the limit state function (g), which can be expanded as follows:

$$ \beta = \frac{{{\text{E}}\left( g \right)}}{{\sigma_{g} }} = \frac{{{\text{E}}\left( {\ln \left( {R_{\text{EOD}} } \right)} \right) + {\text{E}}\left( {\ln \left( {R_{\text{setup}} } \right)} \right) - {\text{E}}\left( {\ln \left( Q \right)} \right)}}{{\sqrt {\sigma_{{\ln \left( {R_{\text{EOD}} } \right)}}^{2} + \sigma_{{\ln \left( {R_{\text{setup}} } \right)}}^{2} + \sigma_{\ln \left( Q \right)}^{2} } }} $$

(11)

where R _EOD is the pile resistance at EOD, R _setup is the pile setup resistance, E(g) is the expected value of the limit state function g, σ _g is the standard deviation of the limit state function g, E(ln(R _EOD)) is the expected value of the natural logarithm of the pile resistance at EOD, and σ _ln(REOD) is the standard deviation of the natural logarithm of the pile resistance at EOD, which can be similarly defined for other random variables. To express Eq. (11) in terms of simple means (i.e., $\overline{R}$ and $\overline{Q}$) and coefficients of variation (COV) for resistances and loads of the normal distributions, the mean and standard deviation of a lognormal distribution for any resistance or load can be transformed using the following general expressions:

$$ {\text{E}}\left( {\ln \left( R \right)} \right) = \ln \left( {\overline{R} } \right) - 0.5\ln \left( {1 + {\text{COV}}_{R}^{2} } \right) $$

(12)

$$ \sigma_{\ln \left( R \right)}^{2} = \ln \left( {1 + {\text{COV}}_{R}^{2} } \right) $$

(13)

Using these expressions for the three random variables (R _EOD, R _setup, and Q) and substituting them into Eq. (11), the reliability index can be expressed as follows:

$$ \beta = \frac{{\ln \left( {\frac{{\mathop {\overline{R} }\nolimits_{\text{EOD}} + \mathop {\overline{R} }\nolimits_{\text{setup}} }}{{\overline{Q} }}} \right) + \ln \left( {\sqrt {\frac{{1 + {\text{COV}}_{Q}^{2} }}{{{\text{COV}}_{\text{RR}} + 2{\text{COV}}_{{R_{\text{EOD}} }}^{2} {\text{COV}}_{{R_{\text{setup}} }}^{2} }}} } \right)}}{{\sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} + 2{\text{COV}}_{{R_{\text{EOD}} }}^{2} {\text{COV}}_{{R_{\text{setup}} }}^{2} } \right)\left( {1 + {\text{COV}}_{Q}^{2} } \right)} \right]} }} $$

(14)

Replacing the simple mean values with their respective bias factors (λ), a ratio between average measured and predicted values (i.e., R _m/R or Q _m/Q), and neglecting the terms that involve multiplying two squared coefficients of variation (i.e., COV²COV²) since their contribution would be insignificantly small, the expression for β is simplified as:

$$ \beta = \frac{{\ln \left( {\frac{{\lambda_{{R_{\text{EOD}} }} R_{\text{EOD}} + \lambda_{{R_{\text{setup}} }} R_{\text{setup}} }}{{\lambda_{Q} Q}}} \right) + \ln \left( {\sqrt {\frac{{1 + {\text{COV}}_{Q}^{2} }}{{{\text{COV}}_{\text{RR}} }}} } \right)}}{{\sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} } \right)\left( {1 + {\text{COV}}_{Q}^{2} } \right)} \right]} }} $$

(15)

using the LRFD strength limit state equation (γQ = φR) and replacing φR with φ _EOD R _EOD + φ _setup R _setup, the equation can be rearranged for R _setup as:

$$ R_{\text{setup}} = \frac{{\gamma Q - \varphi_{\text{EOD}} R_{\text{EOD}} }}{{\varphi_{\text{setup}} }} $$

(16)

substituting Eq. (16) into Eq. (15) and isolating the φ _setup as the variable of interest by rearranging, a preliminary equation for φ _setup can be established as follows:

$$ \varphi_{\text{setup}} = \frac{{\lambda_{\text{setup}} \left[ {\gamma Q - \varphi_{\text{EOD}} R_{\text{EOD}} } \right]}}{{\frac{{\left( {\lambda_{Q} Q} \right)\;\exp{\left({{\beta \sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} } \right)\left( {1 + {\text{COV}}_{Q}^{2} } \right)} \right]} }}\right)} }}{{\sqrt {\frac{{\left( {1 + {\text{COV}}_{Q}^{2} } \right)}}{{\left( {{\text{COV}}_{\text{RR}} } \right)}}} }} - \lambda_{\text{EOD}} R_{\text{EOD}} }} $$

(17)

considering only the dead (Q _D) and live (Q _L) loads, as per the AASHTO [3] “Strength I” load combination, the factored load (γQ) and bias load (λ _Q Q) are expanded to:

$$ \gamma Q = \gamma_{\text{D}} Q_{\text{D}} + \gamma_{\text{L}} Q_{\text{L}} $$

(18)

$$ \lambda_{Q} Q = \lambda_{\text{D}} Q_{\text{D}} + \lambda_{\text{L}} Q_{\text{L}} $$

(19)

As defined in AASHTO [3], γ _D is the dead load factor of 1.25, γ _L is the live load factor of 1.75, λ _D is the dead load bias of 1.05, and λ _L is the live load bias of 1.15. Furthermore, the coefficient of variation of the load (COV _Q ), derived by Bloomquist et al. [6], in terms of dead and live load components is given as:

$$ {\text{COV}}_{Q} = \sqrt {\frac{{\frac{{Q_{\text{D}}^{2} }}{{Q_{\text{L}}^{2} }}\lambda_{\text{D}}^{2} {\text{COV}}_{\text{D}}^{2} + \lambda_{\text{L}}^{2} {\text{COV}}_{\text{L}}^{2} }}{{\frac{{Q_{\text{D}}^{2} }}{{Q_{\text{L}}^{2} }}\lambda_{\text{D}}^{2} + 2\frac{{Q_{\text{D}} }}{{Q_{\text{L}} }}\lambda_{\text{D}} \lambda_{\text{L}} + \lambda_{\text{L}}^{2} }}} $$

(20)

substituting Eqs. [18], [19], and [20] into Eq. (17), the φ _setup can be revised as:

$$ \varphi_{\text{setup}} = \frac{{\lambda_{\text{setup}} \left[ {\gamma_{\text{D}} Q_{\text{D}} + \gamma_{\text{L}} Q_{\text{L}} - \varphi_{\text{EOD}} R_{\text{EOD}} } \right]}}{{\frac{{\left( {\lambda_{\text{D}} Q_{\text{D}} + \lambda_{\text{L}} Q_{\text{L}} } \right)\;\exp{\left({{\beta \sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} } \right)\left( {\kappa } \right)} \right]} }}\right)} }}{{\sqrt {\frac{\left( \kappa \right)}{{\left( {{\text{COV}}_{\text{RR}} } \right)}}} }} - \lambda_{\text{EOD}} R_{\text{EOD}} }} $$

(21)

Normalizing the above expression with respect to the total load (Q _D + Q _L), and further rearrangement of Eq. (21) in terms of the dead load to live load ratio (i.e., Q _D/Q _L) and representing α as the ratio of pile resistance at EOD to the total load (i.e., α = R _EOD/[Q _D + Q _L]), the resistance factor of pile setup at a target reliability index (β _T ) can be expressed as:

$$ \varphi_{\text{setup}} = \frac{{\lambda_{\text{setup}} \left[ {\frac{{\gamma_{\text{DL}} }}{{Q_{\text{DL}} }} - \varphi_{\text{EOD}} } \right]}}{{\frac{{\left( {\frac{{\lambda_{\text{DL}} }}{{Q_{\text{DL}} }}} \right)\exp{\left({{\beta_{\it{T}} \sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} } \right)\left( \kappa \right)} \right]} }}\right)} }}{{\sqrt {\left( {\frac{\kappa }{{{\text{COV}}_{\text{RR}} }}} \right)} }} - \lambda_{\text{EOD}} \alpha }} $$

(22)

The parameter α, a ratio of pile resistance at EOD to total load, noted above is analogous to a safety factor applied to the R _EOD if the traditional allowable stress design (ASD) approach would have been considered. The uncertainties associated with R _EOD have been accounted for in terms of φ _EOD in Eq. (22) to comply with the LRFD approach. Since the uncertainties were addressed, the parameter α is suggested as unity in order to eliminate the redundancy in the safety margin applied to R _EOD, and the resistance factor for pile setup based on the FOSM method can be expressed as:

$$ \varphi_{\text{setup}} = \frac{{\lambda_{\text{setup}} \left[ {\frac{{\gamma_{\text{DL}} }}{{Q_{\text{DL}} }} - \varphi_{\text{EOD}} } \right]}}{{\frac{{\left( {\frac{{\lambda_{\text{DL}} }}{{Q_{\text{DL}} }}} \right)\exp{\left({{\beta_{\it{T}} \sqrt {\ln \left[ {\left( {{\text{COV}}_{\text{RR}} } \right)\left( \kappa \right)} \right]} }}\right)} }}{{\sqrt {\left( {\frac{\kappa }{{{\text{COV}}_{\text{RR}} }}} \right)} }} - \lambda_{\text{EOD}} }} $$

(23)