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9 Field Exploration & Testing Dynamic Analysis Geomaterial Strength & Static Analysis of Superstructure of Driven Piles Deformation Parameters Deep Foundations Loading Evaluation Deep Foundation Type/Construction Method Laboratory Deformation Testing and Bearing Capacity Substructure Settlement Vertical and Loading Lateral Resistance Requirement Single/Group Design · Geometry · Configuration · Installation Criteria No Design Testing Verification/ ? Completed · Material QC Construction OK Modification Substructure · Performance Monitoring · Dynamic testing · Driving · Static testing · Integrity Yes Figure 3. Design and construction process for deep foundations. (i.e., static capacity evaluations) were compared to measured driven piles, and PD/LT2000--are presented in Appendices B pile performance under static load. In the dynamic analysis and C, on the accompanying CD, and discussed in section case, the database was used to identify controlling parame- 2.2. The secondary databases are referred to and used as ters, which were then calibrated. A description of the princi- applicable. ples used for the assessments of the three databases is pro- vided in section 1.4.3.3. Figure 4 presents a flowchart of the research approach for this study. The flowchart outlines the 1.4.3.3 Conceptual Evaluation of Driven Piles framework required for LRFD calibration of design and con- and Drilled Shafts Capacities struction methods of analysis. The stages outlined in Figure 4 are described in the following sections; findings and evalu- Driven Piles--Static Analysis. The vast majority of the ations related to the various stages of the framework are pre- database case histories were related to SPT and CPT field sented in Chapter 2. testing. Four correlations of soil parameters from SPT and CPT were identified. The case histories were divided on the 1.4.3 Principles and Framework basis of soil condition (clay, sand, and mixed) and pile types of the Calibration (H pile, concrete piles, pipe piles). In summary, given field conditions were used via various soil parameter identifica- 1.4.3.1 Determination of Analysis Methods tions and pile capacity evaluation procedures to determine capacities. The capacities were then compared to measured To establish the state of practice, a questionnaire was devel- static capacity. Details of the analyses are presented in sec- oped and distributed to all state highway and federal highway tion 2.3. organizations. The material related to the questionnaire and detailed results are presented in Appendix A, on the accom- Driven Piles--Dynamic Analysis. The dynamic evaluation panying CD, and discussed in section 2.1. of driven piles is the most common way to determine capacity during construction. Existing AASHTO specifications, as 1.4.3.2 Databases described in section 1.3.5.1, are complicated by the use of a factor, v, which convolves the design stage and the construc- Three principal databases and six secondary databases were tion stage. Therefore, a fresh look at the basis for dynamic cal- developed for the evaluation of the analysis methods and inter- ibration was required. Details are described in Paikowsky and pretation procedures. The major databases--drilled shaft, Stenersen (2000) and in section 2.4.

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10 Questionnaire State of Practice Establish Common Database Build-Up Driven Piles and Design Methods Static ­ Driven Piles Drilled Shafts and Procedures for Dynamic ­ Driven Piles Design and Static Analyses Static ­ Drilled Shafts Construction Peripheral Databases Research and Evaluation of the Evaluating Static Establish establish Static Capacity of DP Capacity DP Viable Recommended and DS for all based on Methods and Pf Methods/Correlation Dynamic Controlling Combinations Analyses Parameters for the Dynamic Analyses Calculating the Ratio of Calculating the the Nominal Strength to Resistance Factors and Predicted Capacity Establish a Single Evaluating the Results Evaluate the Method for the Nominal Determination of Strength of Nominal Strength all cases (capacity), its Develop Statistical accuracy and LT Recommended Resistance Parameters for the procedure effect Factors Performance of each Analysis LT-Static Load Test Method/Correlation DP-Driven Piles Combination DS-Drilled Shafts SGP 4/7/02 Figure 4. Stages of the research approach outlining the framework for LRFD calibration of the current study. Drilled Shafts--Static Analysis. Evaluation of the design of where: drilled shafts is difficult as limited data are available for the R = resistance bias factor separation of capacity components (i.e., shaft and tip), and as COVQ = coefficient of variation (the ratio of the standard both components of capacity are affected by the method of deviation to the mean) of the load construction. The following procedure was used for the eval- COVR = coefficient of variation of the resistance uation of the measured skin capacities. The shape of the load- T = target reliability index displacement curves was evaluated, and shafts for which more than 80% of the total capacity was mobilized in a displacement When just dead and live loads are considered, equation 9 can of less than 2% of the shaft diameter were considered as hav- be rewritten as: ing resistance based on friction. Results of these procedures were compared to static analyses as described in section 2.6. (1 + COVQ + COVQ ) 2 2 R + L D QD D L QL (1 + COVR ) 2 = QL + Q exp{ T ln[(1 + COVR )(1 + COVQ + COVQ )]} 1.4.3.4 LRFD Calibration Q Q DD 2 2 2 L D L Existing AASHTO Specifications. Existing AASHTO spec- (10) ifications are based on First-Order, Second-Moment (FOSM) analysis, using = 1 in equation 4, and assuming lognormal where: distributions for resistance. This leads to the relation (Barker et al., 1991), D, L = dead and live load factors QD /QL = dead to live load ratio 1 + COVQ 2 QD, QL = dead and live load bias factors R ( i Qi ) 2 1 + COVR = (9) { Q exp T ln[(1 + COVR 2 )(1 + COVQ2 )] } Present Project Calibration. LRFD for structural design has evolved beyond FOSM to the more invariant First-Order

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11 Reliability Method (FORM) approach (e.g., Ellingwood notation x* and x '* is used to denote the design point in et al., 1980, Galambos and Ravindra, 1978), while geotech- the regular coordinates and in the reduced coordinate nical applications have lagged behind (Meyerhof, 1994). In system, respectively. order to be consistent with the current structural code and the load factors to which it leads, it is necessary for calibra- 2. If the distribution of basic random variables is non- tion of resistance factors for deep foundations to use FORM normal, approximate this distribution with an equiva- (Nowak, 1999). lent normal distribution at the design point, having the Following Ayyub and Assakkaf (1999), the present project same tail area and ordinate of the density function, that calibrates LRFD partial safety factors using FORM, as is with equivalent mean, developed by Hasofer and Lind (1974). FORM can be used X = x * - ( FX ( x *)) X -1 to assess the reliability of a pile with respect to specified limit µN N (13) states and provides a means for calculating partial safety fac- tors and i for resistance and loads, respectively, against a and equivalent standard deviation target reliability level, O. FORM requires only first and sec- ond moment information on resistances and loads (i.e., means ( -1 ( FX ( x *))) N X = and variances) and an assumption of distribution shape (e.g., f X ( x *) (14) normal, lognormal, etc.). The calibration process using FORM is presented in Figure 5. where In design practice, there are usually two types of limit µN X = mean of the equivalent normal distribution, state: ultimate limit state and serviceability limit state. Each can be represented by a performance function of the form, N X = standard deviation of the equivalent normal distribution, g( X ) = g( X1 , X2 , K, Xn ) (11) FX (x*) = original cumulative distribution function (CDF) of Xi evaluated at the design point, in which X = (X1, X2 ,..., Xn ) is a vector of basic random vari- fX (x*) = original PDF of Xi evaluated at the design ables of strengths and loads. The performance function g(X), point, often called the limit state function, relates random variables () = CDF of the standard normal distribution, to either the strength or serviceability limit-state. The limit is and () = PDF of the standard normal defined as g(X) = 0, implying failure when g( X) < 0 (Figures distribution. 2 and 5). The reliability index, , is the distance from the ori- gin of the space of basic random variables to the failure sur- 3. Set x i'* = * i , in which the i are direction cosines. * face at the most probable point on that surface, that is, at the Compute the directional cosines (* i , i = 1, 2,..., n) point on g(X) = 0 at which the joint PDF of X is greatest. This using, is sometimes called the design point and is found by an itera- tive solution procedure (Thoft-Christensen and Baker, 1982). g The relationship of the limit states can also be used to back xi' * calculate representative values of the reliability index, , * i = 2 for i = 1, 2, K, n (15) g n from current design practice. i =1 x i ' * The computational steps for determining using FORM are the following: where 1. In the regular coordinates, assume a design point, x*i, and, in a reduced coordinate system, obtain its corre- g = g N (16) sponding point, xi'*, using the transformation: xi' * xi * Xi x* i - µ Xi 4. With * i , µ Xi , Xi now known, the following equation N N xi' * = is solved for : Xi (12) where [( g µN * N ) ( * * N X1 - X1 X1 , K , µ Xn - Xn Xn = 0 )] (17) µ Xi = mean value of the basic random variable Xi , Xi = standard deviation of the basic random variable. 5. Using the obtained from step 4, a new design point is obtained from, The mean value of the vector of basic random variables is often used as an initial guess for the design point. The xi* = µ N * N Xi - i Xi (18)

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12 Definition Define Limit States at Single Pile of Failure Level: Ultimate & Serviceability Define Statistical Characteristics of Basic Random Variables Resistance Load Determine Model Determine Load Uncertainty for Uncertainties from Strength (from Superstructure to database) Foundation (from ST code) Notes: ST = structural MC Simulation or Probability MC = Monte Carlo Calculation to Get Statistical µ = mean Properties of Scalar R G(x) = performance function of the limit state = limit state function Back-calculated Beta Reliability G(x) = 0 = limit defining failure for vs Load Ratio Curves Assessment G(x)<0 in Practice GL(x) = linearized performance function Assign Target Betas Review Target Betas in the Literature and Practice Calculate Load and Resistance Factors Select Load and Resistance Factors Adjust for Mean/Nominal Parameters Case Study Designs for Comparison G(x)=0 GL(x)=0 Contours of Failure fRS = fX(x) Region Safe Region µS µR Figure 5. Resistance factor analysis flow chart (after Ayyub and Assakkaf, 1999 and Ayyub et al., 2000), using FORM developed by Hasofer and Lind (1974).

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13 6. Repeat steps 1 to 5 until convergence of is achieved. 2. With the mean value for R computed in step 1, the par- This reliability index is the shortest distance to the failure tial safety factor, , is revised as: surface from the origin in the reduced coordinate space. n i µ Li (23) i =1 FORM can be used to estimate partial safety factors such = as those found in the design format. At the failure point µR 1 - ... - L* (R*, L* n ), the limit state is given by, where µLi and µR are the mean values of the load and g = R* - L* * 1 - K - Ln = 0 (19) strength variables, respectively, and i, i = 1, 2,..., n, are the given set of load factors. or, in a more general form by, Load Conditions and Load Factors. The actual load trans- g( X ) = g x1( 2 ) * , x * , K, x * = 0 n (20) ferred from the superstructure to the foundations is, by and large, unknown, with very little long-term research having The mean value of the resistance and the design point been focused on the subject. The load uncertainties are taken, can be used to compute the mean partial safety factors for therefore, as those used for the superstructure analysis. design as, LRFD Bridge Design Specifications (AASHTO, 2000) pro- vide five load combinations for the standard strength limit R* state (using dead, live, vehicular, and wind loads) and two for = (21) the extreme limit states (using earthquake and collision loads). µR The use of a load combination that includes lateral loading may at times be the restrictive loading condition for deep L* i = i (22) foundations design. Pile lateral capacity is usually controlled µ Li by service limit state, and as such, was excluded from the scope of the present study, which focuses on the axial capac- In developing code provisions, it is necessary to follow ity of single piles/drilled shafts. The load combination for current design practice to ensure consistent levels of reliabil- strength I was therefore applied in its primary form as shown ity over different pile types. Calibrations of existing design in the following limit state: codes are needed to make the new design formats as simple as possible and to put them in a form that is familiar to Z = R - D - LL (24) designers. For a given reliability index and probability dis- tributions for resistance and load effects, the partial safety Where R = strength or resistance of pile, D = dead load and factors determined by the FORM approach may differ with LL = vehicular live loads. The probabilistic characteristics of failure mode. For this reason, calibration of the calculated the random variables D and LL are assumed to be those used partial safety factors (PSFs) is important in order to maintain by AASHTO (Nowak, 1999) with the following load factors the same values for all loads at different failure modes. In the and lognormal distributions (bias and COV) for live and dead case of geotechnical codes, the calibration of resistance fac- loads, respectively: tors is performed for a set of load factors already specific in the structural code (see following section). Thus, the load L = 1.75 QL = 1.15 COVQL = 0.2 (25) factors are fixed. In this case, the following algorithm is used to determine resistance factors: D = 1.25 QD = 1.05 COVQD = 0.1 (26) 1. For a given value of the reliability index, , probability For the strength or resistance (R), the probabilistic charac- distributions and moments of the load variables, and teristics are defined in Chapter 3, based on the databases for the coefficient of variation for the resistance, compute the various methods and conditions that are described in mean resistance R using FORM. Chapter 2.