Calibration of Resistance Factor for Design of Pile Foundations Considering Feasibility Robustness Hsein Juang Glenn Professor of Civil Engineering Clemson University 1
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Outline of Presentation Background Traditional Resistance Factor Calibration Calibration Considering Robustness Design Example and Further Discussion Summary 3
Foundations Design Methodologies 19 th Early 20 th century Empirical Design Early 20 th century - now Allowable Stress Design Late 20 th century - now Reliability-based Design development of soil mechanics and analysis methods following the lead of structural design practice 4
Allowable Stress Design (ASD) The factor of safety (FS) is introduced and applied to the geotechnical capacity as: Q R FS ni n FS = 2 3 is adequate for foundations The FS is used to account for all uncertainties in Load and material properties Design models Construction effects etc. 5
FS = True Safety Level? f R (R) or f Q (Q) 0.25 0.20 0.15 0.10 0.05 Q μ Q mean FS = 2.5 P[R < Q] = 0.0002 (β = 3.6) R μ R 0 0 10 20 30 40 50 60 Resistance or Load (R, Q) f R (R) or f Q (Q) 0.25 0.20 0.15 0.10 0.05 Q μ Q mean FS = 2.5 P[R < Q] = 0.073 (β = 1.5) 400 times less safe R μ R 0 0 10 20 30 40 50 60 Resistance or Load (R, Q) The same FS may imply very different safety margins 6
FS = True Safety Level? f R (R) or f Q (Q) 0.25 0.20 0.15 0.10 0.05 Q μ Q mean FS = 2.5 P[R < Q] = 0.0002 (β = 3.6) R μ R 0 0 10 20 30 40 50 60 Resistance or Load (R, Q) f R (R) or f Q (Q) 0.25 0.20 0.15 0.10 0.05 Q μ Q mean FS = 3.0 P[R < Q] = 0.0036 (β = 2.7) 20 times less safe R μ R 0 0 10 20 30 40 50 60 Resistance or Load (R, Q) A larger FS does not necessarily mean a smaller level of risk 7
Reliability-based Design (RBD) Reliability analysis is used and probability of failure (P(R<Q)) is introduced to measure the design risk. Full-probabilistic approach e.g., Expanded RBD (Wang et al. 2011) Semi-probabilistic approach e.g., Partial Factor Approach Load and Resistance Factor Design (LRFD) Multiple Resistance and Load Factor Design (Phoon et al. 2003) Quantile Value Method (Ching and Phoon 2011) Robust-LRFD (Gong et al. 2016) 8
f R (R) or f Q (Q) Load and Resistance Factor Design (LRFD) Under the LRFD approach, design must satisfy the equation: QiQ QiQni Rn R (Eurode 7) or ni Rn (AASHTO) Load factors, γ Qi 1, accounts for variability in loads Resistance factor, γ R =(1/φ) 1, accounts for variabilities in soil properties, design models and construction γ Q Q n =R n /γ R Q R Q n R n Resistance or Load (R, Q) 9
Reliability Concept in LRFD Assuming R and Q are lognormally distributed, performance function can be described as g = ln(r) ln(q), which follows normal distribution: g p f p( R Q) p( g 0) ( ) ( ) 2 2 COV COV ln 1 1 R Q Q R 2 2 COVQ COVR ln 1 1 g 2 2 COV COV ln 1 1 R R Q Q Q R 2 2 COVQ COVR ln 1 1 2 2 COV COV ln 1 1 g R Q Q R 2 2 COV COV g ln 1 Q 1 R Q R, R, Q Q n n R R R n Q Q n β T 10
Selection of β T References Meyerhof 1970 3.1-3.7 Phoon et al. 1995 3.2 Canadian Building Code 1995 3.5 AASHTO 1997 2.0-3.5 Paikowsky et al. 2004 2.33 for redundant piles 3.0 for non-redundant piles β T (U.S. Army Corps of Engineers 1997) 11
Calibration of Resistance Factor Load factors (γ Q ) and load statistics (λ Q and COV Q ) developed in the structural design codes are adopted. Resistance factor (γ R ) is calibrated using: R exp ln 1 1 2 2 COV COV Q T Q R 2 2 1COV 1COV R Q Q R λ R and COV R = the mean and the COV of the resistance bias factor, which are estimated from a load test database 12
Challenges The resistance bias factor statistics are hard to ascertain, uncertainty is inherent in the derived statistical parameters of the resistance bias factor The resistance factor calibrated for LRFD is very sensitive to the uncertainty in the resistance bias factor Consequently, a design obtained using the calibrated resistance factor may not achieve β T (i.e., the design is not feasible) if the variation in the resistance bias factor is underestimated. 13
Goal of This Study To propose a new approach for resistance factor calibration that considers explicitly the feasibility robustness of design Feasibility robustness is a measure of robustness, indicating the extent that a system remains feasible even when it undergoes variation. Resistance factor is re-calibrated considering variation in the resistance bias factor. Design using the re-calibrated resistance factor will always satisfy the β T requirement to the extent defined by the designer in the face of uncertainty in the computed capacity 14
Outline of Presentation Background Traditional Resistance Factor Calibration Calibration Considering Robustness Design Example and Further Discussion Summary 15
Traditional Resistance Factor Calibration Li, J.P., Zhang, J., Liu, S.N., & Juang, C.H. (2015). Reliability-based code revision for design of pile foundations: Practice in Shanghai, China. Soils and Foundations, 55(3), 637-649. Design Equation in Shanghai is written as: R n R Q Q D Dn L Ln Pile Types Design Methods Uncertainty in Capacity driven piles bored piles load test-based method design table method within-site variability cross-site variability CPT-based method 16
Uncertainty Analysis of Design Methods Computed capacity (R n ) subjected to within-site variability cross-site variability the variation of the soil properties within a site the construction error associated with the site-specific workmanship the regional variation of the soil properties the construction error associated with workmanship in a region The actual capacity (R) can be expressed as: R NR N N R n 1 2 where N 1 and N 2 are bias factors accounting for within-site variability and cross-site variability, respectively; and N is lumped bias factor. n 17
Statistics of Resistance Bias Factors As the uncertainties associated with N 1 and N 2 are from different sources, it might be reasonable to assume that N 1 and N 2 are statistically independent. It can be show that: R R1 R2 COV COV COV 2 2 R R1 R2 where λ R, λ R1, and λ R2 are the means of N, N 1, and N 2, respectively; COV R, COV R1, COV R2 are the COVs of N, N 1, and N 2, respectively. The within-site variability can be characterized by comparing the measured and the predicted bearing capacities of piles within a site, while the crossvariability can be characterized by comparing the measured and the predicted bearing capacities of piles from different sites. 18
Calibration Database A database consisting of 146 piles from 32 sites and another database comprising 37 piles from 10 sites were used to characterize the within-site variability for driven piles and bored piles, respectively. bored piles driven piles DATABASE 19
Characterization of Within-site and Cross-site Variabilities Characterization of within-site variability λ R1 =1 since within-site is unbiased The COV R1 values vary from site to site, and the computed mean of the COV R1 is used, i.e., COV R1 = 0.087 and COV R1 = 0.093 are adopted for the analysis of driven and bored piles, respectively. Characterization of cross-site variability The values of λ R2 and COV R2 are taken based on the previous design code SUCCC (2000) SUCCC. (2000). Foundation Design Code, Shanghai Urban Construction and Communications Commission (SUCCC), Shanghai (in Chinese). 20
Load Statistics and Load Factors Load Statistics Typical load statistics used in different studies (after Li et al. 2015) References λ D λ L COV D COV L Ellingwod et al. (1980) 1.00 1.05 0.10 0.18 Ellingwood and Tekie (1999) 1.05 1.0 0.1 0.25 Nowak (1999) and ASSHTO (2007) 1.08 1.13 1.15 0.18 Nowak (1994) and FHWA (2001) 1.03-1.05 0.08-0.10 1.1-1.2 0.18 Li et al. (2015) based on MOC (2002) 1.00 1.00 0.07 0.29 Load Factors γ D =1.0 and γ L = 1.0 used in MOC (2002) are adopted MOC, (2002). Code for Design of Foundations (GB 50007-2002). Ministry of Construction (MOC) of China, Beijing. (In Chinese). 21
Calibration of Resistance Factor Calibration Equation R 2 2 COV COV exp ln 1 1 Q T Q R 2 2 1COV 1COV R Q Q R R 2 2 exp ln T 1 COVR 1 COV Q D L R D L 2 2 1 COVQ 1 COVR Q DQDn LQLn D L = = Q Q Q D Dn L Ln D L = Q Q =0.2 Ln Dn 1 COV = COV COV 1+ 2 2 2 Q D L Calibration Results Calibrated resistance factors (γ R ) for load-carrying capacity (β T = 3.7) Driven piles Bored piles Load factors LT method DT method CPT method LT method DT method γ L =1.0 1.53 1.93 1.72 1.56 2.26 γ D =1.0 22
Cumulative Frequency Cumulative Frequency Variation in COV R1 (Background study) Sort the COV R1 values in ascending order; Rank the values from i = 1 to n; Compute the cumulative probability p i = i/(n+1); Establish cumulative distribution function. 1.0 1.0 0.8 0.8 0.6 Lognormal Distribution 0.6 Lognormal Distribution 0.4 0.2 Driven Piles 0.4 0.2 Bored Piles 0 0 0 0.05 0.10 0.15 0.20 0.25 0.30 0 0.05 0.10 0.15 0.20 0.25 0.30 COV R1 COV R1 Cumulative frequency of the observed COV R1 with fitted lognormal CDF 23
Relative Frequency Cumulative Frequency Effect of the Variation in COV R1 5000 random samples of COV R1 are generated and the corresponding β values are computed with calibrated γ R using: 1 COV ln = 2 2 ln 1COVR1 COVQ 2 R R D L Q 2 D L 1COVR 0.08 0.06 0.04 0.02 Driven Piles LT Method Histogram Cumulative Frequency 0.466 0 0 1 2 3 4 5 6 Reliability index, β β T 1.00 0.75 0.50 0.25 Relative and cumulative frequency of β associated with calibrated γ R The β values distribute in wide ranges and many of the designs cannot achieve β T = 3.7 (i.e., the designs are not feasible); the probability of (β < β T ) can be obtained from the cumulative frequency curve of β. 24
How to deal with the effect of variation in COV R1? 25
Outline of Presentation Background Traditional Resistance Factor Calibration Calibration Considering Robustness Design Example and Further Discussion Summary 26
Robust Design Robust design, originated from the field of Quality Engineering (Taguchi 1986), seeks an optimal design by selecting controllable design parameters so that the system response of the design is insensitive to, or robust against, the variation of noise factors. Robust design has recently been applied to geotechnical problems (Juang et al. 2013), and examples of geotechnical design with LRFD approach considering robustness have been reported (Gong et al. 2016). This study is aimed at introducing the robustness concept into the LRFD calibration. 27
Robustness Measures (from Khoshnevisan et al. 2014) 28
Feasibility Robustness The feasibility robustness (Parkinson et al. 1993) is adopted herein to measure the robustness of partial-factor design with respect to uncertain parameters (i.e., COV R1 ), and is defined as the probability that β T can still be satisfied even with the variation in COV R1. Feasibility robustness is formulated as (Juang et al. 2013): P[( ) 0] T P 0 where P[(β β T ) 0] is the probability that β T is satisfied; and P 0 is a predefined acceptable level of this probability (i.e., feasibility robustness). 29
Relative Frequency Cumulative Frequency Calculation of Feasibility Robustness Monte Carlo Simulation (MCS) 0.08 0.06 0.04 0.02 Driven Piles LT Method Histogram Cumulative Frequency Normal Distribution 0.466 1.00 0.75 0.50 0.25 0 0 1 2 3 4 5 6 Reliability index, β For driven piles with LT method, when using calibrated γ R = 1.53 P[(β β T ) 0] = 1-0.466 = 0.534 β T 30
Calculation of Feasibility Robustness Point Estimation Method (PEM) Assuming β follows normal distribution: T P[( T ) 0]= P 0 By using PEM (Zhao and Ono 2000): P[( ) 0]= G( ) T 7 7 2 2 = Pi i, = Pi ( i ) i1 i1 R Feasibility robustness of calibrated partial factors in Li et al. (2015) obtained from MCS and PEM Approach Driven piles Bored piles LT method DT method CPT method LT method DT method MCS 0.534 0.489 0.515 0.558 0.500 PEM 0.541 0.479 0.506 0.578 0.464 31
Resistance Factor Calibration Considering Feasibility Robustness The procedure to evaluate feasibility robustness of a design using the existing γ R actually is the inverse of the task of resistance factor calibration considering robustness, which is a process of determining value of γ R such that the resulting design can achieve the pre-defined feasibility robustness level. Trail-and-error approach P[( ) 0]= P A trail γ R MCS P[(β β T ) 0] = P 0? T 0 Solving equation P[(β β T ) 0] = G(γ R ) = P 0 based on PEM 32
Resistance Factor Calibration Results Calibrated resistance factors (γ R ) for load-carrying capacity at different feasibility robustness levels (γ L = 1.0 and γ D = 1.0; β T = 3.7) 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 2.30 0.6 1.55 1.98 1.75 1.57 2.34 0.7 1.59 2.01 1.79 1.62 2.38 0.8 1.64 2.05 1.83 1.69 2.44 0.9 1.72 2.11 1.90 1.81 2.52 0.99 2.04 2.27 2.10 2.39 2.76 To achieve the same P 0, the DT method requires larger γ R, as it is associated with greater uncertainties. On the other hand, the required γ R for the LT method is smaller due to the lower uncertainties involved with the LT method. 33
Values of μ β and σ β at various feasibility robustness levels P0 Driven piles Bored piles LT method DT method CPT method LT method DT method μβ σβ μβ σβ μβ σβ μβ σβ μβ σβ 0.5 3.70 0.67 3.70 0.30 3.70 0.43 3.70 0.83 3.70 0.30 0.6 3.88 0.70 3.79 0.30 3.80 0.44 3.90 0.89 3.79 0.31 0.7 4.11 0.74 3.87 0.31 3.95 0.46 4.18 0.93 3.86 0.31 0.8 4.38 0.79 3.98 0.32 4.10 0.47 4.54 1.01 3.98 0.32 0.9 4.81 0.87 4.13 0.33 4.36 0.50 5.14 1.13 4.12 0.33 0.99 6.32 1.13 4.53 0.36 5.04 0.56 7.56 1.65 4.54 0.36 For the same P 0, piles designed with the LT method have the largest μ β ; while piles designed with the DT method have the smallest μ β. Both the values of μ β and σ β increase with increasing P 0. Thus feasibility robustness is primarily affected by μ β. 34
Outline of Presentation Background Traditional Resistance Factor Calibration Calibration Considering Robustness Design Example and Further Discussion Summary 35
A Bored Pile Design Example Q Dn = 2500 kn, Q Ln = 500 kn, pile diameter D=0.85m, pile length L needs to be determined by using DT method Rn Rs Rt U p fsili qt At R Q Q n R D Dn L Ln Suggested side and toe resistances of bored piles in different soil layers (after SUCCC 2010) Soil description Depth (m) ƒs (kpa) qt (kpa) Grayish yellow clay 0-4 15 - Very soft gray clay 4-20 15~30 150~250 Gray silty sand 20-35 55~75 1250~1700 Gray fine sand with silt 35-60 55~80 1700~2550 Gray fine, medium or coarse sand 60-100 70~90 2100~3000 36
Pile length, L (m) Bored Pile Design Results 54 52 50 48 46 44 0.5 0.6 0.7 0.8 0.9 1.0 Feasibility robustness level, P 0 Design results at various feasibility robustness levels Design with high robustness against variation in the computed capacity can always be achieved at the expense of cost efficiency. 37
Optimization between Robustness and Cost A design with higher feasibility robustness and relatively lower cost is desired. A tradeoff decision is thus needed based on an optimization of γ R performed with respect to two objectives, design robustness and cost efficiency. Find: An optimal γ R compatible with γ L = 1.0 and γ D = 1.0 Subject to: G(γ R ) = P 0 {0.5, 0.51, 0.52,, 0.99} Objectives: Maximizing the feasibility robustness (in terms of P 0 ) Minimizing the construction cost (in terms of γ R ) 38
Optimization between Robustness and Cost Resistance factor, γ R 2.4 Driven Piles LT Method 2.2 2.0 1.8 1.6 1.4 0.5 0.6 0.7 0.8 0.9 1.0 Feasibility robustness, P 0 A tradeoff exists between design robustness and cost efficiency; the tradeoff relationship is presented here as a Pareto front. The knee point on the Pareto front conceptually yields the best compromise among conflicting objectives. 39
Objective 2 Approaches: Determination of Knee Point Reflex Angle approach, Normal Boundary Intersection approach, Marginal Utility Function approach and Minimum Distance approach Most optimal with respect to Objective 1 Minimum distance Objective 1 Minimum Distance approach Gong et al. (2016) Feasible designs Knee point Utopia design Most optimal with respect to Objective 2 Perform a single-objective optimization with respect to each objective function of concern Determine the corresponding maximum value of each objective function Normalize the objective functions into values ranging from 0.0 to 1.0 Compute the distance from the normalized utopia point to the normalized objective functions The point that has the minimum distance from the utopia point is taken as the knee point. 40
Determination of Knee Point Resistance factor, γ R 2.4 Driven Piles LT Method 2.2 2.0 1.8 1.6 Knee Point 1.4 0.5 0.6 0.7 0.8 0.9 1.0 On the left side of knee point, a slight reduction in γ R (i.e., cost) would lead to a large decrease in design robustness P 0. On the other side of the knee point, a slight gain in robustness P 0 requires a large increase in γ R, rendering it cost inefficient. Feasibility robustness, P 0 The knee point represents the best compromise between design robustness and cost efficiency. 41
Calibrated partial factors obtained from knee point of Pareto front Piles Driven piles Bored piles Design method Knee point Suggested P0 γr P0 γr LT method 0.86 1.68 0.85 1.67 DT method 0.84 2.07 0.85 2.08 CPT method 0.85 1.86 0.85 1.86 LT method 0.88 1.78 0.85 1.74 CPT method 0.84 2.46 0.85 2.47 (γ L = 1.0 and γ D = 1.0 ; β T = 3.7) 42
Outline of Presentation Background Traditional Resistance Factor Calibration Calibration Considering Robustness Design Example and Further Discussion Summary 43
Summary A new approach for calibration of resistance factors for design of pile foundations considering feasibility robustness is proposed. Design using re-calibrated resistance factors is robust against the uncertainty in computed capacity. The methodology is demonstrated effective for design of pile foundations in Shanghai, China. 44
Thank You 45