Load and Resistance Factor Design Considering Design Robustness: R-LRFD

Load and Resistance Factor Design Considering Design Robustness: R-LRFD Hsein Juang, PhD, PE, F.ASCE Glenn Professor Glenn Department of Civil Engineering Clemson University 1

Outline 1. Background (Robust design) 2. Methodology of R-LRFD (Robust Load and Resistance Factor Design) 3. Illustrative Example: Drilled Shaft in Clay 4. Conclusions 2

Outline 1. Background 2. Methodology of R-LRFD 3. Illustrative Example: Drilled Shaft in Clay 4. Conclusions 3

Robust design concept Background Robust design aims to make a product or design insensitive to hard-to-control input parameters q (called noise factors ) by carefully adjusting easy-to-control input parameters d (called design parameters ). --- Taguchi (1986) Wayne Taylor http://www.va riation.com/te chlib/val- 1.html 4

Frequency of Occurrence Frequency of Occurrence Background Taguchi method (originated in Industrial Engrg) 1.Reduce variance of system response 2.Bring mean of system response to target Moving mean Minimizing variability Target Value (a) response Target Value (b) response 5

Background Transforming Robust Design Concept into a Novel Geotechnical Design Tool (National Science Foundation grant No. CMMI-1200117) 6

Background Current geotechnical design methods Factor of safety (FS)-based approach (Coping with uncertainties by means of experience and engineering judgment; calculated risk concept) Reliability-based design (RBD) (Incorporating uncertainties explicitly in the analysis; however, difficult to characterize uncertainties of soil parameters, model errors & construction variation) Load and resistance factor design (LRFD) (Current trend; however, uniform risk unattainable with single resistance factors for each analysis model with wide ranges of COV in the input soil parameters) 7

Background Robust geotechnical design (RGD) Offers a new design perspective in the field of geotechnical engineering It is not to replace existing design methods (FSbased design, RBD, or LRFD approach) Complements traditional design approaches (FSbased approach, RBD, or LRFD approach) 8

Outline 1. Background 2. Methodology of R-LRFD 3. Illustrative Example: Drilled Shaft in Clay 4. Conclusions 9

Methodology of R-LRFD Load and Resistance Factor Design Considering Design Robustness: R-LRFD (Robust design plus LRFD) Seeks an optimal design (d) that is insensitive to, or robust against, variation in noise factors (q) such as uncertain soil parameters, model errors, and construction variation. Considers simultaneously safety, robustness, and cost by means of optimization, it is a multiobjective optimization problem. 10

Methodology of R-LRFD Key concepts in R-LRFD Design parameters d (easy-to-control) versus noise factors q (hard-to-control) Measure of design robustness Optimization, Pareto front, and knee point 11

Methodology of R-LRFD Design parameters versus noise factors (1) Easy-to-control design parameters o Geometry parameters o Construction parameters Hard-to-control noise factors o Geotechnical parameters o Loading conditions o Model parameters/model errors 12

Methodology of R-LRFD Design parameters versus noise factors (2) (Using diaphragm-wall supported excavation as an example) Clay -1 m -3 m -5 m -7 m Clay GL -2 m GL -4 m GL -6 m GL -8 m GL -10 m Design parameters: Wall length (L), Wall thickness (t), Vertical spacing of the struts (S), Strut stiffness (EA) Noise factors: Undrained shear strength ( ), horizontal subgrade reaction ( kh v ), and surcharge behind the wall (q s ) 13

Methodology of R-LRFD Key concepts in R-LRFD Design parameters d (easy-to-control) versus noise factors q (hard-to-control) Measure of design robustness Optimization, Pareto front, and knee point 14

Methodology of R-LRFD Measure of design robustness in R-LRFD (slide 1) The system performance, in the context of LRFD approach, may be presented as: f ( d, k ) R( d, k ) S( d, k ) q q R S q where R(d, k θ ) and S(d, k θ ) are the resistance term and load term, respectively; R and S are the resistance factor and load factor, respectively; and, k θ are the characteristic values of noise factors θ. 15

Methodology of R-LRFD Measure of design robustness in R-LRFD (slide 2) Intuitively, the design robustness or the sensitivity of the system response to the noise factors can be mathematically measured using its gradient f ( dk, q ), expressed as follows: f (, ) f (, ) f (, ) d kq d kq d kq f ( dk, q ),,, kq k k 1 q 2 qn q k k k q q q q q 16

Lower vs. higher gradient Measure of design robustness in R-LRFD (slide 3) Design response, f (d,q ) Sensitive Robust Lower degree of design robustness, signaled by higher gradient f (d 1,q ) f (d 2,q ) Noise factors, q 17

Methodology of R-LRFD Measure of design robustness in R-LRFD (slide 4) Sensitivity index (SI) of the system response to the noise factors is defined based on the gradient f ( dk, q ), J kq f ( d, k ) k (, ) k (, ) 1 q q f d k 2 q q f d k 3 q,,, f ( d, k ) k f ( d, k ) k f ( d, k ) k q q qk q q qk q q q k 1 2 n q q q SI J JJ T A higher SI value signals a lower degree of design robustness, as it would suggest a greater relative variation of the system response due to the variation in the noise factors. 18

Methodology of R-LRFD Key concepts in R-LRFD Design parameters d (easy-to-control) versus noise factors q (hard-to-control) Measure of design robustness Optimization, Pareto front, and knee point 19

Methodology of R-LRFD Optimization, Pareto front (Slide 1) Find d to optimize: [C(d), SI(d,q)] Subject to: g i (d,q) 0, i = 1,..,m d - design parameters; q - noise factors; C - cost; SI - robustness measure; g - design constraint. 20

Methodology of R-LRFD Optimization, Pareto front (Slide 2) (In the context of R-LRFD) Find: d (Design parameters) Subject to: d S (Design space) f ( dk, q ) > 0 ( Design constraint) Objectives: Min ( SI) (Sensitivity index) Min ( C) (Cost) 21

Methodology of R-LRFD Optimization, Pareto front, and knee point (Slide 3) Multi-objective optimization may not yield a single best design with respect to all objectives. Rather, a set of nondominated designs may be obtained. This set of designs collectively forms a Pareto front. 22

Objective 2, f 2 (d) Methodology of R-LRFD Optimization, Pareto front, and knee point (4) If no preference is assigned in the robust design optimization, the knee point on the Pareto front that yields the best compromised solution can be selected as most preferred design in the design space. Pareto front Knee point Utopia point Feasible domain Infeasible domain Objective 1, f 1 (d) 23

Outline 1. Background 2. Methodology of R-LRFD 3. Illustrative Example: Drilled Shaft in Clay 4. Conclusions 24

Illustrative Example: Drilled Shaft in Clay L =? F d + F l Stiff clay Design parameters: Pile diameter (D) and pile length (L) Noise factors: Onsite diameter (D T ), onsite length (L T ), normalized undrained shear strength (c n = c u /z), dead load (F d ), and live load (F l ). System response of concern: D = 0.45 m The ultimate limit state (ULS) Schematic diagram of performance (Orr et al. 2011) a drilled shaft in clay (after ETC10) 25

Illustrative Example: Drilled Shaft in Clay Characterization of noise factors in R-LRFD (#1) If a noise factor follows the lognormal distribution, the corresponding characteristic value can be estimated as: 1 exp ln ln 1 1.645 ln 1 2 2 2 k q q q q i i i i where k is the characteristic value of i th q noise factor; and i i qi are the mean and coefficient of variation (COV) of i th noise factor, respectively. Note that the number of 1.645 is adopted in above Equation to ensure that there is 95% likelihood of i th noise factor not greater than (for the load term) or less than (for the resistance term) the characteristic value of. k qi q 26

Illustrative Example: Drilled Shaft in Clay Characterization of noise factors in R-LRFD (#2) 27

Illustrative Example: Drilled Shaft in Clay Characterization of noise factors in R-LRFD (#2) Other assumption can be made Characterized using test data Specified in ETC 10 28

Illustrative Example: Drilled Shaft in Clay Construct the system response in R-LRFD (#1) The system performance in the context of R-LRFD, in terms of ULS performance, can be presented as: LT DT cu1 1dz A 0 bnc cu2 2 f (, ) dk q 5 Fd 6 Fl 7 3 4 where 1, 2, 3, 4, 5, 6, and 7 are the partial factors on undrained shear strength along the pile length (c u1 ), undrained shear strength at the pile base (c u2 ), slide resistance (Q s ), end resistance (Q b ), selected geotechnical model (Equation 9), dead load (F d ), and live load (F l ), respectively. 29

Illustrative Example: Drilled Shaft in Clay Construct the system response in R-LRFD (#2) 30

Illustrative Example: Drilled Shaft in Clay Construct the system response in R-LRFD (#2) Illustrative example 31

Sensitivity index, SI Illustrative Example: Drilled Shaft in Clay Results of the R-LRFD optimization (For a given design space S 1, where D = 0.45 m and L {10.0 m, 10.3 m, 10.6 m,, 24.7 m, 25.0 m}) 250 200 150 Pareto front Least cost Preferred design 100 50 Knee point Most robust 0 18 19 20 21 22 23 24 25 Cost, C (m) 32

Sensitivity index, SI Illustrative Example: Drilled Shaft in Clay The optimization results of R-LRFD, (kn) (For a given design space S 1, where D = 0.45 m and L {10.0 m, 10.3 m, 10.6 m,, 24.7 m, 25.0 m}) 250 200 150 Pareto front Least cost Preferred design 100 50 Knee point Most robust 0 18 19 20 21 22 23 24 25 Cost, C (m) 33

Feasibility, Pr [ f (d,kq)] Illustrative Example: Drilled Shaft in Clay Effect of COV of input noise factors 1.0 0.8 0.6 0.4 0.2 Least cost design Knee point Most robust design 0.0 0 5 10 15 20 25 30 Coefficient of variation of kq, COV(%) Least-cost design cannot withstand the variation; most robust design would still be feasible even at high COV but is costly. 34

Sensitivity index, SI Sensitivity index, SI Illustrative Example: Drilled Shaft in Clay Effect of selected design space 250 250 200 150 Pareto front in S1 Pareto front in S2 200 150 Pareto front in S1 Pareto front in S3 100 Knee point in S2 (D = 0.45 m, L = 19.34 m) 100 Knee point in S1 (D = 0.45 m, L = 19.5 m) 50 Knee point in S1 (D = 0.45 m, L = 19.5 m) 50 Knee point in S3 (D = 0.45 m, L = 19.7 m) 0 18 19 20 21 22 23 24 25 Cost, C (m) 0 18 20 22 24 26 28 30 Cost, C (m) 35

Sensitivity index, SI Sensitivity index, SI Illustrative Example: Drilled Shaft in Clay Effect of selected design space 250 250 200 150 Pareto front in S1 Pareto front in S2 200 150 Pareto front in S1 Pareto front in S3 100 Knee point in S2 (D = 0.45 m, L = 19.34 m) 100 Knee point in S1 (D = 0.45 m, L = 19.5 m) 50 Knee point in S1 (D = 0.45 m, L = 19.5 m) 50 Knee point in S3 (D = 0.45 m, L = 19.7 m) 0 18 19 20 21 22 23 24 25 Cost, C (m) 0 18 20 22 24 26 28 30 Cost, C (m) 36

Outline 1. Background 2. Methodology of R-LRFD 3. Illustrative Example: Drilled Shaft in Clay 4. Conclusions 37

Conclusions R-LRFD, a new design paradigm, has been demonstrated as an effective tool to obtain optimal designs that are robust against variation in noise factors (e.g., uncertain soil parameters, model errors, and construction variation). R-LRFD consider safety, cost, and robustness simultaneously and is shown as an effective tool. Pareto front and knee point concepts can aid in making informed decision in the design. The proposed gradient-based robust design methodology complements all existing design methods, including the FSbased approach, RBD, or LRFD approach. 38

Selected papers on robust design *Gong, W., *Khoshnevisan, S., Juang, C.H., Gradient-based design robustness measure for robust geotechnical design, Canadian Geotechnical Journal, 2014. *Gong, W., *Wang, L., *Khoshnevisan, S., Juang, C.H., Huang, H., and Zhang, J., Robust geotechnical design of earth slopes using fuzzy sets, Journal of Geotechnical and Geoenvironmental Engineering, 2014. Juang, C.H., *Wang, L., Hsieh, H.S., and Atamturktur, S., Robust geotechnical design of braced excavations in clays, Structural Safety, Vol. 49, 2014, pp. 37-44. *Gong, W., *Wang, L., Juang, C.H., *Zhang, J., and Huang, H., Robust geotechnical design of shield-driven tunnels, Computers and Geotechnics, Vol. 56, March 2014, pp. 191-201. Juang, C.H., *Wang, L., *Liu, Z., Ravichandran, N., Huang, H., and Zhang, J., Robust geotechnical design of drilled shafts in sand - A new design perspective, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 139, December 2013, pp. 2007-2019. *Wang, L., *Hwang, J.H., and Juang, C.H., and Sez Atamturktur, Reliability-based design of rock slopes A new perspective on design robustness, Engineering Geology, Vol. 154, 2013, pp. 56-63. 39

Thank You! 40