Robust Geotechnical Design of Shield- Driven Tunnels Using Fuzzy Sets Hongwei Huang, Wenping Gong, C. Hsein Juang, and Sara Khoshnevisan

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1 Robust Geotechnical Design of Shield- Driven Tunnels Using Fuzzy Sets Hongwei Huang, Wenping Gong, C. Hsein Juang, and Sara Khoshnevisan Tongji University 1

2 Outline Introduction Robust geotechnical design (RGD) using fuzzy sets Illustrative example Conclusions 2

3 Outlines Introduction Robust geotechnical design (RGD) using fuzzy sets Illustrative example Conclusions 3

4 Introduction Shield tunnels are widely constructed, the uncertainty in the estimate of soil parameters plays a vital role in designing shield tunnels. Metro network in Shanghai (China) Shield tunnel performance problem in Shanghai (China) 4

5 Outlines Introduction Robust geotechnical design (RGD) using fuzzy sets Illustrative example Conclusions 5

6 Robust geotechnical design methodology What is robust design? Robust design, initiated by Taguchi (1986) in the field of quality engineering, has been reported in many fields. Robust design is to seek an optimal design, represented by a set of easy-to-control design parameters (d), such that the system response is robust against, or insensitive to, the unforeseen variations of hard-to-control or hard-tocharacterize noise factors (θ). Design response, f (d,θ ) f (d 1,θ ) Sensitive Robust f (d 2,θ ) Noise factors, θ 6

7 Robust geotechnical design using fuzzy sets Simulating soil parameters with fuzzy sets Limited data availability at a given site makes it difficult to characterize the soil parameters with certainty; rather, the highest conceivable value (HCV) and lowest conceivable value (LCV) can be easily determined. µ(x) x (degree) The friction angle of a sand is described as about 32 7

8 Robust geotechnical design using fuzzy sets Vertex method for the uncertainty propagation For a system with fuzzy sets as inputs, the system response (i.e., factor of safety Fs) will be a fuzzy set, either. The uncertainty propagation can be realized through vertex method (Dong and Wong 1987). µ(x) µ(fs) 1.0 α i -cut interval 1.0 C αi -level α i α i 0.0 a x α i m x α +i b x 0.0 A Fsa i + Fsa i B Fs α i -cut interval of an input fuzzy number Fuzzy output at α i -cut level 8

9 Robust geotechnical design using fuzzy sets Probabilistic interpretation of the fuzzy factor of safety Inspired by an analogy of finding the joint probability of the occurrence of a series of n independent events each with a chance of α i, the discrete probability mass function of the resulting fuzzy factor of safety can be assessed. µ(fs) p(fs) Fs (i=1) a i Fuzzy output Fs + (i=1) a i Fs p 6 p 5 p 1 Fs (i=1) a i Fs + (i=1) a i PMF of fuzzy output Fs 9

10 Robust geotechnical design using fuzzy sets Probabilistic interpretation of the fuzzy factor of safety Inspired by an analogy of finding the joint probability of the occurrence of a series of n independent events each with a chance of α i, the discrete probability mass function of the resulting fuzzy factor of safety can be assessed. 5 5 Probability density (p) Fs2 Vertex method MCS (Triangular) Fs1 Probability density (p) Fs2 Vertex method MCS (truncated Normal) Fs Factor of safety (Fs1, Fs2) Factor of safety (Fs1, Fs2) Validation of the proposed probabilistic procedure using MCS 10

11 Robust geotechnical design using fuzzy sets Signal-to-noise ratio (SNR)-based robustness measure Based upon the formulated discrete probability mass function of the resulting fuzzy factor of safety, E[Fs] and σ[fs] are computed, and thus, the design robustness, in terms of the signal-to-noise ratio (SNR), is readily obtained. SNR= E [Fs] 10 log 2 10 σ 2 [Fs] where a higher SNR means less variation of the system response (i.e., Fs), and thus higher design robustness. 11

12 Robust geotechnical design methodology Optimization setting of robust geotechnical design? Advanced by Juang et al. (2012), robust geotechnical design (RGD) treats the geotechnical design as a multi-objective optimization problem, in which the design robustness and cost efficiency are optimized while the design (safety) requirements are satisfied. Find: Subjected to: Objectives: d(design parameters) d S(design space) Design (safety) requirements Maximizing the design robustness Minimizing the cost 12

13 Robust geotechnical design methodology Optimization results of robust geotechnical design? The RGD optimization could not derive a single best design; rather, only a set of non-dominated designs can be attained, which from a Pareto front showing the tradeoff between design robustness and cost efficiency. Further, the knee point on the Pareto front that yields the best compromised solution can be selected as most preferred design in the design space. Objective 2, f 2 (d) Infeasible domain Pareto front Knee point Utopia point Feasible domain Illustration of a bi-objective optimization Objective 1, f 1 (d) 13

14 Outlines Introduction Robust geotechnical design (RGD) using fuzzy sets Illustrative example Conclusions 14

15 Illustrative example Robust design optimization setting Noise factors Find: (t, ρ, D j ) Subjected to: t l t t u ; ρ l ρ ρ u ; D jl D j D ju ; β 1 β Τ1 ; β 2 β Τ2 Objective: Maximizing the robustness index of ULS, SNR 1 Maximizing the robustness index of SLS, SNR 2 Minimizing the cost, C (t, ρ, D j ) RGD optimization setting of shield tunnel design o Soil parameters: Ks, c, ϕ, and H GWT o Ground surcharge: q 0 15

16 Illustrative example Parameters setting of illustrative example Membership functions characterization of noise factors Noise factors Soil resistance coefficient (Ks: kn/m 3 )* Lower bound (a) Mode [m = (a + b)/2] Upper bound (b) Soil cohesion strength (c: kn/m 2 )* Soil friction angle (ϕ: )* Ground water table (H GWT : m)** Ground surcharge (q 0 : kn/m 2 ) *** * Data from Shanghai code DGJ **Data from site investigation in Shanghai metro line 13 ***Data from engineering experience or judgment 16

17 Illustrative example Analytical solution of jointed shield tunnels The analytical solution derived by Lee et al. (2001) is adopted in this paper. q 0 Groundwater table HGWT y H p 1 p 3 ϕ ϕ1 p 6 x R p h p 5 p 3 + p 4 p 2 17

18 Illustrative example Factor of safety against ULS and SLS failure ULS performance SLS performance N 2+ M 2 Fs = 0.4% D Fs = Lm Lm 2 max( 1 v,2 ) N2+ M2 h N Axial compression failure v Small eccentricity failure N L Critical failure N M Large eccentricity failure M L Bending failure Ultimate bearing envelope of RC segment M h Convergence deformation of tunnel lining 18

19 Illustrative example RGD results-pareto front and knee point In the RGD optimization using NSGA-II (Deb et al. 2002), the population size is set as 50 while the generation number is set as 100. Marginal utility approach (Branke et al. 2004) Identified knee point Cost (C:USD) Robustness (SNR1) Robustness (SNR2) t = mm ρ = 1.16 % D j = 49.2 mm Corresponding 3-D coordinate SNR 1 = SNR 2 = C = USD 19

20 Illustrative example RGD design, probabilistic design, and current practice Category Design parameters Parameter Segment thickness t (mm) Reinforcement ratio ρ (%) Robust design Probabilistic design Current practice Safety Joint diameter D j (mm) β 1 of ULS β 2 of SLS Robustness SNR 1 of ULS SNR 2 of SLS Cost C (USD) Coefficient of Fs 1 of ULS variation (COV) Fs 2 of SLS

21 Outlines Introduction Robust geotechnical design (RGD) using fuzzy sets Illustrative example Conclusions 21

22 Conclusions A fuzzy sets-based RGD method is advanced and shown effective in producing a robust design. The Pareto front obtained from RGD shows the tradeoff between design robustness and cost efficiency; further, the knee point on the Pareto front can be selected as the best compromised design. A new probabilistic procedure is proposed to interpret the resulting fuzzy output, which is proved to be efficient and effective. 22

23 Thank You! 23