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1 ,, Department of Civil Engineering, Chuo University Kasuga , Bunkyo-ku, Tokyo , JAPAN atsu1005@kc.chuo-u.ac.jp kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD , Nihonbashi-Honcho 4-Chome, Chuo-ku, Tokyo , Japan Abstract This paper presents a numerical technique to predict a soft stratum ahead of an excavating face of a tunnel. The functional method is a parameter identification of the elastic property which is important coefficient at the actual site. The parameter identification is an useful technique in the reverse analysis to determine an unknown parameter using the observed data. The parameter identification is utilized to minimize the difference between the computed and observed displacements to determine the Young s modulus of the ground. In order to solve the minimization problem, the conjugate gradient method is applied. As a numerical model, the 3-D linear elastic body with the finite element method is used to excavate the natural ground and to identify the Young s modulus ahead of a tunnel face. Using the 3-D tunnel, the displacement of the axial direction can be considered. Key words: Parameter identification, 3-D tunnel, Excavation, Finite element method, Conjugate gradient method 1 2 σ,j ρb i = 0 (1) 1
2 σ, b i, ρ ε = 1 2 (u i,j + u j,i ) (2) ε u i σ = Dkl e ε kl (3) Dkl e Dkl e = λδ δ kl + µ(δ ik δ jl + δ il δ jk ) (4) δ λ µ λ = νe (1 2ν)(1 + ν) µ = E 2(1 + ν) (5) (6) E ν S S U S T u i =û i on S U, (7) t i = σ n j = ˆt i on S T, (8) û i ˆt i n i 3 K αiβk = ˆΓ αi = N αi 4 F V K αiβk u βk = ˆΓ αi (9) V (N α,j D e kln β,l )dv (10) (N α ρ ˆb i )dv (N α ˆt i )ds S T (11) F (1) u = du (1) (0) σ = dσ (0) (1) (0) (1) σ = σ + dσ p 2 p (2) u = du (1) (2) (1) (2) u (1) + σ = σ + dσ
3 n 1. n =0 2. u (0) i = du (0) i σ (0) = dσ (0) 3. σ (0) p (n+1) αi 4. p (n+1) αi 5. p (n+1) αi 6. du (n+1) i dσ (n+1) 7. u (n+1) i σ (n+1) 8. σ (n+1) p (n+2) αi 9. n = n +1 4 p αi p (n+1) αi = V N α,j σ (n) dv, (12) σ (n) 5 T T Z X B A α β B Y T = cos α cos β 0 cos α sin β 0 sin α 0 0 cos α cos β 0 cos α sin β 0 sin α. 3
4 Z EXCAVATION Y X 6 J J 1 J = 2 (u u ) T Q(u u )dv (13) V u u m E {P } T = {E} T = {E 1,E 2,E 3, E m }, (14) 1. P (0), ε j i =0 2. u(p ) (0) J (0) 3. [ ] (0) u(p ) 4. {d} (0) = { J 5. α = } (0) [ ] T = u(p ) (u(p ) u(p ) ) {d} T [ u(p ) { } {d} T J ] T [ 6. P (i+1) = P (i) + αd (i) u(p ) ] {d} 7. u(p ) (i+1) J (i+1) 8. [ ] (i+1) u(p ) 4
5 9. β = ψ{ J } (i+1), ψ{ J } (i), { { J J } (i+1)! } (i)! 10. {d} (i+1) = { J } (i+1) + β{d} (i) 11. J (i+1) <ε j 12. i = i +1 [5] 7 : ( ) F : ( ) 1. E init F 4. F = γh γ h E init Excavation Computational domain Natural ground Tunnel face 5
6 A (x1,y1,z1) A (x2,y2,z2) Excavation Origin Origin B (x1,y1,z1) B (x2,y2,z2) (a) (b) dux = x2 - x1 duy = y2 - y1 duz = z2 - z1 A dux B duz Excavation dux duy (c) [kn/m 2 ] [kn/m 2 ] 4.0[m] 6
7 (a) (b) [kn/m 2 ] 0m E1 10m E3 E2 16m 14m 12m E4 20m E(kN/m 2 ) E ( 0m - 12m ) E (12m - 14m ) E (14m - 16m ) E (16m - 20m ) 7
8 m - 12m ( E1 ) 12m - 14m ( E2 ) 14m - 16m ( E3 ) 16m - 20m ( E4 ) Youngs modulus [kn/m2] Iteration [kn/m 2 ] 0m E1 10m E3 E2 16m 14m 12m E4 20m Young s modulus Stratum E(kN/m 2 ) E Hard ( 0m - 12m ) E Hard (12m - 14m ) E Soft (14m - 16m ) E Hard (16m - 20m ) 8
9 m - 12m ( E1 ) 12m - 14m ( E2 ) 14m - 16m ( E3 ) 16m - 20m ( E4 ) Youngs modulus [kn/m2] Iteration 9 F = γh 100m 105m Domain1 Domain2 30m 0m 200m 0m 200m Sand Stone Figure 11. Soft layer 9
10 (a) (b) D(m) h(m) γ(t/m 3 ) F (kn/m 2 ) [kn/m 2 ] [kn/m 2 ] 0m E4 E3 E2 200m 105m 150m E1 130m 110m :Hardground Measured point : 100m E(kN/m 2 ) E Hard ( 0m - 110m ) E Hard (110m - 130m ) E Hard (130m - 150m ) E Hard (150m - 200m ) 10
11 Youngs modulus [kn/m2] m - 110m ( E1 ) 110m - 130m ( E2 ) 130m - 150m ( E3 ) 150m - 200m ( E4 ) Iteration [kn/m 2 ] 0m E4 E3 E2 200m 105m 150m E1 130m 110m :Hardground : Soft ground Measured point : 100m E(kN/m 2 ) E ( 0m - 110m ) E (110m - 130m ) E (130m - 150m ) E (150m - 200m ) 11
12 m - 110m ( E1 ) 110m - 130m ( E2 ) 130m - 150m ( E3 ) 150m - 200m ( E4 ) Youngs modulus [kn/m2] Iteration 10 References [1] Sakurai, H. and Tanigawa, M., Study of some problems on the back analysis of measured displacements in tunnelling, Vol.1, Journal of Geotechnical Engineering, pp85-94(1990). [2] Sugimoto, M., Sakurazawa, M. and Kageyama, S., Three dimensional reverse analysis on ground properties with time-dependence, 9th International Congress on Rock Mechanics, Vol.2, pp (1999). [3] Y.M. Hsieh and A.J. Whittle, A computational strategy for solving three-dimensional tunnel excavation problems, Second M.I.T. Conference on Computational Fluid and Solid Mechanics (2003). [4] Nojima, K. and Kawahara, M., Mesh Generation of Three-dimensional Underground Tunnels Based on the Three-Dimensional Delaunay Tetrahedration, Vol.5, Journal of Applied Mechanics JSCE, pp (2002). [5] R.Fletcher and C.M.Reeves., Function Minimization by Conjugate Gradients, Computer J., pp (1964). 12
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