PRAGMATIC PROBABILISTIC MODELS FOR QUANTIFICATION OF TUNNEL EXCAVATION RISK

Transcription

1 PRAGMATIC PROBABILISTIC MODELS FOR QUANTIFICATION OF TUNNEL EXCAVATION RISK Jiří Šejnoha 1,3) Daniela Jarušková ) Eva Novotná 1,3) Olga Špačková 3) 1) Department of Mechanics 1) Department of Mechanics ) Department of Mathematics 3) Centre of Integrated Design of Advanced Structures (CIDEAS) Faculty of Civil Engineering, CTU in Prague May 011

2 A Motivation Presentation Outline B Modeling of successive excavation C1 Categorization of failures and concepts of risk analysis C Failure probability of surface structure C3 Failure due to random cave-in collapse C4 Systems with varying geotechnical parameters D Conclusions

3 A Motivation Fig.1 Courtesy: T. Ebermann et al, Tunel 4/010

4 A Motivation Fig. Fig.3

5 B Modeling of successive excavation Theconvergence confinement method vs quasi-3d FEM Fig.4 Fig.5

6 B Modeling of successive excavation Extenzometricmeasurements measurements as a source of the model data Fig.6 Fig.7

7 B Modeling of successive excavation Fig.8

8 C1 Categorization of failures and concept of risk analysis Most typical failures: Extensive deformations of the tunnel tube Exceeding of acceptable progress of the subsidence trough Cave-in collapse Occurrence of a tunnel segment surrounded by a suddenly weakened rock Occurrence of a resistant overburden the thickness of which randomly tends to diminish Risk assessment: The simplest formula R = P[ Event] D. can be generalized as (1) R = [ ( i) Event] D( i), P[ Event] P Consequence i () where D(i) is the expected financial loss caused by the Consequence.

9 C Failure probability of surface structure The subsidence trough description ( x, y) = W ( x, y) w* ( x, y) ( 3) W + Asimple approximation reads W ( x; y) = W ( 0;0) [ g ( x) + g* ] [ g ( y) + g* ] 1 W ( x; y) + w* [ g ( x) + g ( y) ], ( 4) 1 Fig.9 where W ( x, y) = W g ( x) g ( y), w* = W g*. ( 5) The loading effect In longitudinal direction W d W ; y x 1 0 x ( x y) g ( x) g ( ). In transverse direction W d ; y y 1 0 y y W ( x y) g ( x) g ( ), w* w* g ( y). ( 7) y d x w* w* g x x 1 d ( ) ( 6)

10 C Failureprobability of surface structure Let the structure resistance, R x and/or R y,be a maximum curvature the structure is able to sustain. The failure probability of a segment located at a distance x p f where p p fy fx [ ] ( x ) = P R < W ( x ;0 ) R < W ( x ;0 ) x x y y = p ( x) + p ( x) P R < W ( x;0) R < W ( x;0) fx fy [ ], x x = x x ; Rx 0 1 W* w range of w* x ( x) P[ R < W ( x 0) ] = F ( W + w* ) g ( x) f ( w* ) d * ( x) = P R < W ( x 0) W0 g [ ] = F ( x) y y y d y ; Ry 1 W* w range of w* y ( + w* ) g ( 0) f ( w* ) d *. d (8) (9) The unconditional probability of failure Is p f = L1 1 ( L + B) 1 + B L B 1 p f ( x) dx. (10)

11 C3 Failure due to random cave-in collapse The sequence of collapses is mostly described by means of the Poisson model (Fig. 10). Hence, a random variable distance, U, is exponentially distributed, i.e. Fig.10 = =1, d = +d = d where λ istheintensityoftheprocess., (11)

12 C3 Failure due to random cave-in collapse Modification of Eq. (11) based on an expert s judgment (Fig. 11) Fig.11 The expert s surrogate for Eq. (11) may be written as d = 1 d, 1 (1) Evidently, the normalization condition must be fulfilled d = 1 1 d = 1 1 1

13 C3 Failure due to random cave-in collapse Two scenarios with regard to the cave-in location (Fig.10) a) The cave-in position is sufficiently far away from the selected segment ( ) 1 p f ξ u b) The cave-in occurs in the segment s close vicinity ( ) ( ) 1 p f ξ u x L 0 The unconditional probability of failure in a selected segment ξ = d (13) Fig.10

14 C4Random system with continuously varying geotechnical parameters A lose material randomly separated from the rock overburden(fig. 1) Fig.1 + For a constant level a, the barrier up-crossing rate, ν a, is given by Rice s formula + 1 σ ( a µ ( x)) V V = v f ( a, v) d v a = exp 0 V V π σ σ ν ( 14 ) V V

15 C4Random system with continuously varying geotechnical parameters Introducing the spectral density function S V (ω) along with corresponding relations σ V = SV ( ω )dω, σ = ω S V V ( ω )dω and considering a narrow band process characterized by frequency ω o we arrive at σ V ω o σ V + Evidently, could also be regarded as the intensity of a Poisson process. ν a Hence, the first-passage probability is assessed as where P[ N N f f > 1] = 1 exp( ν a L), is the number of up-crossings on a given length L. L ~ ( ν ( x), L) = ν ( x)dx = ν L a 0 a a ( 15) ( 16) ( 17 ) ( 18)

16 C4 Random system with continuously varying geotechnical parameters Example -A segment of the Blankatunnel (Fig. 3) Interface between the rock overburden and soft material detected by in situ measurements is displayed in Fig. 13. A solid line obtained by the linear regression demonstrates a variable mean function of Gauss process. Fig.13 Fig.3

17 C4 Random system with continuously varying geotechnical parameters The results of a case study depicted in Fig. 14show sensibility of the model to the wave length L 0. The standard deviation of Gauss process σ=0.841m was evaluated numerically and the depth of the rock layer was drawn as h a=6m. Fig.14

18 D Conclusions Probability-based based approaches are an efficient alternative to expert methods such as FTA and ETA. They operate either separately to estimate risks in a direct way or as an auxiliary tool forftaand ETA. The proposed methodology suggests theoretical instruments making it possible to analyze most serious problems tunnel engineering has to face. All the phenomena discussed within the scope of this paper have been recently met during the excavation of the Blanka tunnel in Prague. Of course, there are certain drawbacks that could not be overlooked. The main point is material data which has to be properly predicted both by in situ measurements and laboratory tests. If reliable data is missed, any sophisticated theory whatever becomes pointless and cannot responsibly be implemented.