geostatistics and differ in their spatial correlation structure. ... Probabilistic calculation of differential settlement due to tunnelling. 2. 2 Calculation of tunnelling ...
Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
Probabilistic calculation of differential settlement due to tunnelling Maximilian Huber 1, Michael A. Hicks 2, Pieter A. Vermeer 1, Christian Moormann 1 1 Institute of Geotechnical Engineering, University of Stuttgart, Germany 2 Delft University of Technology, Delft, Netherlands
Abstract: Within this contribution it will be shown that soil variability has a significant influence on settlement predictions. This influence of soil variability is demonstrated in the case study of a tunnel construction affecting a nearby building. Spatial variability of soil properties was investigated with numerical methods taken from geostatistics. The input parameters of these studies have been taken from field investigations, which have been carried out at a tunnelling site in Stuttgart.
1 Introduction According to prognoses of the European Commission, the growth in traffic between Member States is expected to double by 2020. To meet the challenges connected with the increased requirements for efficient traffic infrastructure, the use of underground space often constitutes an efficient and environmentally friendly solution. But there can also be one essential disadvantage in building tunnels, because, especially in an urban environment, large settlements due to tunnelling can cause tremendous consequences. This article will focus on the probabilistic evaluation of settlements incorporating soil spatial variability. Herein, the soil variability is captured via the random field approach. In order to find an appropriate random field representation of the mechanical parameters, two random field generators are compared. These random field generators are taken from geostatistics and differ in their spatial correlation structure. In situ measurements of stiffness parameters, which were carried out for a tunnelling site in Stuttgart (Germany), offer a good basis for the calibration of the model. The random fields are used with the well known Random Finite Element Method (RFEM), in order to focus on the settlements due to the construction phases of a tunnel.
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Probabilistic calculation of differential settlement due to tunnelling
2 Calculation of tunnelling induced settlements Especially in urban areas, the calculation of differential settlements catches the attention of civil engineers. These settlements can cause damage to buildings in the neighbourhood of an urban tunnelling site. SCHMIDT [25] and PECK [19] were the first to show that the transverse settlement trough, taking place after construction of a tunnel, in many cases can be well described by the Gaussian function. Among others, KOLYMBAS [15] and VERRUIJT [27] derived the settlement curve analytically. This practical approach, as well as numerical methods such as the Finite Element Method (FEM) [17], are often based on the assumption of a homogeneous soil. However, O’REILLY & NEW [18] and MAIR & TAYLOR [16] offered empirical formulae derived from case studies to take layered soils into account. Engineers should also take account of the spatial variability of soils. Up to now; this has been done using global or partial safety factors offered in the national standards. However, researchers like FENTON [12], PHOON [20] and POPESCU ET AL. [23] have proposed the use of probabilistic methods and random fields to take distributed and spatially correlated parameters into account.
3 Stochastic soil properties Many researchers, e.g. PHOON & KULHAWY [21, 22], BAECHER & CHRISTIAN [4] and BAKER ET AL. [5], have investigated the variability of soil properties. These researchers described soil properties via their mean values and standard deviations, as well as the spatial fluctuation via random fields. This approach is also present in geostatistical literature [7, 8, 14, 26]. The random field of a material property can be described using the mean value μ, standard deviation σ and correlation distance θ, which describes the spatial correlation. The spatial correlation is low if the correlation distance is short and vice versa, if the distance is larger [9, 13]. According to CHILES & DELFINER [7] and BAKER ET AL. [5], the correlation distance can be evaluated via the autocorrelation function ρ(τ), as well as via the semicovariance function γ(τ) shown in Figure 1. The spatial correlation of mechanical properties was evaluated for a tunnelling building site in a research project [9]. For this purpose, 45 horizontal core borings with a mutual distance of approximately 2 m and a depth of 2 m were carried out within a layer of mudstone (Figure 2). According to the geotechnical report and the engineering judgement of the planning engineer, this layer was assumed to be homogeneous. However, the results of the borehole jacking test according to DIN 4094-5 [2] showed different results. It can be clearly seen in Figure 3 (right) that there is a variation in the modulus of elasticity along the tunnel. The spatial variation is depicted in Figure 1 (right). A horizontal correlation length of approximately θ = 15 ~ 20 m can be deduced by fitting the semivariogram function γ(τ) as well as the autocorrelation function ρ(τ) by eye. A more detailed description of the results can be found in HUBER ET AL. [13]. 2
γ(τ) ρ(τ)
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semicovariance function γ(τ)
autocorrelationfunction ρ(τ)
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Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
x 10 2
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Figure 1: Results of the modulus of elasticity for all borehole locations (left) and autocorrelation function ρ(τ) together with semi-covariance function γ(τ) of EB,2(right)
data acquisition pressure control pressure hose Lias α
hydraulic pump sediment catcher
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Figure 2: Details of the borehole jacking test in a geological cross section of the tunnel (left) and test equipment according to DIN 4094-5 (right)
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Figure 3: Typical result for a borehole jacking test (left) and histogram of the evaluated modulus of elasticity EB,2 based on 45 tests (right)
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Probabilistic calculation of differential settlement due to tunnelling
Mapping the random fields onto the FEM mesh
Generation of random fields
Performing a FEM calculation
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Evaluation of the probability of damage pdamage
Evaluation of the system response
Figure 4: Evaluation scheme of the probability of damage pdamage due to differential settlements
Tabel 1: Material properties used in the FE calculation Soil Unit weight Friction angle Cohesion Modulus of elasticity Poisson ratio Shotcrete lining Modulus of elasticity Poisson ratio
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Linear elastic, perfectly plastic soil model γ = 20 kN/m³ ϕ' = 20 ° c' = 40 kN/m² μ = 60,000 kN/m² lognormally distributed COV =10% , 50% , 75% θv = θv = 10 m , 15 m , 20 m , 30 m , 50 m ν = 0.35 Linear elastic material model E = 7,500 MN/m² ν = 0.20
Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
4 Random Finite Element Method Within the concept of the Random Finite Element Method (RFEM), a case study was carried out to investigate the influence of spatial variability of the modulus of elasticity on the settlements induced by tunnelling excavation. The various steps of RFEM, as described by FENTON & GRIFFITHS [12], are shown in Figure 4. The spatial variability of the modulus of elasticity is represented via the random field approach by using a mean value, a standard deviation and a spatial correlation structure. The correlation structure is simulated in this investigation via two different algorithms, called “Sequential Gaussian Simulation” and “Sequential Indicator Simulation”. In order to show the influence of the spatial variability, both the correlation length and coefficient of variation were varied. After mapping the random field onto the FEM mesh, the settlements due to the tunnelling excavation process are calculated. The evaluated differential settlements of points X and Y (Figure 4) are used for evaluating the limit state function Z and the corresponding probability of damage pdamage.
4.1 Random field generators Different kinds of random field generators have been developed, e.g. in geostatistical [7, 9, 10, 24] or in geotechnical literature [4, 12, 20]. For the present case study, two different random field generators are used to generate unconditional random fields with an isotropic correlation structure. That is, in this simple implementation of spatial correlation, the horizontal and vertical correlation lengths are the same. The criteria of stationarity and ergodicity of the random field are fulfilled by the Sequential Gaussian Simulation (SGSIM) algorithm as well as by the Sequential Indicator Simulation (SISIM) algorithm. A sequential algorithm means that all points of the random field are visited by following a random path and sequentially the area around each step of the random path is analysed, by following SGSIM or SISIM algorithms as described in detail in the appendix. The random field is not treated as a single entity, but is subdivided into smaller areas, which offers a faster generation in comparison to other methods according to CHILES & DELFINER [7]. The main difference between SGSIM and SISIM is the spatial correlation of the fields. When using the SGSIM algorithm, the extreme tails of the simulated distribution are not correlated [9]. In Figure 5 (left) the correlation of each section of the cumulative distribution function for both algorithms is shown, which has been evaluated by using the indicator approach described in [7, 13]. It can be clearly seen that the extreme values have a lower spatial correlation than those values near to the centre of this distribution. This low correlation can also be seen in Figure 5 (right). The zones of extreme low and high values are less clustered in SGSIM because of the negligible correlation of the extreme values.
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cumulative distribution function
Probabilistic calculation of differential settlement due to tunnelling
SGSIM
SISIM
modulus of elasticity
200 250 300 modulus of elasticity [MN/m²] SGSIM SISIM
indicator correlation length qind maximum indicator correlation length qind,max
100 MN/m²
500 MN/m²
Figure 5: Comparison of the spatial correlation of SGSIM and SISIM algorithms (left) and realisations of SGSIM and SISIM algorithms (right) These two algorithms are used for modelling the spatial variability within the scheme, shown Figure 4. For each algorithm 300 realisations of the random fields have been analysed. More random field realisations were not found to change the mean value and the standard deviation of the output significantly. With respect to evaluated probabilities in the range pdamage ≈ 10-1 -10-2, the number of realisation seemed to be adequate. The random fields generated by the SGSIM and SISM algorithms have a mean value of μ = 0 and a standard deviation of σ = 1. These random fields have been transformed into a lognormal target distribution with a target mean value and a target coefficient of variation. This was done using the normal score transformation, as described in DEUTSCH & JOURNEL [9] and CHILES & DELFINER [7]. This algorithm transforms the random field via transformation of the quantiles. The quantiles of the cumulative distribution function of the simulated random field is substituted by the quantiles of the target distribution function. The big advantage of this rank preserving transformation is the non-parametric way of transforming the quantiles of the simulated cumulative distribution into the quantiles of the target distributions. 4.2
Mapping the random field onto the FEM mesh
The mapping of the random field onto the FEM mesh is carried out via a process of averaging. That is, the elements of the finer random field are averaged within each FEM element. Using this simple discretisation method it is possible to combine a non structured FEM mesh with a rectangular random field, so that the stochastic properties of the random field are preserved.
4.3 Finite Element Method A 2D FEM Model, as pictured in Figure 4, was used to calculate the surface settlements. This Plaxis 2D [3] FEM model consists of 2326 15-noded elements. Table 1 summarises
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Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
the material parameters for the linear elastic, perfectly plastic soil model and the linear elastic material model for the concrete lining. The conventional tunnelling excavation was simulated via the “stress reduction method” [17]. In this 2D method the 3D excavation problem is captured through a stress relaxation factor. Using this, the stress relaxation of the ground due to the delayed installation of the shotcrete lining and the load sharing between soil and lining are nicely addressed. A full faced excavation with a stress relaxation of 35% was chosen according to MOELLER [17]. The stiffness of the building was not taken into account in this evaluation of surface settlements.
4.4 Estimation of the probability of damage A linear limit state function Z was defined to quantify the consequences of spatial variability. This function describes the difference between the rotation α, which was evaluated by RFEM, and an ultimate rotation α ultimate, that is Z = α ulitmate − α
(1)
The ultimate rotation α ultimate = 1/500 due to differential settlements was taken from the German standard DIN 4019 [1] to avoid cracks in masonry. Various methods can be used to calculate the probability pdamage of exceeding the ultimate rotation α ultimate , as explained in BUCHER [6]. In this contribution, the First-Order-Second Moment (FOSM) method [6] and the Monte-Carlo approach have been combined to evaluate pdamage. That is, FOSM was used to approximate the probability of damage, based on the esults of 300 Monte Carlo simulations. It was found out, for 300 realisations, the reliability index only changed slightly. The reliability index βz and probability of damage pdamage are calculated via equation (2), assuming that the limit state equation Z is normally distributed supported by the results shown in Figure 6. In equation (2), the mean value μZ and standard deviation σz of the limit state function (1) are used to estimate the probability of damage pdamage . (2)
100 %
50 %
0%
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Probability denstiy function
Cumulative distribution
⎛ μ ⎞ p damage = Φ ⎜ − Z ⎟ = Φ ( −β Z ) ⎝ σZ ⎠
0 0.002 0.004 Limit state function Z
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fitted normal distribution of Z (μΖ=0.00176;σΖ =0.00076)
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empirical distribution of Z
0
0.004 0.002 0 Limit state function Z
Figure 6: Cumulative distribution function (left) and histogram with fitted probability density function (right) of the normally distributed limit state function Z, based on an underlying SGSIM field (μ = 3,000 kN/m², COV = 50% , θ = 2·D) 7
reliability index β
Probabilistic calculation of differential settlement due to tunnelling
B = 0.5 D B = 1.0 D B = 2.0 D B = 3.0 D B = 4.0 D
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Figure 7: Influence of the width B of the structure for a correlation length of θ = 2D and COV = 50 % for the SGSIM random fields 20
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COV = 10 % COV = 50 % COV = 75 %
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Figure 8: Influence of the coefficient of variation of the SGSIM random field for a correlation length of θ = 2 D and a width of the building of B = 2 D
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reliability index β
Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
θ = 0.5 D θ= 1D θ= 2D θ= 5D
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Figure 9: Influence of the correlation length θ for a structure of width B = 2 D and a COV = 50% for SGSIM random fields
GSSIM SISIM
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Figure 10: Comparison of the consequences of the SGSIM and the SISIM algorithms for a correlation length θ = 2 D and a coefficient of variation COV = 75% 9
Probabilistic calculation of differential settlement due to tunnelling
4.5 Results of the RFEM case study For this RFEM case study the width of the building B, the location of the building and the parameters of the random field (coefficient of variation and correlation distance θ) have been varied to study their influence on the probability of damage pdamage. To show the influences in detail, the parabolic curves (Figure 8) are only shown for a reliability index β10-4 within Figure 7, 9 and 10. In Figure 7 the influence of the width of the structure B and its location relative to the tunnel axis on the probability of damage pdamage are studied. The results are symmetrically to the tunnel axis. The probability of damage pdamage and the reliability index β indicate that the width of the building B is directly related to the probability of damage pdamage. For small widths damage is expected at the distance of x/D ≈ 1 to the tunnel axis. A wider building has a lower probability of damage than a smaller one. In Figure 8 the influence of the coefficient of variation of the SGSIM random fields is shown. The results show that a higher coefficient of variation in the random field causes a higher probability of damage pdamage. For a very high coefficient of variation COV = 75 % a damage is expected at the distance of x/D ≈ 1 to the tunnel axis. The impact of the spatial correlation structure is shown in Figures 9 and 10. In Figure 9, the correlation distance θ of the SGSIM random field was varied. The longer the correlation length, the higher the probability of failure. The differences in results through using SGSIM and SISIM algorithms can be seen in Figure 10. For a correlation length of θ = 2 D and a coefficient of variation COV = 75%, one can see that the SISIM algorithm offers slightly higher probabilities of damage pdamage than the GSSIM algorithm. This is linked to the different spatial correlation structures as described before.
5 Summary and conclusions Within this contribution soil heterogeneity has been addressed. Random fields are used within the framework of RFEM to represent the spatial variability of the modulus of elasticity for a tunnelling case study. A combination of the FOSM and Monte Carlo methods has been used to evaluate the probability of damage due to differential settlements. This approximation of the tails of the limit state function is an aid to understanding the system response. The influence of width of the building B, the location of the building and the parameters of the random field (coefficient of variation and correlation distance θ) have been investigated. It can be deduced that this nonlinear problem is mainly influenced by the coefficient of variation of modulus of elasticity and by the location and the width of the building. In contrast, the influence of the correlation length and different random field generators is rather small.
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Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
6 References [1] DIN 4019-1 Beiblatt 1: Baugrund; Setzungsberechnungen bei lotrechter, mittiger Belastung, Erläuterungen und Berechnungsbeispiele. [2] DIN 4094-5: Geotechnical field investigations - Part 5: Borehole deformation tests, June 2001. [3] R. Al-Khoury, K.j. Bakker, P.G. Bonnier, H.J. Burd, G. Soltys, and P.A. Vermeer. PLAXIS 2D Version 9. R.B.J. Brinkgreve and W. Broere and Waterman, 2008. [4] G.B. Baecher and J.T. Christian. Reliability and statistics in geotechnical engineering. John Wiley & Sons Inc., 2003. [5] J. Baker, E. Calle, and R. Rackwitz. Joint committee on structural safety probabilistic model code, section 3.7: Soil properties, updated version. Technical report, Joint Committee on Structural Safety, August 2006. [6] C. Bucher. Computational Analysis of Randomness in Structural Mechanics: Structures and Infrastructures Book Series. Taylor & Francis, 2009. [7] J.P. Chiles and P. Delfiner. Modeling spatial uncertainty. New York: Wiley, 1999. [8] G. Christakos. Random field models in earth sciences. Academic Press, 1992. [9] C.V. Deutsch and A.G. Journel. GSLIB: Geostatistical software library and user’s guide: Oxford University Press, volume 340. Deutsch, C.V. and Journel, A.G., 1992. [10] P. Dowd. A review of recent developments in geostatistics. Computers and Geosciences, 17:1481–1500, 1991. [11] X. Emery. Properties and limitations of sequential indicator simulation. Stochastic Environmental Research and Risk Assessment, 18(6):414–424, 2004. [12] G.A. Fenton and D.V. Griffiths. Risk assessment in geotechnical engineering. John Wiley & Sons, New York, 2008. [13] M. Huber, A. Moellmann, A. Bárdossy, and P.A. Vermeer. Contributions to probabilistic soil modelling. In H. Vrijling, P.H.A.J.M. van Gelder, D. Proske (editors), 7th International Probabilistic Workshop, pages 519–530, 2009. [14] A.G. Journel and C.J. Huijbregts. Mining geostatics. Academic Press, London, 1978. [15] D. Kolymbas. Tunnelling and tunnel mechanics: a rational approach to tunnelling. Springer Verlag, 2005.
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Probabilistic calculation of differential settlement due to tunnelling
[16] R.J. Mair and R.N. Taylor. Theme lecture: Bored tunneling in the urban environment. In Proceedings of the Fourteenth International Conference on Soil Mechanics and Foundation Engineering, pages 2353–2385, 1997. [17] S.C Möller. Tunnel induced settlements and structural forces in linings. PhD thesis, Universität Stuttgart, Institut für Geotechnik, 2006. [18] M.P. O’Reilly and B.M. New. Settlements above tunnels in the United Kingdom-their magnitude and prediction. In Tunnelling, volume 82, pages 173–181, 1982. [19] R.B. Peck. Deep excavations and tunnelling in soft ground. 7th Int. In Conf. on Soil Mech. and Found. Engrg., Mexico, pages 225–290. Sociedad Mexican de Mecanica de Suelos, 1969. [20] K.K. Phoon. Reliability-Based Design in Geotechnical Engineering - Computations and Applications. Taylor & Francis, 2008. [21] K.K. Phoon and F.H. Kulhawy. Characterization of geotechnical variability. Canadian Geotechnical Journal, 36:612–624, 1999. [22] K.K. Phoon and F.H. Kulhawy. Evaluation of geotechnical property variabaility. Canadian Geotechnical Journal, 36:625–639, 1999. [23] R. Popescu, G. Deodatis, and A. Nobahar. Effects of random heterogeneity of soil properties on bearing capacity. Probabilistic Engineering Mechanics, 20(4):324 – 341, 2005. [24] N. Remy, Alexandre. Boucher, and Jianbing. Wu. Applied geostatistics with SGeMS. Cambridge University Press, 2009. [25] B. Schmidt. Settlements and ground movements associated with tunneling in soil. PhD thesis, University of Illinois., 1969. [26] E.H. Vanmarcke. Random fields: analysis and synthesis. The M.I.T., 3rd edition, 1983. [27] A. Verruijt and J.R. Booker. Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique, 46(4):753–756, 1996.
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Gucma, van Gelder & Proske: Proceedings of the 8th International Probabilistic Workshop, Szczecin 2010
7 Appendix-Simulation of random fields Deutsch & Journel [9] propose to following scheme for simulation of random fields using Sequential Gaussian Simulation (SGSIM) and Sequential Indicator Simulation (SISM). The following steps describe the algorithms of SGSIM and SISIM roughly DOWD [10]: Sequential Gaussian Simulation (1)
Define a random path through all n grid points on which values are to be simulated.
(2)
At each simulation grid point krige a value from all other values (conditioning and simulated).
(3)
The kriged value and the associated kriging variance are the parameters of the conditional.
(4)
Gaussian distribution at the given grid point given the conditioning data and all previously simulated values. Draw a value at random from this distribution and add it to the set of simulated values.
(5)
Return to step (4) until values have been stimulated at all grid points.
Sequential Indicator Method According to Dowd [10] and Emery [11] SGSIM and SISIM are essentially is the same except that at each location the conditional distribution is estimated directly from the indicator variables defined for specified threshold values: (1)
Approximate the range of values taken by the attribute z by K discrete threshold values zk, k = l , . . . , K
(2)
Code each conditioning value into a vector of K indicator values.
(3)
Model the variograms of the indicator variables.
(4)
Define a random path through all n grid points on which values are to be simulated.
(5)
At each simulation grid point obtain a kriged estimate of the indicator variable for each threshold. Each of these K indicator estimates is regarded as an estimate of: Pr{Z(x) ~ zk}.
(6)
Select a value at random for the pseudodistri-bution obtained in step (5) and add this value to the set of simulated values.
(7)
Add the simulated value obtained in step (6) to the set of conditioning values, code it into a vector of K indicator values and go to step (5).
(8)
Repeat until all simulation grid points have been visited.
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