contributions to probabilistic soil modelling

Huber, Moellmann, Vermeer & Bárdossy: Proceedings of the 7th International Probabilistic Workshop, Delft 2009

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Contributions to probabilistic soil modelling

Maximilian Huber1, Axel Moellmann1, András Bárdossy2, Pieter A. Vermeer1

1)University of Stuttgart, Institute of Geotechnical Engineering, Pfaffenwaldring 35, 70569 Stuttgart, Germany

2)University of Stuttgart, Institute of Hydraulic Engineering, Department of Hydrology & Geohydrology Pfaffenwaldring 35, 70569 Stuttgart, Germany

Abstract: In order to capture the varying soil properties, the EUROCODE 7 offers different methods. The most general possibility to incorporate soil vari-ability is the fully probabilistic approach. However, the probabilistic charac-terisation of the soil is crucial. Within this contribution, basics of describing spatial soil variability are presented. The results of experiments to evaluate soil variability are compared to results presented in literature. Furthermore, a case study has been carried out using the random finite element method.

1 Soil Variability

Geological processes and man-made influences cause fluctuations of properties within soil layers. Several researchers have investigated this soil variability in their work. PHOON & KULHAWY categorised the uncertainty of soil properties in their work [21, 22] in order to model soil variability. Similar investigations have been carried out by other researchers like BAECHER & CHRISTIAN [4], ORR [18], ASOAKA & GRIVAS [3], POPESCU [23], VANMARCKE [33] and many others.

1.1 Random fields

In this contribution random fields are used to describe spatial variability as presented by VANMARCKE [32] or BAECHER & CHRISTIAN [4]. According to BAECHER & CHRISTIAN [4] the application of random field theory to geotechnical issues is based on the assumption that the spatially variable of concern is the realization of a random field, which can be defined as a joint probability distribution [8].


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The spatial dependency of a random field can be expressed via an autocorrelation function ρ(τ). Herin τ is the lag between the points. If a random field Xi has mean μX and variance σX

2 then the definition of the autocorrelation ρ(τ) is as shown in equation (1). Herein is E(X) is the expected value operator.

( ) ( ) [ ]τiXiX

XτiXi

σσ)μ(X)μ(XEτρτρ

+

+

⋅−⋅−

=−= (1)

To make things more general, on can also compare the ranks of a series instead of the single values of a random field. This concept is a more general concept in analyzing spatial patterns. The interested reader is referred to HOLLANDER & WOLFE [14] and JOURNEL & HUIJBREGTS [16].

If the autocorrelation function only depends on the absolute separation distance of xi and xj the random field is called isotropic. Another assumption is ergodicity. Ergodicity means that the probabilistic properties of a random field can be completely estimated from observing on realization of that field [26]. Like for many approaches in Natural Sciences, stationarity is an assumption of the model, and may only be approximately true. Also, stationarity usually depends upon scale. According to BAECHER & CHRISTIAN [4], within small region, such as a construction site, soil properties may behave as if drawn from a stationary process; whereas, the same properties over a larger region may follow this assumption. By definition, autocorrelation functions are symmetric and bounded. Another assumption is the separability of the autocovariance function according to VANMARCKE [32]. Separable autocovariance function can be expressed as a product of autocovariance of lower dimension fields. VANMARCKE [32] as well as RACKWITZ [26] offer various models for autocorrelation functions.

1.2 Geostatistics

In the field of geostatistics the spatial dependence is described via a so called variogram. The so-called semivariance γ is defined as the expected squared increment of the values between two locations according to WACKERNAGEL [35]. For practitioners BAKER ET AL. [5] describes explains the variogram in equation 2.

( ) ( ) ( )( )2n

iii τxfxf

n21τγ ∑ +−⋅

= (2)

In Fig. 1 the correlation distance is introduced. Within the correlation distance two points are correlated according to the autocorrelation function or variogram. VANMARCKE [32] quantifies a correlation structure via the so-called scale of fluctuation. This scale of fluctuation is defined as an integral over the whole length from minus to plus infinity.


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Fig. 1: Relationship between semivariance and autocorrelation according to CLARK [9]

1.3 Consideration of variability in geotechnical design

EUROCODE 7 [19] offers different methods of calibrating partial safety factors for solicita-tion and resistance to incorporate soil variability as shown in Fig. 2. The full probabilistic analysis is the most general and accurate method.

Fig. 2: Overview of the use of reliability methods in EUROCODE 7 [20]


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2 Evaluation of spatial variability

In order to apply probabilistic models for reliability analysis in geotechnics one has to characterize soil in a probabilistic way. PHOON & KULHAWY [21, 22] evaluated geotechni-cal properties on an exhaustive data set. Their emphasis was on describing a coefficient of variation. The coefficient of variation describes the relationship between standard devia-tion and mean of property. A fruitful literature review resulted in the Tab. 1. Specifications from BAKER ET AL. [5] have been updated with other sources as shown in Tab. 1.

Tab. 1: Summary of autocorrelation distances

Source Correlation distance θ

ASOAKA & GRIVAS [3] Undrained shear strength θv = 2.5 – 6.0 m

MULLA [17] Penetrometer resistance θh =40.0 – 70.0 m

RONOLD [28] Shear strength θv = 2.0 m

UNLU ET AL. [31] Permeability θh = 12.0 – 16.0 m

SOULIE ET AL. [29] Shear strength θv = 2.0 m θh = 20.0 m

REHFELD ET. AL. [27] Permeability θv = 3.2 m θh = 25.0 m

HESS ET AL. [12] Permeability θv = 0.2 – 1.0 m θh = 2.0 – 10.0 m

CHAISSON ET AL. [7] Cone resistance θv = 1.5 m

VROUWENVELDER & CALLE [34] Cone resistance θh = 20 – 35 m

POPESCU ET AL. [25] Cone resistance θv = 0.8 – 1.8 m

JAKSA ET AL. [15] Dilatometer θh = 0.5 – 2.0 m


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2.1 Experiments for evaluating variability

One can deduce from studying Tab. 1 that most of the investigations have been carried out in terms of cone resistance and permeability. By studying the publications it turns out that the investigations have mainly been carried out in clay and in sandy soils. In order to gain more knowledge in probabilistic characterisation of soil and rock and to apply fully probabilistic calculations as proposed in the EUROCODE [19], experiments have been carried out.

2.1.1 Experimental setup

The experiments have been conducted within an urban tunnelling site. During the tunnel-ling construction process at the Fasanenhoftunnel in Stuttgart 45 horizontal borings have been carried out as shown in Fig. 3. These horizontal borings were grouped within a geological homogeneous layer of mudstone with a separation distance of 2.5 m. The elevation of the boreholes is varied according to the gradient of the tunnel. Therefore the first and the last borehole have a difference in the elevation approximately 2.5 m.

Fig. 3: Testing scheme in the tunnel

In every borehole a borehole deformation test has been carried out at an approximate depth in the borehole axes of 1.35 m. Fig. 4 shows the equipment of the borehole deformation test as described in DIN 4094-5 [1]. Within this test to half-shells are pressed diametrically against the walls of a borehole. Three different loading cycles have been executed. The pressure was raised up to three different levels of 1,000 kN/m², 2,000 kN/m² and 3,000 kN/m². During this loading process the deformation of the half-shells was measured as illustrated in Fig. 5.


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Fig. 4: Testing Equipment as described in DIN 4094-5 [1]

2.1.2 Experimental Results

The evaluation of the modulus of elasticity has been carried out according to DIN 4094-5 [1]. When looking at the results shown in Fig. 5, one would not conclude that the layer of mudstone was defined as homogeneous. The broad spectrum of results can be deduced from the spatial variability of the stiffness as well as to the varying content of CaCO3 and MgCO3 in the mudstone. In order to capture these results correctly statistical methods have been used as recommended in JOURNEL & HUIJBREGTS [16].

The results of the statistical evaluation of the measurements are shown in Fig. 6. The lognormal distribution function was accepted to represent the measurements after the Kolmogorov-Smirnov-Test as described in FENTON & GRIFFITHS [10].

To evaluate the correlation structure of the stiffness properties different approaches have been used. According to the very little skewed distribution shown in Fig. 6 one can deduce a similar correlation length. In Fig. 7 the variogram shows at a distance between 15 m and 20 m a nearly horizontal plateau. A similar answer can be drawn from the correlation function and from the rank correlation function in Fig. 7 (bottom). The results of the correlation distances are summarized in Tab. 2. The comparison with the results of the literature review is difficult. As mentioned above, there is only a very limited knowledge of stochastic soil properties about the modulus of elasticity presented in scientific litera-ture. Only JAKSA ET AL. [15] did comparable tests in sandy soil. The outcome is very different from the findings of the test carried out in mudstone. The correlation distance has nearly the same dimension as the tunnel diameter. Studying FENTON & GRIFFITHS [10], one can be seen that the critical correlation length has nearly the same dimension as the building.


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Fig. 5: Experimental results of all experiments

(a) probability density (b) cumulative probability

Fig. 6: Statistical results of the measurements of the third loading cycle


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Fig. 7: Analysis of the covariance structure of the third loading cycle for the spatial distribution of the modulus of elasticity (top) using semi-covariance (middle), cor-

relation function and rank correlation function (bottom)

Tab. 2: Results of experiments with assumed lognormal distribution for the modulus of elasticity

mean coefficient of variation

correlation distance

1st cycle loading 124 MN/m² 0,54 10 m

reloading 660 MN/m² 0,64 10 - 15 m

2nd cycle loading 229 MN/m² 0,53 10 - 15 m

reloading 432 MN/m² 0,57 10 - 15 m

3rd cycle loading 229 MN/m² 0,59 15 - 20 m

reloading 397 MN/m² 0,61 15 - 20 m


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3 Case study on probabilistic modelling

Several scientists are working in the field of developing and improving probabilistic soil modelling procedures. FENTON & GRIFFITHS [10] are very present through their publication in Random Finite Element modelling. Herein random fields are mapped on a finite element mesh in order to detect the behaviour of a geotechnical structure under spatially varying properties. In terms of random field representation as well as mapping strategies other authors have to be mentioned like POPESCU ET AL. [24], BREYSSE [6] or HICKS & SAMY [13].

In order to perform RANDOM FINITE ELEMENT modelling, an implementation was done on this into the commercial finite element code Plaxis [2]. The generation of correlated random fields with connected mean values via Cholesky decomposition was adapted from BAECHER & CHRISTIAN [4] and POPESCU ET AL. [25]. The mapping of the random field onto the Finite Element mesh was performed via the midpoint method detailly described by SUDRET & KIUREGHIAN [30].

This program was used for a case study of a strip footing on spatially varying soil with the above mentioned implementation. With this program similar results as presented in FENTON & GRIFFITHS [10] and POPESCU ET AL. [24] could be achieved. For this case study an isotropic, normally-distributed random field was chosen. The resulting bearing capacity is also normally-distributed as shown in Fig. 8b. If one varies the correlation length of the random field, it can be found for which value that the bearing capacity is the lowest as published by FENTON & GRIFFITHS [10].

(a) Geometry of a realization random field for the spatially varying subsoil

(b) probability density and cumulative density function for the bearing capacity

Fig. 8: Input and output of the case study on the behaviour of a strip-footing on random soil


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4 Summary and Conclusions

In this article a contribution to probabilistic soil modelling was presented. Experiments have been carried out to investigate in-situ soil properties. These properties have been ana-lyzed with random fields. Moreover, an implementation of random finite element method into a finite element code has been made and was presented through a small case study.

Further research has to be done in both fields of probabilistic soil modelling. On the one hand additional investigations have to be carried out in order to gain more knowledge in describing mechanical soil properties. On the other hand further research has to be done with respect to random field modelling. By now geotechnical scientists just use random fields with connected mean values. It may be useful to perform investigations with random field generations with connected extreme values as proposed by GOMEZ-HERNANDEZ & WEN [11] in order to describe the behaviour of soil in a better way.

5 Acknowledgements

These investigations were friendly supported by Prof. Dr.-Ing. habil H. Schad (Material-prüfungsanstalt Universität Stuttgart - Division 5 Geotechnics) through consulting, support and fruitful discussions. Moreover we thank Dipl.-Ing. C.-D. Hauck (Stadt Stuttgart, Tiefbauamt), who allowed us to carry out the experiments on-site.

6 References

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[22] K.-K. Phoon and F.H. Kulhawy. Evaluation of geotechnical property variability. Canadian Geotechnical Journal, 36:625–639, 1999.

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[29] M. Soulié, P. Montes, and V. Silvestri. Modeling spatial variability of soil parame-ters. Canadian Geotechnical Journal, 27:617–630, 1990.

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[32] E.H. Vanmarcke. Random fields: analysis and synthesis. The M.I.T., 3rd edition, 1983.

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