Determination of Input Parameters for a Fully Probabilistic Geotechnical Design

Stanovení vstupních parametrů pro plně pravděpodobnostní geotechnický návrh

Lumír MIČA1, Roman KOIŠ

2, Jiří BUČEK

3, Radoslav RUSÍNA

4

Abstract: A deterministic analysis is mainly used for a design of engineering structures nowadays. Fully

probabilistic design is a new trend for analysing engineering structures. Determination of input parameters is a very

important part for both types of analyses. Unlike to a deterministic analysis, the input parameters for a fully

probabilistic design are defined as random values in stochastic approach. It means that statistical data set of an input

parameter is needed from which probability distribution function (pdf) can be derived. This function can be then

defined by various statistical parameters such as the mean value, the variance, etc. The paper deals with a

determination of the pdf type and of its statistic parameters for an oedometric modulus, which is the input

parameter for a modelling the interaction of a structure with a subsoil in Soilin software package. The statistical

analysis is done for two cases. In the first case direct laboratory measurements of an oedometric modulus is used.

This case shows an example of a data evaluation of loess soil at Brno region. The second case shows an evaluation

done for an indirect measurement of the oedometric modulus – cone penetration test (locality – Klobouky u Brna).

Abstrakt: Pro navrhování stavebních konstrukcí se v současné době nejvíce používá deterministická analýza.

V posledních letech se však stále více rozvíjí plně pravděpodobnostní přístup. Obecně však pro oba přístupy je

alfou a omegou stanovení vstupních parametrů. Na rozdíl od deterministického přístupu jsou u plně

pravděpodobnostního přístupu vstupní parametry náhodné veličiny. To znamená, že je nutné mít k dispozici

statistický soubor dat, z kterého může být určeno rozdělení pravděpodobnosti (pdf), které je definováno

statistickými parametry jako je např. střední hodnota, rozptyl, atd. Z takto rozsáhlé problematiky se článek zabývá

stanovením typu pdf a jeho statistickými parametry pro geomechanický parametr oedometrický modul.

Oedometrický modul byl zvolen z toho důvodu, že je vstupním parametrem pro modelování interakce podloží se

základovou konstrukcí pomocí podprogramu SOILIN. Statistická analýza je provedena pro dva případy. V prvním

případě jsou použity výsledky z oedometrických laboratorních zkoušek (přímé měření). Tato analýza je ukázána na

příkladu naměřených dat pro spraš. Druhý případ získání dat pro statickou analýzu vychází z nepřímého testování

oedometrického modulu pomocí statické penetrační zkoušky. Jako příklad byla zvolena lokalita Klobouky u Brna.

Keywords: Oedometric modulus, Oedometric laboratory test, Cone penetration test, Statistical analysis.

Klíčová slova: Oedometrický modul, Oedometrická zkouška, Statická penetrační zkouška, Statistická analýza.

1 Ing. Lumír Miča, Ph.D., Brno University of Technology, Faculty of Civil Engineering, Veveří 95, 602 00 Brno,

Czech Republic, +420541147234, mica.l@fce.vutbr.cz 2 Ing. Roman Koiš, Statika Olomouc, s.r.o., Balbínova 374/11, 779 00 Olomouc, Czech Republic, +420585700702,

kois@statika.iol.cz 3 Ing. Jiří Buček, Ph.D., FEM consulting, s.r.o., Veveří 95, 602 00 Brno, Czech Republic, +420541147374,

bucek@fem.cz 4 Ing. Radoslav Rusina, Ph.D., FEM consulting, s.r.o., Veveří 95, 602 00 Brno, Czech Republic, +420541147374,

rusina@fem.cz

1 Introduction

An objective when defining an engineering structure is ensuring its reliability as well as

its economy. This task is most distinct when solving foundation structures. Those structures are

in interaction with both subsoil and superstructure and they generate so-called interaction

system. For ensuring reliability of engineering structures at first there were used an allowable

stress method or a factor of safety method. With new knowledge and application of theory of

probability there was created an ultimate limit state method or partial reliability coefficient.

Although this method is specified as probabilistic it is needed to keep in mind that it is about

deterministic calculation because all magnitudes are given with particular value. On the other

hand it is not corresponding with full reality because most of events yield to randomness. It is

because input parameters do have specific variability. In addition in geotechnics this is

amplified by fact that soil is not made by man (it was not dosed in exact recipe) but by nature,

hence there is also developing of space variability (but it is not a subject of the article). If we

want this randomness to be involved in the design then we have to apply some of the stochastic

methods (essential magnitude as a random magnitude, random function or by fuzzy sets theory).

We have to apply so-called fully probabilistic method. Although the application of this method

is in-process for longer time its signification has not been well appreciated. It was mainly

because of chances of getting input parameters as same as the efficiency of computers. In recent

years this tendency is changing and in near future the usage of this method will be common way

of designing engineering structures. A research project SISMO is also in a spirit of this

tendency. The project deals with stochastic analysis of interaction system “subsoil-foundation-

superstructure”. The part of this project is also statistic analysis of geomechanical input

parameters.

2 Statistic analysis of geomechanical data

Stochastic interaction problem is being solved by usage of computing core FEM of

system SCIA ENGINEER with a program unit SOILIN [1], [11] and probabilistic FReET [4].

For analysis of the interaction between subsoil and foundation structure the model called surface

model of subsoil is being applied. For this model is given that during its strain the same virtual

work as in 3D subsoil is being carried out whereas it is possible to establish all hierarchy of

parameters C1, C2, C3. This model of subsoil is described in detail in the book [2]. However the

parameters C are not characteristic of soil because they are dependant also on the geometry of

the foundation. It is needed to figure them out either experimentally (almost unrealistic) or by

calculations. For this reason the program SOILIN has been created. This program is able to find

out the process of settlement wherever and from which to find out the wanted parameters C.

This procedure is carried out on the basis of state of stress of elastic homogenous half-space and

standard model of soil. Physical model of subsoil in program unit SOILIN results from ČSN 73

1001 [9], where the settlement of surface of subsoil is being calculated from given state of stress

from the formulae:

, ,

1 ,

nz i i or i

i

i oed i

ms h

E

(1)

where z,i is vertical normal stress component in elastic isotropic homogenous infinite

half-space (or in stratum), or,i is analogous component of original geostatic state of stress, mi is

correcting coefficient of surcharge (the coefficient of structural rigidity) and n is the number of

layers with a thickness hi and oedometric modulus Eoed,i, in which so-called effective stress is

non-negative:

, , , , 0zú z i s i z i i or im (2)

Zones where the effective stresses are negative also the deformation is zero. It is mainly

in a greater depths where the subsoil is no more being deformed. The conditions of zero

effective stress determine so-called depth of deformation zone of subsoil.

Input parameters to the calculation is Eoed, which can be replaced by Edef, at present

definition of Poisson ratio . The usage of Eoed or Edef is dependent on its assignment.

Oedometric modulus is being defined by laboratory testing while deformation modulus is most

often gain from the results of in-situ tests. The next input parameter is a bulk density of soil , which is also being defined by laboratory testing (very difficult to determine) it is being taken

from the tables. The last parameter is a coefficient of structural rigidity which is not possible to

determine by measuring but only by use of tables.

In order to carry out of stochastic analysis of construction it is needed to carry out the

randomness of the problem. It means to think of the input geomechanical parameters as random

with specific theoretical distribution of probability „pdf“(normal, lognormal, Weibull etc.). The

type „pdf“ is commonly determining on the basis of statistic analysis of random selection Xi of

great range and on the basis of physical principle of a given effect. The most difficult task is to

have the data set of plotting parameter. Generally we do have two options available, which is

either the laboratory (direct) or field (indirect) testing of geomechanical data.

2.1 Statistic analysis form laboratory testing data

In the case of our solving problems it concerns particularly with determination of

oedometric modulus of deformability. Oedometric modulus characterises the state when soil

owing to vertical surcharge cannot be deformed to the sides. It is called one-axial deformation.

Oedometric modulus is being calculated for single intervals of loading according to the formulae

(3). Hence there must be stated a given range of stresses. This modulus is being determined by

laboratory test which must be carried out by given procedure according to ČSN CEN ISO/TS

17892-5 [8]. For its determination it is needed to remove undisturbed sample.

z

zoedE

(3)

In order to carrying out of statistic evaluation we must have data set available. In the

work experience it is very difficult because the range of site investigation is rather limited,

which is being reflected on the number of sufficient data. Hence the other option how to get

those data is archive. (e.g. Czech geological service or the archives of investigative

organisation). From those sources it is possible to get the data which can be comprised to

statistic analysis. The utilisation of those data is shown on the example of Brno town for the

loess which is most frequent soil you can find in this area.

Statistic characteristics of oedometric modulus of this kind of soil has been evaluated

from 40 results of oedometric test by usage of program IDENT. The results are summarised in

table 1.

Load

interval

MPa

Ēoed

MPa SEoed vEoed aEoed eEoed

Type of

pdf

0,05-0,10 6,56 2,69 0,41 0,40 -1,14 Bouned

normal

0,10-0,20 7,02 1,97 0,28 1,08 0,34 Weibull

0,20-0,30 8,00 1,85 0,23 0,24 -0,98 Weibull

0,30-0,40 19,21 5,38 0,28 0,25 -0,96 Weibull

0,40-0,50 22,60 5,04 0,22 -1,14 -0,68 Pearson III

Table 1. Statistical characteristics for Eoed [3]

In the table there is evaluated mean value Ēoed, standard deviation sEoed, variation

coefficient VEoed, coefficient of inclination aEoed and taperness eEoed and also appropriate type of

distribution of probability. All the time we have to keep in mind that each statistic

characterisation is valid for appropriate interval of loading.

In conclusion it is needed to mention, that the size of this modulus may be significantly

influenced during the extracting but also during the test of itself [7].

2.2 Statistical analysis with using penetration testing

The second possibility, how to get data set for statistical analysis, is using indirect

methods of geotechnical survey. The penetration testing is one of them. In these days four

testing methods are standardized. Those are:

Cone Penetration Test,

Dynamic Probing Test,

Standard Penetration Test,

Weight Sounding Method

The first and second methods are being used in the Czech Republic and from these tests

we can obtain information about subsoil:

Delineating soil stratigraphy

Type of soil

Mechanical properties of soil by using correlation formula

Compaction quality

and the others

Cone penetration test (CPT)

The principle of test consists of pushing cone tip into the ground continuously at 20

mm/s. The depth interval between readings is usually each 20 mm or 50 mm and the maximum

interval is 200 mm. The base CPT probe measures the resistance on the cone tip qc and the

sleeve friction fs but other probes with specific sensors can be used. The qc and fs is calculated

from equation (4) and (5):

qc = Fc / Ac (4)

fs = Fs / As (5)

Where Fc = pushing force, Ac = cone plan area, Fs = shear force on friction sleeve, As =

area of friction sleeve. The soil engineering properties are derived from these results. In our

case the relationship was derived for deformation modulus:

cdef AqE (6)

Where parameter “A” depend on type soil. The parameter “A” is for sand from 2 to 2.5

and fro clay from 3 to 7 or form own experience. For more detail see [6]. CPT test is generally

suitable to all soil type. The problem can be in coarse-grained soils (dense) because the

penetrating machine will not achieve required pressure. In this case it is suitable to use dynamic

penetration test.

Dynamic penetration test (DP)

Comparing CPT to DP test the cone is driven to the ground by blow of hammer, which

falls from constant high. The rate of driving is from 15 to 30 blows/min. The depth interval

between readings is usually each 100 mm. In the practise three types DP are used:

Dynamic probing typ A (DPA)

Dynamic probing typ B (DPB)

Dynamic probing light (DPL)

The difference is in drop high, weight of hammer and cone shape. The results is dynamic

penetration resistance qdyn and then the deformation modulus is calculated from equitation (7):

dyndef nqE (7)

Where parameter n depends on the type of soil. For more detail see STN EN ISO 22476-

2 (721032) [10].

2.3 Statistical evaluation from CPT

In civil engineering practice, calculation models of structures are mostly created from

planar and beam (finite) elements, i.e. 2D and 1D. The subsoil as soil environment is, however,

a typical 3D medium and it should be analysed that way (i.e. 3D). In general, the system

(structure + foundation + subsoil) is 3D in nature and if we wanted to know in detail the stress-

state and deformation below the foundation, we would have to model the subsoil using 3D finite

elements. This would, on the other hand, disproportionally increase the number of unknown

parameters of deformation and – in practical models – we would exceed the time and capacity

limits of contemporary computers. Moreover, if the application of 3D finite elements was driven

by the attempt to perform a more detailed analysis, such a solution would make

disproportionally big demands on the physical input data and, as a result, the geological survey

would strongly increase the total costs of the whole project. Fortunately enough, the primary

goal is the design of the structure and foundations and we are interested in the conditions in the

subsoil only to be able to determine its effect on the response of the structure. In that situation

we can swap to a solution in which the whole 3D subsoil is represented just by its 2D surface.

The statistical evaluation has been done in probability software FReET that is being

developed at the Institute of Structural Mechanics at Brno University of Technology. Among

other options, FReET features Latin Hypercube Sampling (LHS), which is the stratified

simulation of Monte Carlo type, as the preferred sampling technique (see [4]). Realizations of

random variables are selected from predefined intervals of probability distribution and they are

ranked by advanced strategies to deliver match between the desired and actual dependency

pattern. LHS is very effective variance reduction technique, i.e. sufficient accuracy is usually

achieved with small number of realizations. The programme allows sampling of random

variables from about 30 theoretical probability distribution functions (e.g. normal, log-normal,

Weibull etc.) and also from empirical distribution functions obtained from field measurements.

The Kolmogorov-Smirnov test for goodness of fit can be applied to obtain the best-fit of

probability distribution function.

A statistical evaluation has been performed with results of CPT measuring for foundation

of oil tank. For each tank were done five CPT probes. The typical output of CPT is shown on

Fig. 1.

(a) Penetration and (b) geotechnical data

neutron-gamma log data

Fig. 1 Interpretation of cone penetration test (Geotrend Slany)

From these five CPT probes the deformation modulus was determined. The number of

values varied from 80 to 180 in dependence on the thickness of layer. Using the goodness of fit

test, Beta distribution (Fig. 2) was selected as the most suitable for deformation modulus. The

reason for selection of this four-parameter distribution is that it is very flexible and it can fit

wide range empirical histograms. The bounds of the theoretical distribution had to be corrected

because, physically, Edef cannot be negative. The parameter correction of the distribution was

performed such that both the mean value and variance remain unchanged. The correction is

summarized in Tab. 2. The table contains information on statistical moments of both, the

original parametric Beta distributions and the corrected empirical distributions.

Fig. 2 Beta probability distribution function of the parameter Edef for geotechnical type Ia

NAME DISTRI

BUTION

DESCRI

PTORS MEAN STD COV SKEWNESS

KURTOSIS

EXCESS STATUS

Edef Ia Beta Moments 3.8658e+6 1.1977e+6 0.30983 0.2474 -0.61927 O.K.

Edef Ia Empirical Moments 3.8658e+6 1.1977e+6 0.30983 0.2474 -0.61927 O.K.

Edef Ib Beta Moments 4.7385e+6 1.9038e+6 0.40178 0.34673 -0.4423 O.K.

Edef Ib Empirical Moments 4.7385e+6 1.9038e+6 0.40178 0.34673 -0.4423 O.K.

Edef IIa Beta Moments 1.5985e+7 4.9678e+6 0.31079 0.80344 0.21765 O.K.

Edef IIa Empirical Moments 1.5985e+7 4.9678e+6 0.31079 1.442 3.5183 O.K.

Edef IIb Beta Moments 3.0546e+7 9.3753e+6 0.30692 0.20952 -0.6451 O.K.

Edef IIb Empirical Moments 3.0546e+7 9.3753e+6 0.30692 0.23778 -0.64849 O.K.

Edef IIc Beta Moments 3.4833e+7 1.023e+7 0.29369 0.42587 -0.53101 O.K.

Edef IIc Empirical Moments 3.4833e+7 1.023e+7 0.29369 0.72132 0.0012231 O.K.

Table 2 Modified and empirical probability distribution functions

2.4 Conclusion

We believe that fully probabilistic calculations are going to be employed more frequently

in civil engineering practise. In order to use it, adequate data bank available for input data

selection will be needed.

SISMO project deals with stochastic interaction of foundation with subsoil. A part of this

project deals with the possibility of determination of input data for subsoil.

In the case of selected physical model of subsoil the concern is the probabilistic

modelling of oedometric modulus. This paper presents two options how to get the data set of

oedometric modulus for subsequent statistical analysis:

• The first option is the utilization of results from laboratory tests. The advantage of this

procedure is smaller error when determining it in comparison with the second option

(penetration testing – correlation relation). On the other hand we have to work with

limited number of data available in the range of survey. In case of the particular

construction it usually concerns about two up to three values.

• The second option described in this paper is the utilization of cone penetration testing

(CPT). The advantage of this testing is getting more representative sample of data set

due to methodology of testing.

For evaluation of data set the universal probabilistic software FreET has been used. This

software communicates with FEM analysis (SCIA ENGINEER) through very efficient interface

for solving fully probabilistic analysis of the interaction of foundation with subsoil [5]. The

interface is created within SISMO project.

Acknowledgements

This research was financially supported by the project of the Czech Science Foundation

(GA ČR) No. GA103/09/1262, No. GAČR č. 103/08/0752 and by the research project of The

Ministry of Education, Youth and Sports (MŠMT ČR) No. MSM0021630519. Authors

appreciate this support.

References

[1] Buček J., Kolář V., Obruča J. (1993): SOILIN Výpočet sedání a parametrů interakce

podle norem ČSN, DIN a zásad EC7 (SOILIN Calculation of settlement and interaction

parameters in accordance with ČSN, DIN and EC7 – in Czech). Brno, FEM consulting,

s.r.o, 56s.

[2] Kolář V., Němec I. (1989) Modelling of Soil-Structure Interaction. New York, Oxford,

London, Amsterdam, Tokyo, ELSEVIER, 334p.

[3] Koiš, R. (1995) Zpřesněné výpočetní modely při navrhování základových pasů, Vysoké

učení technické v Brně (Improvement computational models for designing of shallow

foundation – in Czech), Brno, 122s.

[4] Novák, D., Vořechovský, M., Rusina, R. (2007): FREET version 1.5 – User's Manual

and Theory Manual. Brno/Červenka Consulting, http://www.freet.cz

[5] Novák, D., Miča, L., Teplý, B., Vořechovský, M., Buček, J., Rusina, R. and Němec, I. (2009): Soil-structure interaction stochastic modelling. In: Furuta, Frangopol and

Shinozuka, M. (Eds.), Safety, Reliability and Risk of Structures, Infrastructures and

Engineering Systems, proc. of ICOSSAR 2009, 10th

International Conference on Structural

Safety and Reliability, held in Osaka, Japan. Taylor & Francis Group, London, pp. 3972–

3977. ISBN: 978-0-415-47557-0.

[6] Matys, M., Ťavoda, O., Cuninka, M. (1990): Poĺné skušky zemín (Field testing of soils

– in Slovak), Alfa, Bratislava, 303 p

[7] Syček, J. (1982) Problematika správného stanovení modulu přetvoření zemin (The

problem of correct determination of modulus of soils – in Czech), Inženýrské stavby 9, str.

413-421

[8] ČSN CEN ISO/TS 17892-5 (2005) Geotechnický průzkum a zkoušení - Laboratorní

zkoušky zemin - Část 5: Stanovení stlačitelnosti zemin v edometru (Geotechnical

investigation and testing - Laboratory testing of soil - Part 5: Incremental loading

oedometer test – in Czech), 75 p.

[9] ČSN 73 1001 Základová půda pod plošnými základy (Subsoil under spread foundations –

in Czech), 1988, 75 p.

[10] STN EN ISO 22476-2 (721032) (2005) Geotechnický prieskum a skúšanie. Terénne

skúšky. Časť 2: Dynamické penetračné skúšky (Geotechnical investigation and testing -

Field testing - Part 2:Dynamic penetration test – in Slovak)

[11] SCIA ENGINEER (2008). Software System for Analysis, Design and Drawings of Steel,

Concrete, Timber, Aluminium and Plastic Structures. SCIA Group nv, Industrieweg 1007,

B-3540 Herk-de-Stad, Belgium.

Reviewer: doc. RNDr. Eva HRUBEŠOVÁ, Ph.D., VŠB - Technical University of

Ostrava, Faculty of Civil Engineering, Ostrava, Czech Republic