**A Collection of Fluctuation Scale Values and Autocorrelation Functions of **

**Fine Deposits in Emilia Romagna Plain**

_{, Italy }

_{, Italy }

Joanna M. Pieczyńska-Kozłowska, Ph.D.1; Wojciech Puła, Ph.D.2; and Giovanna Vessia, Ph.D.3

1_{Dept. of Civil Engineering, Wroclaw Univ. of Science and Technology, }

Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland. E-mail: [email protected]

2_{Dept. of Civil Engineering, Wroclaw Univ. of Science and Technology, }

Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland. E-mail: [email protected]

3_{Dept. of Engineering and Geology, Univ. of Chieti-Pescara, Italy }

Campus Universitario, Via dei Vestini 31, 66100 Chieti. E-mail: [email protected]
**Abstract **

The spatial variability structure of soils strongly influences the shallow foundation designing
when it is taken into account. As a matter of fact, in reliability-based design the estimated value
*of the scale of fluctuation θ and the choice of the autocorrelation function ρ(z) can lead to *

different estimations of the bearing capacity of shallow foundations. Many different methods and
types of measurements can be used to estimate the preceding quantities. In this paper, seven cone
penetration tests CPTs performed in the first 10m of the fluvial deposits located in the Emilia
*Romagna plain (Italy) have been analyzed. Different θ values have been drawn by using two *
*numerical techniques and two ρ(z) have been assumed as the best fitting models to the sample *
*autocorrelation functions. Results show that differences in θ values slightly affect the design *
*values of the bearing capacity whereas they are mostly influenced by the shape of the ρ(z). *

**INTRODUCTION **

Spatial variability structure of physical and mechanical soil properties can be characterized by its statistics, that are the mean value , the variance 2, the probability density function, the scale of fluctuation , the autocorrelation function and the variance reduction function Γ. The first three statistics are easy to be calculated from a sample of measurements (provided an adequate sample size); the remainder parameters have been introduced by Vanmarcke (1984) as the pillars of the stochastic field theory that describes the realizations of variables named random fields.

The scale of fluctuation is the distance beyond which two random variables that describe a physical property can be considered independent, the autocorrelation function

represents the mathematical shape of the spatial self-correlation of a random variable. Several studies have been addressed to measure the preceding two variables. Two expressions are commonly used for modelling the autocorrelation functions, that are the Gaussian equation:

( )= {− (| |) } (1)

and the Markov one:

( )= { | |} (2)

*where τ is the distance between two points of a field. *The selection of these two types of

autocorrelation functions is due to their wide use in many computer codes carrying out reliability computations.

*Let us consider a continuous reading of the cone tip resistance Qc *from a cone penetration

*test CPT. The spatial waving values of a continuous profile of measures m can be divided into a *
*mean trend t and a fluctuation or residual profile values e according to the following relation *
(Vanmarcke, 1977):

( ) = ( ) + ( )* * (3)

*The t(z) is commonly represented by an interpolation function selected through the mean square *
root (LSM) method. If the whole trend content within the measurement profile has been removed
the de-trended values ( ), named fluctuations or residuals, will be randomly distributed about
the zero value and their summation (the mean) alongside depth will be approximately zero.

*Furthermore, their variance σ2* will be constant. If the aforementioned conditions are
fulfilled the fluctuations can be defined a central stationary random field (Vanmarcke, 1984). A
random field is spatially correlated meaning that adjacent values do not differ that much each
other but they are independent of those values that are further apart, lying beyond a distance
named scale of fluctuation.

In order to use the fluctuation profiles as one dimensional random field the preceding
*stationarity conditions in weak-sense (second-order stationarity) must be checked. This means *
*that only the first two moments (the mean and the variance) of ε(z) sample values are requested *
to be constant.

The aim of the study is to investigate the influence exerted by (1) the numerical methods
*used to estimate θ in vertical direction, hereafter θy*, and (2) the autocorrelation functions varying

*between Eq. (1) and (2) on the design values of the bearing capacity of shallow foundations qd*.

*In this study the horizontal scale of fluctuation θx* will be treated as a constant parameter equal to

20m The horizontal scale of fluctuation was considered as a much larger value than the vertical
*one and then the RFEM mesh domain. For each value of θy the undrained shear resistance su* will

*where su* is the random field derived from Qc*, Nk *is the cone factor that varies between 9 and 20

*depending on the geometry of the tip of the penetration device. In this study Nk *is assumed 15.

*Thus, in the sections below, at first the variability structure of Qc* will be calculated

according to the methods illustrated in section 2, then the target values of the reliability index
to be used in performing reliability-based design of shallow foundations (ISO 2394, 2015) are
*briefly introduced. After that, the case study of the Emilia Romagna plain is shown: the qd* is

hereafter estimated by a numerical code based on the Random Finite Element Method (RFEM), (Griffith & Fenton 1993, Fenton & Griffiths 2008). In the analytical way of thinking the bearing capacity of the shallow foundation will be calculated according to the following formula:

= 5.14 ⋅ (5) However, in RFEM the ultimate bearing capacity is estimated by classical finite element method

*with given displacement step. The su* is modeled as a random field with lognormal distribution

function, mean value and standard deviation. Each random field is characterized also by the correlation function (Eq. (1) or (2)) and the scale of fluctuation, calculated according to

section 2. To generate the random field for each FEM realization the local average subdivision
(LAS) method is used (Fenton & Vanmarcke 1990). The results of the presented RFEM analyses
are the mean and the standard deviation values of the bearing capacity. The results in terms of
*the variation of qd versus and qd versus θy* values are discussed hereafter.

**METHODS TO CALCULATE THE SCALE OF FLUCTUATIONS **

From a set of fluctuations and their spatial distribution along vertical direction (1D random field)
the scale of fluctuation can be derived through several methods. These methods can be divided
into two groups: graphical (Rice 1944, Vanmarcke, 1984) and analytical (Lloret-Cabot, Fenton,
*and Hicks, 2014, Baecher and Christian, 2003). The graphical approaches derive θ from the *
distances between two subsequent crossings of the zero-fluctuation axis.

*As the graphical method is concerned, named di* the distance between two adjacent

crossings whose width is detected by the operator, the scale of fluctuation can be calculated through the Rice formula (1944), namely

### ≈

### |

( )### |

### ,

(6)*where d is the mean value between above mentioned crossings. By subsequent substitution of *
*autocorrelation function ρ(z) given by eqs (1) and (2) one can obtain corresponding values of *
*fluctuations scale θ. Hence the scale of fluctuation was estimated using the equations shown in *
Table 1.

**Table 1. Rice equations for scale of fluctuation (after Pieczyńska-Kozłowska 2015). **

Correlation model Scale of fluctuation
Gaussian
θ*= d* 2
π
Markov
θ=*2d*
π

This method was utilized by (Pieczyńska-Kozłowska, 2015). Past applications can be found in Puła (2004), Cherubini et al. (2007), among others. Furthermore, the graphical approaches gave the advantage of the straightforward estimate of the scale of fluctuation directly from the visual insight of fluctuations.

The scale of fluctuation can be calculated also by analytical methods. These methods
*calculate at first the autocorrelation function ρ(*_{Δ}) that represents the shape of the spatial
correlation of the sample of measures within the scale of fluctuation distance. As the analytical
*methods are considered, a widely used approach to calculate ρ(*_{Δ}) is the method of moments. It
consists of the following expression (Baecher and Christian, 2003):

( ) = _{⋅(} _{)}⋅ ∑ ( ) − ( ) ⋅ ( ) − ( *) * (7)

*where σ2 is the variance of the sample of measures, Δ is the separation distance between adjacent *
*measures, n-Δ is the number of data pairs at a constant separation distance Δ, t(xi*) is the value of

*the trend function calculated at xi and xi+Δ and z(xi*) are the measured values of the soil

*parameter. The sample autocorrelation functions ρ(Δ) can be fitted by smooth functions known *
as model autocorrelation functions (see Cherubini et al. 2007; Fenton and Griffiths, 2008). Two
*commonly used ρ(*_{Δ}), namely given by eqs (1) and (2) will be employed in the present study.

**TECHNICAL CODE REQUIREMENTS FOR RELIABILITY INDEX **

In design practice, the safety margin of a structure is assumed to be strictly related to the target reliability level. Diamantidis et al. (2016) proposed to relate the target reliability level of a structure to a balanced combination of the costs of the failure and of the safety countermeasures. From the economic point of view, the objective is to minimize the total working-life cost. These instructions have been reported in ISO 2394 (2015).

The parameter generally used as a measure of the reliability in codes is the reliability
*index β. β values for a geotechnical design has been recommended for different failure *

*probabilities in many international documents. In this paper the reference values of β are taken *
from ISO 2394 (2015), shown in Table 2.

**Table 2. Values of β index according to ISO 2394 (2015) **

Standard Reliability index

ISO 2394:2015 (economic optimization) 3.3 / 4.2 / 4.4* ISO 2394:2015 (LQI) 3.1 / 3.7 / 4.2*

**High / moderate / low relative costs of safety measures. The definition of economic optimization and Life Quality Index (LQI) can be found in Rackwitz (2000).

**THE CASE STUDY OF THE EMILIA ROMAGNA PLAIN **

The Emilia Romagna plain is located in the north of Italy and it is one of the largest valley filled of Holocenic alluvial deposits of the whole Europe (the biggest in Italy). It was shaped after the uplift of the Italian Apennine and Alp ranges the water of the Adriatic Sea was pulled away and the uplift land formed the ancient “Padano gulf”. During the last 600.000 years the gulf was dried out and cropped out the alluvial sediments brought by the numerous tributaries of the Po river and by itself. Thus, these deposits are thick (several tens of meters) and spatially

heterogeneous. In order to investigate the spatial variability of these soil deposits, in this study
seven piezocone penetration tests CPTUs drilled in an area of about 25km2 and 30km far from
Bologna city center have been analyzed through the aforementioned methods (Fig. 1). The
horizontal distances among the CPTUs are about 5-7km thus no horizontal spatial correlation has
been considered. At first, the classification of the soils under study has been carried out through
the criteria by Eslami-Fellenius chart (2004), shown in Table 3. As can be seen the investigated
surficial soils are mixtures of silt and clay meaning that the bearing capacity design must refer to
the Eq. (4). This is the reason why, in Table 3, from column four to six the first and second
*moments of the undrained shear strengths su* are reported.

**Table 3. Soil classification and characterization based on seven CPTUs: (column three) **
**lithotypes according to Eslami-Fellenius chart (2004); measured undrained shear strength **
**s****u**** (column four) mean value, (column five) standard deviation and (column six) Coefficient **

**of Variation (CoV). **

**Identification **
**number of **
**piezocone **
**penetration tests **

**Depth (m) ** **Lithotype ** **Su mean value **

**[kPa] **
**Su Standard **
**deviation value **
** [kPa] **
**Su coefficient of **
**variation CoV **

**202060U501 ** 1.7-26.7 clay silt/silt clay 92,92 20,77 0,22

**202070U501 ** 1-16 clay silt/silt clay 86,36 21,70 0,25

**202070U502 ** 1-4 clay silt 42,57 7,22 0,17

**202070U504 ** 2.7-6 clay silt 23,99 5,51 0,23

**202070U505 ** 1.2-5 clay silt 71,75 36,39 0,51

**202070U506 ** 2.4-8.9 clay silt 39,13 9,41 0,24

**Figure 1. Location of the set of seven CPTUs performed nearby Bologna city center. The **
**number of the CPTUs are reported in Table 4. **

**Table 4. Trend equations (column 3) and θ values estimated according to different ****methods: moment method with Markov model of ρ(z) (MMM), moment method with ****squared exponential model of ρ(z) (MMSE), Rice method with Markov model (RM) and ****Rice method with squared exponential model of ρ(z) (RSE). **

**Item CPTU **
**identification **
**number **

**Depth (m) ** **Trend equation ** **MMM ** **MMSE ** **RM ** **RSE **

**1 ** _{202060U501 1.7-26.7 y= 0.51⋅z}0.462 _{0,75 0,66 0,82 }_{1,03 }
**2 ** 202070U501 1-16 y = 0.55⋅z0.48 _{0,87 0,73 0,41 }_{0,52 }
**3 ** 202070U502 1-4 y = -0.11⋅z2_{ + 0.49}_{⋅z + 0.24 } _{0,15 0,13 0,34 }_{0,43 }
**4 ** _{202070U504 2.7-6 y = -0.07⋅z + 0.73 } 0,27 0,22 0,32 0,40
**5 ** 202070U505 1.2-5 y = -0.47⋅z + 2.61 0,47 0,39 0,38 0,47
**6 ** _{202070U506 2.4-8.9 y = 0.026⋅z}2_{ - 0.3346}_{⋅z + 1.67 0,40 0,25 0,41 }_{0,52 }
**7 ** 202080U501 1-3 y = 0.04⋅z2_{ - 0.16}_{⋅z + 1.1 } _{0,13 0,20 0,11 }_{0,14 }

After that, Table 4 shows results of the variability structure analyses (consisting on the
calculation of _{θ}* and ρ(*_{Δ})) carried out through the procedures illustrated in section 2. The study
shown that results, in terms of _{θ} values are practically coincident between Rice’s and
Vanmarcke’s graphical rule. Thus, in Table 3 only Rice methods is mentioned in lieu of the

method (MM) and the Rice formula (R). In addition, _{θ} is determined through two models of

*ρ(*_{Δ}): Markov (M) and the Squared Exponential (SE). As plotted in Fig. 2, among the four
*models of ρ(*_{Δ}) the above mentioned M and SE are the best fitting models in all the cases studied.

**Figure 2. Examples of the best fitting models for sample autocorrelation functions ρ(**_{Δ}**) ****calculated for two CPTUs out of the eight studied. **

In Table 4, from the values of _{θ} listed in columns 4-7 it can be drawn that larger
differences in _{θ} values can be sometimes (items 1-5, first column) attributed to the estimation
procedure, that is the moment or the Rice method, but some other times it is related to the
autocorrelation function model adopted (items 6-8, first column). All the methods used are valid
and they do not introduce a systematic bias in the calculation of the scale of fluctuation _{θ}.
Nevertheless, calculating the differences of _{θ} values drawn through the 4 combination of
methods, it is clear that the largest differences are related to the different method employed for
the autocorrelation function estimation _{θ}. These values vary from 2% to 112%, as shown in
Table 4.

On the contrary, when the same method for _{θ} estimation is used, especially for the case
of Rice one, the calculated differences are almost constant, varying from 18% to 21%.

Furthermore, in the case of Rice method, it is worth noticing that Markov model always
underestimates the scale of fluctuation _{θ} compared with the squared exponential.

This evidence allows to go further on, to analyze the results of the bearing capacity (BC)
*calculations, in terms of the mean value of the BC and the reliability index β. The reliability *
index considered to the bearing capacity analyses are taking into account according the values
shown in Table 1.
0.
0
0.
5
1.
0
0 0.1 0.2 0.3 0.4
**ρ****Depth [m]**
**CPT: 202070U504**
empirical
Markov R^2=0.995
square exponential R^2=0.990
0.
0
0.
5
1.
0
0 0.2 0.4 0.6 0.8 1 **Depth [m]**1.2
**CPT 202070U505**
empirical
Markov R^2=0.990
Square exponential R^2=0.993

**Figure 3. Bearing capacity design value versus β index for the investigated CPTUs: from ****(a) to (g) the results are related to the seven CPTUs. **

160
170
180
190
200
210
220
230
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
**q****d****[kPa]**
**β**
**CPT: 202060U501** MMM [θy =0.75m]
MMSE [θy =0.66m]
RM [θy =0.82m]
RSE [θy =1.03m]
160
170
180
190
200
210
220
230
3.0 3.5 4.0 4.5
**q****d****[kPa]**
**β**
**CPT: 202070U501** MMM [θy =0.87m]_{MMSE [θy =0.73m]}

RM [θy =0.41m]
RSE [θy =0.52m]
130
135
140
145
150
155
160
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
**q****d****[kPa]**
**β**
**CPT: 202070U502** MMM [θy =0.15m]
MMSE [θy =0.13m]
RM [θy =0.34m]
RSE [θy =0.43m]
(c)
80
90
100
110
120
130
140
3.0 3.5 4.0 4.5
**q****d****[kPa]**
**β**
**CPT: 202070U504** MMM [θy =0.27m]
MMSE [θy =0.22m]
RM [θy =0.32m]
RSE [θy =0.4m]
(d)
80
90
100
110
120
130
140
3.0 3.5 4.0 4.5
**q****d****[kPa]**
**β**
**CPT: 202070U505** MMM [θy =0.47m]MMSE [θy =0.39m]

RM [θy =0.38m]
RSE [θy =0.47m]
(e)
80
90
100
110
120
130
140
3.0 3.5 4.0 4.5
**q****d****[kPa]**
**β**
**CPT: 202070U506**
MMM [θy =0.4m]
MMSE [θy =0.25m]
RM [θy =0.41m]
RSE [θy =0.52m]
(f)
234
235
236
237
3.0 3.5 4.0 4.5
**q****d****[kPa]**
**β**
**CPT: 202080U501** MMM [θy =0.13m]
MMSE [θy =0.2m]
RM [θy =0.11m]
RSE [θy =0.14m]
(g)
(a) (b)

In all the studied cases the probability distribution is acceptable to the lognormal fit. According to the preceding assumption the design value of the bearing capacity is calculated as follows:

= exp − (8)

where the μ*lnq f* _{is the mean value of the underlying normal distribution in relations to the }

lognormal distribution of bearing capacity and σ*lnq f*_{is the standard deviation of the underlying }

normal distribution (Fenton & Griffiths 2008; Pieczyńska-Kozłowska et al. 2015).

Figures 3a-g show the relationships between the bearing capacity design values and the β
index for the seven CPTUs under study. Figure 3a shows that the highest value of the design BC
has been achieved for the minimum value of the scale of fluctuation estimated through the
*MMSE. Accordingly, the maximum scale of fluctuation value (from RSE) gave the minimum qd*.

In Figure 3b, one more time the smallest value of the scale of fluctuation gave the

maximum bearing capacity value. However, in case of the maximum _{θ}* the results are not similar *

*to the prevised one. The MMSE case gave smaller qd* value than the MMM although the scale of

MMM was higher. In Figure 3c and 3d the same trend can be observed but with the small values of the scales. Although the MMSE case has the smallest scale value the highest bearing capacity design value corresponds to e MMM case.

Figure 3e is the hardest to explain. The scales of fluctuation are almost the same for all
CPTUs. The mean values of the BC are again close each other. However, when Markov
correlation function model is adopted higher design values of BC are drawn. Results shown in
Figures 3e and 3g could be interpreted as affected by the choice of the autocorrelation function
model. If the differences among the _{θ} values are large the BC design values decrease with the
scale of fluctuation increasing. But when the difference among the _{θ} values are narrow the
autocorrelation function models affect the results: in this latter case, the Markov correlation
function gives higher design values than the Squared exponential.

**CONCLUSIONS **

In this paper, seven continuous readings of tip resistances from CPTUs performed in Italian alluvial deposits have been studied in terms of random field variability structure. Graphical and analytical methods have been employed to estimate the scale of fluctuations and two

autocorrelation function models have been adopted to carry out shallow footing bearing capacity
through RFEM numerical method. Results show that the variations in _{θ} values affected the
mean values of qd: as θ increases qd decreases. The methods used to estimate θ values not

always predict similar values. A larger database of scale of fluctuations is needed to investigate
and characterize the bias of the _{θ} values induced by the choices of the best fitting models for the
sample autocorrelation functions and the methods used to measure the scale of fluctuations.

**REFERENCES **

Baecher, G. B., and Christian, J. T. (2003). “Reliability and Statistics in Geotechnical Engineering”. West Sussex: John Wiley & Sons.

Bagińska, I., Kawa, M., and Janecki, W. (2016). “Estimation of spatial variability of lignite mine
*dumping ground soil properties using CPTu results”. Studia Geotechnica et Mechanica, *
38(1), 3-13.

Cherubini, C., Vessia, G., and Pula, W. (2007). “Statistical soil characterization of Italian sites for reliability analyses”. In Tan, Phoon, Hight, & Leroueil (Eds), Characterisation and Engineering Properties of Natural soils (pp. 2681-2706). London: Taylor and Francis Group. ISBN: 978-0-415-42691-6.

Diamantidis, D., Holicky, M., and Sykora, M. (2016). “Risk and Reliability Acceptance Criteria
*for Civil Engineering Structures”. Proc. Int. Conf. on Structural Reliability and *

*Modelling in Mechanics, At Ostrava, Czech Republic, pp. 1-11. *

Eslami, A., and Fellenius, B.H. (2004). “CPT and CPTu data for soil profile interpretation: review of methods and a proposed new approach”. Iranian Journal of Science & Technology, Transaction B, 28(B1), 69-86.

*Fenton, GA., and Griffiths, D.V. (2008). Risk assessment in geotechnical engineering. Hoboken, *
N.J: John Wiley & Sons.

Fenton, G.A., and Vanmarcke, E. (1990). Simulation of random fields via local average subdivision. ASCE J Geotech Eng, 116(8):1733–49.

Griffiths, D., and Fenton, G. (1993). Seepage beneath water retaining structures founded on
*spatially random soil. Geotechnique , 43 (6), 577-587. *

Lloret-Cabot, M., Fenton G.A., and Hicks, M.A. (2014). “On the estimation of scale of fluctuation in geostatistics”. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 8(2), 129-140. doi: 10.1080/17499518.2013.871189.

Pieczyńska-Kozłowska, J., Puła, W., Griffiths, D., and Fenton, G. (2015). “Influence of
embedment, self-weight and anisotropy on bearing capacity reliability using the random
*finite element method”. Computers and Geotechnics, 67, 229-238. *

Pieczyńska-Kozłowska, J.M. (2015). “Comparison between two methods for estimating the
*vertical scale of fluctuation for modeling random geotechnical problems”. Studia *

*Geotechnica et Mechanica, 37(4), 95-103. *

Puła, W. (2004). “Zastosowanie teorii niezawodności konstrukcji do oceny bezpieczeństwa
*fundamentów”. Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław. *

*Rackwitz, R. (2000). “The basic of code-making and reliability verification”. J. Struct. Saf., 22 *
(2/21), 27-60.

*Rice, S. (1944). “Mathematical Analysis of Random Noise”. Bell System Technical Journal, *
23(3), 282-332.