Aleatory Uncertainty

In geotechnical engineering, natural variability can be used interchangeably with inherent or spatial variability, stemming from a combination of various geologic, environmental and physio-chemical processes (e.g. Lumb 1966; Phoon and Kulhawy 1999a). Various geological processes that take place with time will lead to changes of the geotechnical properties (Bjerrum 1967). As a result of the glacio-tectonic disturbance, for example, Pliocene deposits exhibit an extremely pronounced spatial variability (Jamiolkowski 2014). Natural variability is routinely discussed in standard texts (see Chapter 10 in Look

  • 2014) and design guidelines (e.g. DNV 2017; JCSS 2006). For the simplest example of a test profile varying with depth within a single soil layer, it is customary to separate this profile into a trend function and a fluctuating component (Phoon and Kulhawy 1999a). In the literature, this fluctuating component or natural variability is modelled, whenever possible, as a statistically homogeneous random field (Vanmarcke 1983). All soil/rock properties in situ will vary vertically and horizontally, most often with more pronounced variability in the vertical direction. While a random field model is easy to describe probabilistically, it is noteworthy that a full statistical characterization of this model in 3D based on actual sparse and incomplete site investigation data is very difficult and a practical solution only emerged recently (Ching et al. 2020a,
  • 2021). The characterization of site stratigraphy is another major missing feature of random field studies until quite recently (e.g. Wang et al. 2013, 2016, 2017, 2018, 2019a, b; Ching et al. 2015; Li et al. 2016; Qi et al. 2016; Cao et al. 2019; Shuku et al. 2020; Zhao et al. 2020). Phoon et al. (2021) proposed three challenges in data-driven site characterization.

As suggested in Annex D of ISO 2394 (ISO 2015), the characterization of natural variability is the most important element to develop risk-informed decision making in the form of RBD (e.g. Nadim 2017; Phoon and Ching 2019; Phoon et al. 2019). Notwithstanding the evolving research on this issue, it is useful and practical to present statistics that are (1) based on actual databases and (2) aligned with existing design practice. The key features are as follows: (1) COV of a geotechnical parameter found in Phoon and Kulhawy (1999a, b), who undertook a comprehensive compilation of soil data, does not take a unique value; (2) multivariate nature of geotechnical data can be exploited to reduce the COV; and (3) spatial variability affects the limit state beyond the reduction in COV arising from Vanmarcke’s spatial averaging (Phoon et al. 2016a). An extensive characterization of COV for specific sites can be found elsewhere (e.g. Soulie et al. 1990; Rehfeldt et al. 1992; Cherubini et al. 2007; Chiasson and Wang 2007; Jaksa 2007; Uzielli et al. 2007; Stuedlein 2011; Sruedlein et al. 2012a). Tables 2.1, 2.2 and 2.3 summarize typical natural variability of soil strength properties, index parameters and field measurements, respectively. Table 2.4 summarizes the statistics of index, strength and deformation properties of igneous, sedimentary and metamorphic rocks that are taken from Aladejare and Wang

(2017). The number of tests is presented as an indicator of the degree of statistical uncertainty associated with the mean and the COV. Based on the compiled soil data, approximate guidelines for inherent soil variability were proposed in Table 2.5 by Phoon and Kulhawy (1999a). Similar guidelines are available for properties of rocks (e.g. Phoon et al. 2016a; Aladejare and

Table 2.1 Summary of the natural variability of soil strength parameters

Property

Soil type

n

N

Property mean

Property COV (%)

Range

Mean

Range

Mean

Range

Mean

Fine

grained

38

2-S38

101

6-412

100

6-56

33

Clay, silt

13

14-82

33

15-363

276

1 1-49

22

Clay

10

12-86

47

130-713

405

18-42

32

Clay

42

24-124

48

8-638

112

6-80

32

Sand

7

29-136

62

35-41

37.6

5-1 1

9

Clay, silt

12

5-51

16

9-33

15.3

10-50

21

Clay, silt

9

17-41

33.3

4-12

9

Clay, silt

4

  • 0.24-
  • 0.69

0.51

6-46

20

Clay, silt

3

0.62

6-46

23

Sand

13

6-1 1 1

45

  • 0.65-
  • 0.92

0.74

5-14

9

(Source: Table I in Phoon and Kulhawy 1999a)

Note: su = undrained shear strength, ф' = effective friction angle, UC = unconfined compression test, UU = unconsolidated-undrained triaxial compression test, CIUC = consolidated-isotropic-undrained compression test.TC = triaxial compression test, DS = direct shear test, n = no. of data groups and N = no. of tests per group. For the two greyed rows, laboratory test type was not reported.

Table 2.2 Summary of the natural variability of soil index parameters

Index

Soil type0

n

N

Index mean

Index COV (%)

Range

Mean

Range

Mean

Range

Mean

Fine

grained

40

17-439

252

13-105

29

7-46

18

Fine

grained

38

15-299

129

27-89

51

7-39

18

Fine

grained

23

32-299

201

14-27

22

6-34

16

Fine

grained

33

15-299

120

12-44

25

9-57

29

Clay, silt

2

32-118

75

0.094

60-88

74

Fine

grained

6

5-3200

564

14-20

17.5

3-20

9

Fine

grained

8

4-315

122

13-18

15.7

2-13

7

Sand

5

30-70

50

1 1-36

19

Sand

5

30-70

50

49-74

61

(Source: Table 2 in Phoon and Kulhawy 1999a)

Note: wn = natural water content, wL = liquid limit, wp = plastic limit, PI = plasticity index, LI = liquidity index, у = total unit weight, yd = dry unit weight and Dr = relative density.‘Fine-grained materials derived from a variety of geologic origins (e.g. glacial deposits, tropical soils and loess). bTotal variability for the direct method of determination. ‘Total variability for the indirect determination using SPT values.

Wang 2017) and cemenr-mixed soils (e.g. Liu et al. 2015; Pan et al. 2018,

2019). They are more empirically grounded than the indicative standard deviations of some soil strength and stiffness properties noted in Table 2.6, which are given in Section 3.7 - Soil Properties of the JCSS Probabilistic Model Code (JCSS 2006). Phoon et al. (2016a) made the following observations:

The highest COV values seem to be associated with measurements of soil stiffness.

With respect to field measurements, the COV values of natural variability for sand are higher than that for clay.

The COV values of natural variability for index parameters are the lowest, with the possible exception of the relative density and liquidity index.

Intuitively, the more information an engineer has that can provide different insights into the properties of the geomaterials and the behaviour of geotechnical structures, the more reliable judgement he or she can make, such as the selection of design parameters and calculation methods. Ching et al. (2010) utilized a Bayesian analysis to reduce the uncertainties in the estimation of undrained shear strength by incorporating multivariate

Table 2.3 Summary of the natural variability of field measurements

Test

type

Property

Soil type

n

N

Property value

Property COV (%)

Range

Mean

Range

Mean

Range

Mean

CPT

Sand

57

10-2039

1 15

0.4-29.2

4.1

10-81

38

CPT

Silty clay

12

30-53

43

0.5-2.1

1.59

5-40

27

CPT

Clay

9

0.4-2.6

1.32

2-17

8

VST

Clay

31

4-31

16

6-375

105

4-44

24

SPT

Sand

22

2-300

123

7-74

35

19-62

54

SPT

Clay,

loam

2

2-61

32

7-63

32

37-57

44

DMT

Sand to clayey sand

15

12-25

17

64-1335

512

20-53

33

DMT

Clay

13

10-20

17

119-455

358

12-32

20

DMT

Sand to clayey sand

15

12-25

17

346-2435

1337

13-59

37

DMT

Clay

13

10-20

17

502-876

690

12-38

20

DMT

Sand to clayey sand

15

10-25

15

9.4-46.1

25.4

9-92

50

DMT

Sand, silt

16

10.4-53.4

21.6

7-67

36

DMT

Sand to clayey sand

15

10-25

15

0.8-8.4

2.85

16-130

■ 53

DMT

Sand, silt

16

2.1-5.4

3.89

8-48

30

DMT

Sand to clayey sand

15

10-25

15

1.9-28.3

15.1

20-99

44

DMT

Sand, silt

16

1.3-9.3

4.1

17-67

38

PMT

Sand

4

17

1617-3566

2284

23-50

40

PMT

Cohesive

5

10-25

428-2779

1084

10-32

15

PMT

Sand

4

5.2-15.6

8.97

28-68

42

(Source: Table 3 in Phoon and Kulhawy 1999a)

Note: CPT = cone penetration test,VST = van shear test, SPT = standard penetration test, DMT = dilatometer test, PMT = pressuremeter test, qc = CPT tip resistance, qt = corrected CPT tip resistance, NSPT = SPT blow count (number of blows per 305 mm), A and В = DMT A and В readings, ED = DMT modulus, lD = DMT material index, KD = DMT horizontal stress index, pL = PMT limit stress and EPMT = PMT modulus.

Rock type

Property

No. of data groups

No. of tests per group

Range of data

Property mean value

Property COV (%)

Type

Parameter

Range

Mean

Range

Mean

Range

Mean

Igneous

Index

12

9-40

13

1.60-3.06

2.49-2.75

2.64

0.7-5.6

2.7

6

5-172

38

1.45-3.04

2.65-2.88

2.73

0.7-1.7

1.2

2

5-8

7

55.10-97.12

60.50-96.88

78.69

0.4-10.7

5.6

13

5-517

84

10.90-30.30

21.60-28.34

23.39

0.9-13.7

4.9

3

4-1 1

8

0.09-4.00

0.45-2.76

1.34

29.9-93.1

61.5

19

9-172

26

0-42.50

0.20-16.56

4.82

35.7-107.1

69.7

16

5-517

62

17.00-76.00

  • 32.19
  • -75.75

46.28

0.8-48.9

16.8

4

9-108

38

1 1.00-88.00

39.20-68.50

59.65

3.1-67.1

24.2

Rock mass

2

57-70

64

37.00-77.00

52.1-53.0

52.55

17.5

2

6-57

32

30.00-100.00

65.00-97.03

81.02

2.9

Strength

42

5-164

55

0.75-380.02

  • 24.78-
  • 329.70

103.90

8.9-103.9

38.7

16

5-517

60

0.26-34.14

4.39-26.05

10.70

20.7-87.5

36.0

14

5-517

101

0.15-17.40

2.49-12.53

7.26

14.1-98.1

40.6

3

7-29

7

0-176.00

5

7-29

7

0-66.80

2

3

8.00-35.00

13.00-32.00

18.83

22.9

Deformation

25

5 -108

40

0.50-97.38

4.10-71.90

38.40

4.0-91.2

30.7

18

5-172

20

0.04-0.44

0.17-0.33

0.27

9.4-23.5

14.1

Sedimentary

Index

27

6-120

28

1.35-3.61

1.73-3.00

2.38

0.4-13.0

5.6

10

10-49

17

1.79-2.95

2.63 - 2.71

2.67

0.4-3.4

1.4

4

19-49

35

1.30-99.00

71.10-92.30

85.17

5.4-84.4

38.4

21

2-778

92

8.80-31.60

19.38-27.20

24.07

0.1-12.7

6.2

7

1 1-121

28

0.02-16.00

0.37-7.00

3.13

48.0-188.0

97.0

47

7-262

31

0-67.88

0.26-39.94

10.68

1.0-181.6

53.4

29

6-510

76

9.00-76.00

15.00-70.00

40.26

0.8-33.7

18.7

12

6-44

18

4.20-96.00

12.87-60.16

46.89

5.4-39.2

20.3

Rock mass

4

17-120

41

15.00-89.00

  • 22.60-
  • 60.40

46.13

17.1- 27.0

21.5

5

15-120

49

4.50-99.30

57.40-80.80

70.00

1 1.2-52.1

29.0

Strength

73

6-470

50

0.63-345

4.40-264.00

62.80

0.4-109.6

42.8

26

3-77

25

0.06-76.60

1.20-17.00

7.90

1.6-59.3

31.5

37

6-1305

140

0.05-14.60

0.23-16.21

3.52

2.9-91.7

40.9

9

13-58

30

0-96.00

2.57-31.82

21.23

15.7-79.0

42.8

12

9-58

25

7.00-66.00

24.93-58.31

41.71

3.9-30.6

14.1

5

8-58

25

2.00-41.00

4.00-21.00

17.94

14.2-27.5

20.9

Deformation

50

6-121

30

0.06-196.26

0.59-73.17

23.7

7.0-128.1

43.0

13

2-62

16

0.03-0.45

0.17-0.39

0.24

4.0-75.4

25.6

(Continued)

Rock type

Property

No. of data groups

No. of tests per group

Range of data

Property mean value

Property COV (%)

Type

Parameter

Range Mean

Range

Mean

Range

Mean

Metamorphic

Index

9

9-24

16

2.58-3.1 1

2.58-2.82

2.70

0.3-2.5

1.2

3

10-25

16

2.18-2.92

2.66-2.72

2.71

1.7-3.0

2.4

2

10-1 1

1 1

95.87-99.33

98.12-98.44

98.28

1.0-1.1

1.0

4

5-92

43

24.71-28.15

25.50-26.56

26.10

1.5-3.1

1.8

4

9-13

1 1

0-6.60

0.05-3.85

1.39

  • 49.8-
  • 130.4

1 18.4

1 1

2-32

16

0.06-22.40

0.32-1.88

0.78

12.5- 1 13.6

61.8

9

2-92

16

31.66-63.00

37.47-56.50

51.63

2.7-16.3

10.0

3

2-9

7

46.00-82.00

57.83-64.00

61.35

9.0-39.8

20.8

Strength

23

2-151

25

4.62-320.00

32.81 — 150.00

81.45

7.9-70.4

43.1

10

2-151

53

0.55-19.00

3.18 -12.10

10.54

9.2-59.0

31.5

14

2-92

25

0.52-13.30

2.79-6.56

4.61

6.9-88.0

51.2

3

0-76.00

3

13-20

1 1

15.00-60.60

40.87

3.4

1

3.00-33.00

3.00-29.00

Deformation

9

2-41

8

1.00-88.40

1 1.57-38.50

24.33

22.6- 82.1

43.9

3

4-37

14

0.02-0.40

0.24

86.1

(Source: data taken from Tables 2-9 in Aladejare and Wang 2017)

Note: p = bulk density, Gs = specific gravity, ld2 = slake durability index, у = unit weight, wn = water content, n = porosity R = Schmidt hammer hardness (RL = L-type Schmidt hammer hardness), S„ = Shore scleroscope hardness, GSI = geological strength index, RQD = rock quality designation, oc = uniaxial compressive strength, ctm = Brazilian tensile strength, ls = point load strength index (ls50 = ls for diameter SO mm),c = cohesion, ф = friction angle, mi = Hoek-Brown constant, E = Young’s modulus, and о = Poisson ratio.

Test type

Property

Soil type

Mean

COV (%)

Lab strength

Clay

10-400 kPa

20-55

Clay

10-350 kPa

10-30

Clay

150-700 kPa

20-40

Clay and sand

20°-40°

5-15

Lab index

Clay and silt

13%-100%

8-30

Clay and silt

30%-90%

6-30

Clay and silt

15%-25%

6-30

Clay and silt

10%-40%

_a

Clay and silt

10%

_a

Clay and silt

13-20 kN/m3

< 10

Sand

30%-70%

10-40; 50-70b

CPT

Clay

0.5-2.5 MPa

< 20

Clay

0.5-2.0 MPa

20-40

Sand

0.5-30 MPa

20-60

VST

Clay

5-400 kPa

10-40

SPT

Clay and sand

10-70

25-50

DMT

Clay

100-450 kPa

10-35

Sand

60-1300 kPa

20-50

Clay

500-880 kPa

10-35

Sand

350-2400 kPa

20-50

Sand

1-8

20-60

Sand

2-30

20-60

Sand

10-50 MPa

15-65

PMT

Clay

400-2,800 kPa

10-35

Sand

1,600- 3,500 kPa

20-50

Sand

5-15 MPa

15-65

(Source: Table 7 in Phoon and Kulhawy 1999a)

Note: aCOV = (3%-l2%)/mean. bThe first range corresponds to the total variability of the direct method of determination, and the second range is the total variability of the indirect determination by SPT values.

information. Ching and Phoon (2015) used a multivariate database CLAY/7/6310 to estimate the mobilized undrained shear strength from a variety of undrained shear strengths obtained using different test procedures (e.g. direct shear, unconfined compression, field vane, consolidated undrained compression and extension and isotropically consolidated undrained compression). Muller et al. (2014) extended the method of Ching et al.

(2010) to reduce the total uncertainty of spatially averaged values of undrained shear strength. This extended multivariate approach was then utilized by Muller et al. (2016) to assess the representative average values and reduce the associated uncertainties of undrained shear strength for staging the construction of embankments on soft clay.

Compared to the mean and COV value of a single quantity, the characterization of correlation can be more challenging. Correlation exists between different soil/rock parameters, such as between the undrained shear strength

Table 2.6 Indicative standard deviations of soil properties (Source Table 3.7.4.2 in JCSS 2006)

Soil property

Standard deviation (% of the expected value)

Unit weight (kN/m3)

5%-l 0%

Effective friction angle (°)

10%-20%

Drained cohesion (kPa)

10%-50%

Undrained shearing strength (kPa)

10%-40%

Stiffness (MPa)

20%—100%

and the OCR. This is called cross-correlation. It is most commonly reported in association with transformation models (Kulhawy and Mayne 1990). Correlation also exists between the same parameter measured at different spatial locations. This is called spatial correlation. It is commonly reported as an autocorrelation function parameterized by a value called the scale of fluctuation. The scale of fluctuation describes the distance over which the

Table 2.1 Summary of the range of the scale of fluctuation (SOF) values for natural soils

Soil type

Horizontal SOF (m)

Vertical SOF (m)

N

Min

Max

Average

N

Min

Max

Average

Alluvial

9

1.07

49

14.2

13

0.07

l.l

0.36

Ankara clay

4

1

6.2

3.63

Chicago clay

2

0.79

1.25

0.91

Clay

9

0.14

163.8

31.9

16

0.05

3.62

1.29

Mixture of clay, sand and silt

13

1.2

1,000

201.5

28

0.06

21

1.58

Hangzhou clay

2

40.4

45.4

42.9

4

0.49

0.77

0.63

Marine clay

8

8.37

66

30.9

9

0.1 1

6.1

1.55

Marine sand

1

15

15

15

5

0.07

7.2

1.43

Offshore soil

1

24.6

66.5

45.6

2

0.48

1.62

1.04

Overconsolidated

clay

1

0.14

0.14

0.14

2

0.063

0.255

0.15

Sand

9

1.69

80

24.5

14

0.1

4

1.17

Sensitive clay

2

l.l

2

1.55

Silt

3

12.7

45.5

33.2

5

0.14

7.19

2.08

Silty clay

7

9.65

45.4

29.8

14

0.095

6.47

1.40

Soft clay

3

22.2

80

47.6

8

0.14

6.2

1.70

Undrained engineered soil

22

0.3

2.7

1.42

Water content

9

2.8

22.2

12.9

8

0.05

6.2

1.70

(Source: data taken from Table 8 in Cami et al. 2020) Note: N = number of studies collated.

References

Test (result)

Mean

COV

SOF (m)

Vertical

Horizontal

Honjo and Kuroda (1991)

Unconfined compressive test (UCS)

0.6-8.0 MPa

0.21-0.36 (clay) 0.32-0.4 (sand)

0.8-8.0

Babasaki et al. (1 996)

UCS

0.22-0.27

Hedman and Kuokkanen (2003)

Penetrometer test (cu)

0.38-1.12

0.07-0.33

Navin and Filz (2005)

UCS

1.0-4.7 MPa

0.34-0.79

«24.0

Larsson et al. (2005)

Penetrometer test (cu)

< 0.60

Radial: < 0.1 3

Orthogonal: < 0.32

Larsson and Nilsson (2009)

CPT (tip resistance)

0.20-0.60

1.8-3.6

Chen et al. (201 1)

UCS (Marina Bay Financial Centre)

2.0-2.7 MPa

0.29-0.46

UCS (Nicoll Highway MRT Station)

3.2-4.5 MPa

0.29

Al-Naqshabandy et al. (2012)

CPT (tip resistance)

0.22-0.67

0.2-0.7

2.0-3.0

Namikawa and Koseki (2013)

UCS

1.7 MPa

0.20-0.40

Bruce et al. (201 3)

UCS

0.7-2.1 MPa

0.34-0.79

Chen et al. (2016)

Binder concentration

29%

0.19

Bergman et al. (201 3)

CPT (tip resistance)

0.08-0.77

< 3.5

Liu et al. (2019)

Centrifuge test (Binder concentration)

1.7-2.1 MPa

0.42-0.44

1.0-3.33

Small scale SOF: Intracolumn:

Radial: 0.I2D-0.28D Circumferential: 67°-l33° Intercolumn:

0.1 2D-0.28D Large scale SOF: 25

(Source: data taken from Table 4 in Pan et al. 2018 and Table 9 in Cami et al. 2020)

Data source: Liu et al. (201S) and Pan et al. (2018,2019).

Note: Small-scale SOF is the fluctuation due to insufficient mixing or positioning error, while large-scale SOF is the fluctuation due to natural variation of water content.

soil/rock parameters are similar or correlated and therefore, it is also known as correlation length. Recently, Cami et al. (2020) collated a comprehensive database of the horizontal and vertical scale of fluctuation values. The results are summarized in Table 2.7 for natural soils and Table 2.8 for cement-mixed soils to provide engineers with a sense of the probable range of the scale of fluctuation values. For natural soils, the horizontal scale of fluctuation ranges from nearly 0 to 100 m with the most probable range of 0 to 60 m, while the vertical scale of fluctuation ranges from nearly 0 to 9 m with the most probable range of 0 to 5 m.

Typically, soil-structure interaction occurs over a finite volume of soil that is called the influence zone. For example, this can be identified from the shearing surface around a pile tip observed by Vesic (1977). The properties of soil mass beyond this zone have an insignificant effect on soil-structure interaction behaviour. It is the mobilized values in this zone that one should consider in design. In EN 1997-1:2004, Clause 2.4.5.2 - Characteristic values of geotechnical parameters - described the influence zone with the following two application rules:

  • 1. The influence zone governing the behaviour of a geotechnical structure at a limit state is usually much larger than a test sample or in an in situ test. The characteristic value of the governing parameter is often the mean of a range of values covering a large surface or volume of the ground.
  • 2. The influence zone at this limit state may depend on the behaviour of the supported structure. For instance, when considering the ultimate limit state of a building on several footings, the governing parameter should be the mean strength over each individual zone of ground beneath a footing if the building is unable to resist a local failure. If, however, the building is stiff and strong enough, the governing parameter should be the mean of these mean values over the entire zone or part of the zone of ground under the building.

In the consideration of the natural or spatial variability of a soil mass, the problem becomes more complicated because the formation of the influence zone is complex in spatially heterogeneous soils. It can depart significantly from the classical symmetrical failure mechanisms developed for homogeneous or simple linearly varying soils. It is advantageous in practice to convert the governing parameter (spatially variable) into a homogeneous spatial average (e.g. Vanmarcke 1977,1983). This is similar to the concept of homogenization (e.g. Arwade and Deodatis 2011; Arwade et al. 2016). Several studies on the characterization of a spatially variable soil mass using the spatial average have been implemented for shear strength (e.g. Ching et al. 2016a), Young’s modulus (e.g. Griffiths et al. 2012; Paiboon et al. 2013; Ching et al. 2017b), bearing capacity of a footing (e.g. Soubra et al. 2010; Honjo and Otake 2013), settlement of a footing (e.g. Fenton and Griffiths 2005) and active lateral force (e.g. Hu and Ching2015).Tabarroki et al. (2021) proposed a mobilization-based characteristic value that accounts for spatial variability, failure mechanism, and their interactions correctly.

 
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