draft report of a joint tc205/tc304 discussion group on...

1TC205/TC304 Excel‐FORM‐RBD‐EC7

Draft report of a joint TC205/TC304 discussion group on “Discussion of statistical/reliability methods for Eurocodes”

Discussion topic EXCEL‐based direct reliability analysis and its potential role to complement Eurocodes

Lead discusser Bak Kong Low ([email protected])

Members Gregory Baecher; Richard Bathurst; Malcolm Bolton; Zuyu Chen; Jianye Ching; Peter Day; Mike Duncan; Herbert Einstein; Robert Gilbert; Vaughan Griffiths; Suzanne Lacasse; Lars Olsson; Trevor Orr; Kok‐Kwang Phoon; Brian Simpson; Yu Wang; Tien H. Wu; Nico De Koker; Karim Karam; Philip Boothroyd; Sónia H. Marques.

Objectives & scope

The similarities and differences between the design points of the first‐order reliability method (FORM) and the Eurocode 7 (EC 7) design approach are studied. Eleven geotechnical examples of reliability analysis and reliability‐based design (RBD) are discussed with respect to parametric correlations, sensitivity info from RBD, ultimate limit states and serviceability limit states, system reliability, spatially autocorrelated properties, and characteristic values and partial factors. Focus is on insights from RBD and how RBD can complement EC7 design approach in some situations, but limitations of RBD will also be mentioned.

Contents Page Session 1: Excel‐based FORM reliability approach; Design points and sensitivity information ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 2

Session 2: RBD of semi‐gravity retaining wall and anchored sheet pile wall; Comparisons with EC7 DA1b designs ‐‐‐ 6

Session 3: RBD of spread footing; RBD of laterally loaded piles for ULS and SLS ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 11

Session 4: An underwater slope failure; Spatial autocorrelations; System slope reliability ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 15

Session 5: Hong Kong 2‐D rock slope; 3‐D rock slope; Roof wedge in rock tunnel ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 19

Session 6: Excel‐based Subset Simulation and its role in EC7 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 23

Session 7: Probabilistic models for geotechnical data ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 26

Conclusions ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 34

References ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 35

Remarks Sessions 1, 2, 3, 4 and 5 by B K Low.Session 6 by Yu WangSession 7 by KK Phoon, J Ching & Y Wang

Draft: 20160907

Session 1: Introduction: practical Excel-based FORM reliability analysis.

• Excel‐Solver automatic constrained optimization for implementing first‐order reliability method (FORM)

• Intuitive perspective of expanding dispersion ellipsoid (or equivalent ellipsoid for non‐Gaussian distributions) in the original space of random variables.

• Excel‐Solver can in principle deal with more than hundred constraints and hundred changing cells

• The Excel VBA programming platform adds versatility, enabling FORM analysis involving implicit and iterative numerical model

• MATLAB and its optimization toolbox can also be used, but with bigger learning curve and less transparency to practitioners than Excel.

• Some Excel files for hands‐on reliability analysis are available for download at http://alum.mit.edu/www/bklow

References:Low, B.K. and Wilson H. Tang (2004). "Reliability analysis using object‐oriented constrained optimization." Structural Safety, Elsevier

Science Ltd., Amsterdam, Vol. 26, No. 1, 69‐89.Low, B.K. (2005). "Reliability‐based design applied to retaining walls." Geotechnique, Vol. 55, No. 1, 63‐75. Low, B.K. and Wilson H. Tang (2007). "Efficient spreadsheet algorithm for first‐order reliability method." Journal of Engineering

Mechanics, ASCE, Vol. 133, No. 12, 1378‐1387.Low, B.K. (2015). Chapter 9: Reliability‐based design: Practical procedures, geotechnical examples, and insights, (pages 355‐393) of the

book Risk and Reliability in Geotechnical Engineering, CRC Press, Taylor & Francis group, 624 pages, edited by Kok‐Kwang Phoon, Jianye Ching.

2TC205/TC304 Excel‐FORM‐RBD‐EC7: Session 1

Design points of FORM and EC7 compared

Fig. 1.1 Illustration of the reliability index β in the plane when c’ and φ’ are negatively correlated. This perspective is also valid for non‐normal distributions, when viewed as “equivalent ellipsoids”.

3

Fig. 1.2 Characteristic values, partial factors, design point, and design approaches (DA) in Eurocode 7. (Low and Phoon, 2015)

Coh

esio

n, c

'

Friction angle, φ ' (degrees)

UNSAFE

SAFE

μc'

μφ′

βσφσφ one-sigma dispersion ellipse

σc'

Mean-valuepointDesign

point

β-ellipse

Limit state surface: boundary between safeand unsafedomain

rR

β = R/r General concepts of ultimate limit state design in Eurocode 7:

Diminished resistance (ck / γc, tanφk / γφ) > Amplified loadings

Characteristic values

Partial factors

Based on characteristic values and partial factors for loading parameters.

“Conservative”, for example, 10 percentile for strength parameters, 90 percentile for loading parameters

(if “=“, then “design point”)

The three sets of partial factors (on resistance, actions, and material properties) are not necessarily all applied at the same time.

In EC7, there are three possible design approaches:

● Design Approach 1 (DA1): (a) factoring actions only; (b) factoring materials only.

● Design Approach 2 (DA2): factoring actions and resistances (but not materials).

● Design Approach 3 (DA3): factoring structural actions only (geotechnical actions from the soil are unfactored) and materials.

• The design point in FORM reflects parametric uncertainties, sensitivities, and correlations, and is obtained without relying on partial factors.

• EC7 design point values, obtained by applying partial factors to characteristics values, are in general different from FORM design point values.

• The FORM reliability index β affords an estimation of the probability of failure. Design can aim at higher target β if consequence of failure is high.

tanφ′ instead of φ′ can be used in the figure above without affecting the bulleted statements below.

TC205/TC304 Excel‐FORM‐RBD‐EC7: Session 1

4

Sensitivity info in the design point of FORM reliability analysis

Ref.: Low, B.K. and Wilson H. Tang (2007). "Efficient spreadsheet algorithm for first‐order reliability method." Journal of Engineering Mechanics, ASCE, Vol. 133, No. 12, 1378‐1387. (See short extracts in the attached PDF file)

• Hypothetical statistical inputs using various non-Gaussian distributions to test Excel FORM and to discuss insights.

• The mean value of spring stiffness k3 at point 3 is 10, and that of rotational restraint λ1 at point 1 is 500. Design point values from reliability analysis are indicated under the column labelled “xi*”.

• Reliability analysis reveals that the design values of λ1 and k3hardly deviate from their mean values. This means the buckling load Pcrt is insensitive to these two parameters at their mean values of 500 and 10.

• In this case, the strut becomes a full sine curve, i.e. arching upwards on one side of point 3, and downwards on the other side, when k3 = 4. Higher stiffness of k3 serves no purpose once the full sine wave is formed.

• This case may suggest characteristic values of λ1 and k3 equal to their mean values, and partial factors for both equal to 1.0, for the case in hand with inputs as shown.

• For the case in hand, sensitivity of k3 increases at lower values (k3 < 4) when the strut has not gone into full sine wave yet.

• Reliability-based design of P, E and/or I can be done for target β index of 2.5 or 3.0.

Fig. 1.3 Excel‐Solver reliability analysis of a strut with complex supports. Performance function is implicit.


Distributions Para1 Para2 Para3 Para4 xi* units ni

Lognormal P 700 140 973.95 N 1.7667Triangular L 800 1000 1200 1109.50 mm 1.2681Lognormal a 500 50 539.42 mm 0.8107BetaDist E 3 3 150000 250000 174306 N/mm2 -1.3027

Lognormal I 200 20 174.50 mm4 -1.3173PERTDist λ1 350 500 650 499.99 Nmm/rad -0.0002Gamma k3 100 0.1 9.97 N/mm -0.0002

β Φ(−β ) Pcrt. PerFunc2.6513 0.40% 973.95 0.000

Correlation Matrix1 0 0 0 0 0 0

0 1 0.7 0 0 0 0

0 0.7 1 0 0 0 0

0 0 0 1 0.5 0 0

0 0 0 0.5 1 0 0

0 0 0 0 0 1 0.6

0 0 0 0 0 0.6 1

M1 = λ1θ1

P P

R1

R3 = k3v3

R2

3 2v3

a

L

E, I

Monte Carlo:Pf = 0.36%

(250,000 trials)

1

5

Rupture strength Qu

Applied load Q

Spring stiffness k3

z = 0

Stiff claycu = 150 kPa

z = 23 m

water

y = 60 mm

y = 1 mPH = 421 kN

steel pipe pile, d=1.3m

e = 26 m

seabed

water

Soil

Seabed

23 m

Spring stiffness k

However, similar (but stiffer) spring and/or rotational restraint, if applied to the simple system on the left, or near the cantilever pile head of the laterally loaded pile on the right, would be important and sensitive parameters to the serviceability limit states of vertical displacement (left) and pile head deflection (right) and the ultimate limit states of spring rupture (left) and bending failure (right).

• Hence direct FORM reliability analysis and reliability‐based design (RBD) are preferred. Partial factors back‐calculated from FORM will NOT be pursued in our discussion group, except when discussing the limitations of partial factors.

• Partial factors of spring stiffness k back‐calculated from reliability analysis are of different values within the same problem and across different problems.

Fig. 1.4 Spring suspending a vertical load.Low, B.K. (2010). "Slope reliability analysis: some insights and guidance for practitioners", Proceedings of the 17th Southeast Asian Geotechnical Conference, Taipei, Taiwan, May 10‐13, 2010, Vol. 2, p231‐234.(PDF file available at http://alum.mit.edu/www/bklow)

Fig. 1.5 A cantilever steel tubular pile in a pile group of a breasting dolphin.

Limitations of partial factors back-calculated from reliability analysis


Session 2: Reliability-based design of retaining walls• It is assumed that rotating mode is one of the ULS to be checked.

Tanφ′, tanδ and ca are normally distributed, with mean values (μ) and standard deviations (σ) as shown, and with correlation coefficient 0.8 between tanφ′ and tanδ.

• x* are the design point values obtained in FORM reliability analysis. The column labelled n shows the values of (x*−μ)/σ.

• For the statistical inputs shown, RBD obtains a design base width b= 1.83 m for a target β index of about 3.0, corresponding to a probability of rotation failure Pf ≈ Φ(‐β) = 0.122%,.

• For comparison, Monte Carlo simulation with 200,000 realizations using @RISK (www.Palisade.com) yields a Pf of 0.120%.

• The n value of 0.0 for ca means the design value of ca (under the x* column) stays put at its mean value, because rotating mode is not affected by ca at all. Reliability analysis reveals input sensitivities.

• Reliability analysis with respect to the ultimate limit states of sliding and bearing capacity failure can be done, and system reliability for multiple limit states can be evaluated readily (for example using the Low et al. 2011 system reliability procedure).

• Shown in the next page is EC7 design for the base width b with respect to the overturning ULS, via characteristic values and partial factors, starting from the same statistical inputs of mean values and standard deviations, but without considering correlations in EC7.

6TC205/TC304 Excel‐FORM‐RBD‐EC7: Session 2

Low, B.K. (2005). "Reliability‐based design applied to retaining walls." Geotechnique, Vol. 55, No. 1, 63‐75. Low, B.K. (2015). Chapter 9: Reliability‐based design: Practical procedures, geotechnical examples, and insights, (pages 355‐393) of the book Risk and Reliability in Geotechnical Engineering, CRC Press,

Taylor & Francis group, 624 pages, edited by Kok‐Kwang Phoon, Jianye Ching.

H γw all λ α γsoil a b6 24 10 90 18 0.4 1.83

β Pf PerFn1 PerFn2

3.031 0.122% 0.0000 56.71

Rotating mode

Sliding mode

µ σx* mean StDev n

tanφ ' 0.492 0.7 0.07 -2.977

tanδ 0.262 0.36 0.036 -2.725ca 100 100 15 0.000

Correlation matrixtanφ ' 1 0.8 0

tanδ 0.8 1 0

ca 0 0 1tanφ ' tanδ ca

σμ−

=*x

Fig. 2.1

TC205/TC304 Excel‐FORM‐RBD‐EC7: Session 2 7

EC7 DA1b design of base width b for rotation ULS

• Even though partial factors are specified, EC7 does not produce a unique design, but depends on how conservative the characteristic values are determined. This is not objectionable, for it allows flexibility in design to match the consequence offailure, in the same way that target reliability index can be higher or lower depending on the consequence of failure. Analogous situation exists for LRFD’s nominal values and load and resistance factors.

• For a design width b obtained via EC7, the value of the corresponding reliability index β is not unique, but depends on whether parametric correlations (if any) are modelled. To compare with the target β of 3.0 in RBD (previous page), correlations should be modelled.

• EC7 DA1a requires characteristic values of resistance and actions, on which partial factors are applied. If characteristic values are based on percentiles, one needs to know the probability distributions of actions and resistance in order to estimate the upper tail (e.g. 70 percentile) characteristic value of actions and lower tail characteristic values of resistance (e.g. 30 percentile). Whether based on 5%/95% or 30%/70%, DA1a is satisfied; DA1b governs. Are there other ways of estimating characteristic values of actions and resistance for EC7 DA1a?

Frank R, Bauduin C, Driscoll R, Kavvadas M, Krebs Ovesen N, Orr T, and Schuppener , B. (2005). Designers’ guide to EN 1997‐1 Eurocode 7: geotechnical design – general rules. Thomas Telford.

0.7 0.07

0.36 0.036100 15

tanφ '

tanδ ca

Normalµ σ

xk Partial F x* xk Partial F x*tanφ ' 0.5849 1.25 0.4679 tanφ ' 0.6633 1.25 0.5307

tanδ 0.3008 1.25 0.2406 tanδ 0.3411 1.25 0.2729

ca 75.33 1.4 53.80 ca 92.14 1.4 65.81

EC7 DA1b, 5/95 percentiles for xk

xk = Characteristic valuex* = Design valueComputed EC7design b = 1.90 mfor which β = 3.47 if φ‐δcorrelation is modelled, and β = 4.36 if not.

EC7 DA1b, 30/70percentiles for xk

Computed EC7 design b = 1.755 mfor which β = 2.53 if φ‐δcorrelation is modelled, and β = 3.18 if not.

Fig. 2.2

Correlation matrix Rx* μ σ n γ γsat qs tanφ' tanδ z

Normal γ 16.13 17 0.9 -0.971 γ 1 0.5 0 0.5 0 0

Normal γsat 17.32 19 1 -1.677 γsat 0.5 1 0 0.5 0 0

Normal qs 10.28 10 2 0.140 qs 0 0 1 0 0 0

Normal tanφ' 0.663 0.78 0.05 -2.336 tanφ' 0.5 0.5 0 1 0.8 0

Normal tanδ 0.310 0.35 0.023 -1.760 tanδ 0 0 0 0.8 1 0

Normal z 2.921 2.4 0.3 1.736 z 0 0 0 0 0 1

Ka Kah Kp Kph H d* PerFn0.261 0.2492 5.897 5.6333 12.31 2.989 0.00

Forces Lever arm Moments β P f

(kN/m) (m) (kN-m/m) 3.01 0.13%

-31.54 4.655 -146.81-82.31 2.767 -227.73

-152 7.855 -1194.1

-32.75 8.840 -289.47

189.34 9.81 1858.11

1

2

3

4

5

Boxed cells contain equations

Water table

T

Surcharge qs

A1.5 m

6.4 m

z

d5

1

2

3

4

γ, φ', δ

γsatDredge level

d* = H − 6.4 − z*

σμ−

=*x

8

RBD of sheet pile total length H via FORM

References:Low, B.K. (2005). "Reliability‐based design applied to retaining walls." Geotechnique, Vol. 55, No. 1, 63‐75. Low, B.K. (2015). Chapter 9: Reliability‐based design: Practical procedures, geotechnical examples, and insights,

(pages 355‐393) of the book Risk and Reliability in Geotechnical Engineering, CRC Press, Taylor & Francis group, 624 pages, edited by Kok‐Kwang Phoon, Jianye Ching.


• Free‐earth support method. Ka based on Coulomb formula; Kpbased on Kerisel‐Absi.

• For the statistical inputs shown, RBD for a target β =3.0 results in design H (= 6.4 + z* + d*) of 12.31 m, and Pf ≈ Φ(‐β) = 0.13%. For comparison, Monte Carlo simulation with 200,000 realizations gives Pf =0.14%

• For the statistical inputs shown, tanφ′ and z are sensitive inputs, based on n values. The n values of tanδ and γare due largely to correlations with tanφ′, revealed if uncorrelated analysis is done.

• The design value of γ, 16.13, is lower than its mean value of 17, an apparent paradox which can be understood due to the logical positive correlation of γ to γsat and tanφ′ which both have design values below their respective mean values. If all six parameters are uncorrelated, the design value γ* will be bigger than mean γ.

• Soil on either side is assumed to be same source, hence one γsatwith action‐resistance duality. Reliability analysis yields γsat*= 17.32, which is less than the mean γsat of 19.

• Mean embedment depth d = 12.31 – 6.4 – 2.4 = 3.51 m. Design embedment depth d* = 2.99 m, i.e., “overdig” = 0.52m, which is determined automatically as a by‐product of RBD.

• Reliability analysis with respect to other ultimate limit states and system reliability analysis can be done.

• One can back‐calculate partial factors for the six material and geometric inputs, and also load/action and resistance factors of LRFD and EC7 DA1a, based on presumed nominal/ characteristic values of loads/actions and resistance. Not pursued here due to limitations of such back‐calculated partial factors.

Fig. 2.3

9

• EC7 has an “unforseen overdig” allowance for z, to account for the uncertainty of the dredge level. The design value of z is obtained from μz + 0.5 m = 2.4 + 0.5 = 2.9 m, where 0.5 m is the “overdig”.

• Although EC7 partial factor of soil unit weight is specified to be 1.0, conservative characteristic values of γ and γsat still need to be estimated, and if same source, cannot increase unit weight on the active side while decrease unit weight on the passive side.

• Based on 5/95 percentiles for characteristic values, γ* > γsat*, which violates reality.

• With characteristic values at 30/70 percentiles and EC7 partial factors from DA1b, one obtains a design H of 12.87 m, closer to the RBD design H of 12.31 m for a target β of 3.0. A less critical design H of 12.64 m is obtained if one wrongly set characteristic value of γsat at 19.52 instead of 18.48 kN/m3.

• FORM reliability analysis based on the H from EC7 design will give different β index depending on whether correlations are modeled (correlation matrix, previous page) or not.


EC7 DA1b design of sheet pile total length H

Water table

T

Surcharge qs

A1.5 m

6.4 m

z

d5

1

2

3

4

γ, φ', δ

γsatDredge level

μ σ17 0.9

19 1

10 2

0.78 0.05

0.35 0.023

2.4 0.3

γγsat

qs

tanφ'tanδ

z

Normal xk Partial F x*γ 18.48 1.00 18.48

γsat 17.36 1.00 17.36

qs 13.29 1.30 17.28tanφ' 0.698 1.25 0.558tanδ 0.312 1.25 0.250

z NA NA 2.90


xk = Characteristic valuex* = Design valueComputed EC7 design H = 13.95 mNote: Unrealistic that γ* > γsat*

Fig. 2.4

xk Partial F x*γ 17.47 1.00 17.47

γsat 18.48 1.00 18.48

qs 11.05 1.30 14.36tanφ' 0.754 1.25 0.603tanδ 0.338 1.25 0.270

z NA NA 2.90


Computed EC7 design H = 12.87 m.Computed EC7 design H = 12.64 m if γsat,kwrongly set at 70 percentile value (19.52) instead of the 30 percentile value (18.48).

10

Reliability‐based design (RBD) can provide additional insights to EC7 design or LRFD design when the statistical information (mean values, standard deviations, correlations, probability distributions) of the key parameters affecting design are known and one or more of the following circumstances apply:

• When partial factors have yet to be proposed by EC7 to cover uncertainties of less common parameters, for example in situ stress coefficient K in underground excavations in rocks, dip direction and dip angles of rock discontinuity planes, and the smear effect of vertical drains installed in clay.

• When output has different sensitivity to an input parameter depending on the engineering problems (e.g. shear strength parameters in bearing capacity, retaining walls, slope stability).

• When input parameters are by their physical nature either positively or negatively correlated. One may note that EC7 does not specify different characteristic values and partial factors when some parameters are correlated. The same design is obtained in EC7 with or without modeling of correlations among parameters.

• When spatially autocorrelated soil properties need to be modeled.

• When there is a target reliability index or probability of failure for the design in hand. The target reliability index or probability of failure can be set explicitly higher or lower in RBD depending on the importance of the structure or consequence of failure.

• When uncertainty in unit weight of soil needs to be modelled.


How RBD can potentially complement EC7 design

One should not attach too precise a meaning to the results of RBD, because the output depends on input data, the mechanical model, the failure modes considered, etc. For example, in a RBD for target reliability index of 3.0, the resulting design is not to be regarded as having exactly a probability of failure

equal to Φ(‐β)=0.135%, but as a design aiming at sufficiently small probability of failure (<1%, say), in the same spirit that a EC7 design viacharacteristic values and partial factors and different ULS and SLS also aims at sufficiently safe design by implicit considerations of parametric uncertainties and sensitivities. In this regard, the statistical data and correlations are laid bare in RBD, and uncertainties, though imprecise, are explicitly accounted for.

Low, B.K. and Phoon, K.K. (2015). “Reliability‐based design and its complementary role to Eurocode 7 design approach”. Computers and Geotechnics, Elsevier, Vol. 65, 30–44.


• Similar to Fig. 3 of Low and Phoon (2015), except that Qh and Qv are uncorrelated, and the uncertainty in tanφ′ is modelled instead of φ′, in line with EC7 which applies partial factor to tanφ′. The mean values follow those in Tomlinson (2001, page 94)’s Example 2.2.

• For the statistical inputs shown, a base width B of 4.55 m is required to achieve a target reliability index of β = 3.0 against bearing capacity failure.

• The design value of c’, 15.03 kPa, is slightly above the c’ mean value of 15 kPa, due to negative correlation coefficient of ‐0.5 between c’ and tanφ′.

• For the case in hand, the design is much more sensitive to Qh than to Qv, with n values 2.49 versus −0.71, and much more sensitive to tanφ′ than c’, with n values ‐1.37 versus 0.11, where n = (x*‐μN)/σN, in which superscript N denotes equivalent normal mean and equivalent normal standard deviation of lognormal distributions.

Session 3: RBD of base width B of a retaining wall against bearing capacity failure mode

μ σDistr.Name x* mean StDev n Correlation matrix RLognormal c′ 15.03 15 3 0.11 1 -0.5 0 0

Lognormal tanφ′ 0.41 0.47 0.047 -1.37 -0.5 1 0 0

Lognormal Qh/m 429.82 300 45 2.49 0 0 1 0

Lognormal Qv/m 1019.7 1100 110 -0.71 0 0 0 1

B L D γ eB B' eL L' q qu(x*)4.55 25 1.8 21 1.054 2.442 0 25 417.5 417.5

ca po Ω

12.0224 37.8 22.856 Nq 7.967 sq sc sγ iq ic iγNc 17.09 1.021 1.024 0.987 0.578 0.518 0.3235

PerFunc g(x) β Nγ 7.313 dq dc dγ0.00 3.00 1.125 1.158 1

Qv (kN/m)

Qh (kN/m)

D = 1.8 m

c', φ ', γB

2.5 m

γγγγγ idsNBidsNpidscNq qqqqoccccu 2′

++=

(Continue next page)

Low, B.K. and Phoon, K.K. (2015). "Reliability‐based design and its complementary role to Eurocode 7 design approach". Computers and Geotechnics, Elsevier, Vol. 65, 30‐44.

Fig. 3.1


• The mean values of c’, tanφ′, Qh and Qv are 15 kPa, 0.47, 300 kN/m and 1100 kN/m, respectively, based on deterministic Example 2.2 in Tomlinson (2001, 7th ed., Foundation Design and Construction, p94), which computed an Fs of 3.0 against general shear failure of the base of the wall when base width B is 5 m.

• The RBD in the previous page assumes that the coefficients of variation of c’, tanφ′, Qh and Qv are 0.2, 0.1, 0.15 and 0.1, respectively. It is also assumed that c’ and tanφ′ are negatively correlated (as shown in the correlation matrix), but Qh and Qv are uncorrelated (as befitting the horizontal earth thrust and the applied vertical load).

• When Qh = 0, the vertical load Qv is an unfavorable action without ambiguity. However, when Qh is acting and of comparable magnitude to Qv, the latter possesses action‐resistance duality, because load inclination and eccentricity decreases with increasing Qv. RBD automatically takes this action‐resistance (or unfavorable‐favorable) duality into account in locating the design point. Interestingly, RBD reveals that the design value of Qv (1019.7 kN/m) is about 7.3% lower than its mean value of 1100 kN/m, thereby revealing the action‐resistance duality of Qv when Qh is acting.

• Model uncertainty: The bearing capacity equation is approximate even for idealized conditions. Also, several expressions for Nγ exist. The Nγ used here is attributed to Vesic in Bowles (1996). The nine factors sj, dj and ij account for the shape and depth effects of foundation and the inclination effect of the applied load. The formulas for these factors are based on Tables 4.5a and 4.5b of Bowles (1996), which may differ from those in EC7.

• RBD can be done for system reliability with multiple failure modes of ULS and/or SLS, as illustrated for a laterally loaded pile next.

RBD of base width B of a retaining wall against bearing capacity failure mode (continued)


0

5

10

15

20

25

-0.05 0.00 0.05 0.10

Deflection y (m)

0

5

10

15

20

25

-20000 0 20000

Moment M (kNm)

0

5

10

15

20

25

-500 0 500

Soil reaction (kN/m)

z = 0

Stiff claycu = 150 kPa

z = 23 m

water

y = 60 mm

y = 1 mPH = 421 kN

steel pipe pile, d=1.3m

e = 26 mseabed seabed

seabed

-1500

-1000

-500

0

500

1000

1500

-0.3 -0.1 0.1 0.3

p (k

N/m

)

y (m)

γ' = 11.8 kN/m3

cu = 150 kN/m2

J = 0.25B = 1.3 mε = 0.01

z = 22 m

z = 0 m

z = 0 m

z = 22 m

12 m

1.35

1.4

1.45

1.5

30 32 34 36 38 40

Ste e l w all thick ne s s (m m )

Exte

rnal

dia

me

ter

(m)

β = 3, (bending)

β = 3, (def lection)a

b

c

• A steel tubular pile in a breasting dolphin. Soil‐pile interaction was based on the nonlinear and strain‐softening Matlock p‐y curves.

• At mean input values of PH and undrained shear strength cu, the pile deflection y is 0.06 m at seabed, and 1 m at pile head.

• For reliability analysis, the PH was assumed to be normally distributed, with mean value 421 kN and a coefficient of variation of 25%. The mean cu trend is μcu = 150 + 2z, kPa, with a coefficient of variation of 30%. Spatial autocorrelation was modelled for the cuvalues at different depths below seabed.

• The β index obtained was 1.514 with respect to yielding at the outer edge of the annular steel cross section. The sensitivities of PH and cu change with the cantilever length e. The different sensitivities from case to case is automatically revealed in reliability analysis and RBD, but will be difficult to consider in codes based on partial factors.

• A target β of 3.0 can be achieved in RBD for both ULS (bending) and SLS (assuming yLimit = 1.4 m) using steel wall thickness t = 32mm and external diameter d =1.42 m. Configurations along line ac means β = 3 for bending mode, and greater than 3 for deflection mode; configurations along line ab means β = 3 for deflection mode, and greater than 3 for bending mode.

Low, B.K. (2015). Chapter 9: Reliability‐based design: Practical procedures, geotechnical examples, and insights. Risk and Reliability in Geotechnical Engineering, CRC Press, Taylor & Francis.Low, B.K., Teh, C.I, and Wilson H. Tang (2001). "Stochastic nonlinear p‐y analysis of laterally loaded piles." Proc 8th ICOSSAR , Newport Beach, California,, 8 pages.

(c) RBD for both ULS and SLS

(a) Laterally loaded cantilever pile in soil with Matlcok p‐y curves.

(b) depth‐dependent Matlock p‐y curves.

RBD of a laterally loaded cantilever pile for ULS and SLS

Fig. 3.2


Questions and possible discussions on bearing capacity ULS (pages 11 & 12) and ULS+SLS of the laterally loaded single pile with large cantilever length (page 13)

Bearing capacity Ultimate Limit State

How would partial factor design approaches (e.g. EC7 and LRFD) deal with a parameter that possesses action‐resistance duality (i.e., unfavorable‐favorable duality), such as the vertical load Qv in the presence of horizontal load Qh?

Laterally loaded pile, Ultimate Limit State and Serviceability Limit State

• The case in hand (page 13) extends a deterministic example of Tomlinson (1994, Pile design and construction) into reliability analysis and RBD. The pile is one of a group of piles in a breasting dolphin, with 23 m embedment length below seabed and 26 m cantilever length in sea water. For serviceability limit state, it is assumed that the maximum permissible pile head deflection is 1.4 m. For both the bending ULS and the pile head deflection SLS, the design point in RBD shows decreasing sensitivity of cu with depth, i.e., decreasing (cu* ‐ μcu)/σcu with depth, where cu* are the design undrained shear strength values at various depths obtained in RBD. How would partial factor design approaches determine the characteristic (or nominal) values of the

undrained shear strength at different depths? Assuming uniform conservatism with depth in determining the characteristic cu values doe not accord well with the different sensitivities of cu with depth as revealed by RBD, and may alter the behavior of the pile at ULS and SLS.

• If I understand correctly, for SLS design in EC7, all partial factors are equal to 1.0. So only conservative characteristic values need be determined for SLS design. For ULS design (e.g. bending of pile), having obtained the conservative cu characteristic values, should one apply the partial factor for cu uniformly across the entire embedded portion of the pile despite different sensitivities revealed in RBD?


Session 4: Soil slope reliability analysis, i.e. items II(e) and II(f) of Report.Three examples are shown in the next three pages:

• Page 16: Reliability analysis of an underwater slope failure in San Francisco Bay Mud

• Page 17: Slope reliability analysis accounting for spatial autocorrelations

• Page 18: System reliability analysis involving multiple failure modes


Underwater excavated slope failure in San Francisco Bay Mud

Spencer method with search for noncircular slip surface

Fig. 4.1 Underwater excavated slope in San Francisco Bay mud, and the average su profile, described by su = c0 + by, and the ± 1 Std. Dev. lines.

Low, B.K. and Duncan, J.M., 2013. "Testing bias and parametric uncertainty in analyses of a slope failure in San Francisco Bay mud". Proceedings of Geo‐Congress 2013, ASCE, March 3‐6, San Diego, 937‐951.

( )bycsu += 0ξ

• The results of both the deterministic and the probabilistic analyses are affected by biases in the strength measurements and interpretations, and by extrapolating the measured strengths to the full depth of the deposit.

• The computed lumped factors of safety are 1.20, 1.16 and 1.00, based on field vane, trimmed 35 mm diameter and untrimmed 70 mm diameter specimens in UU triaxial tests. The FORM analyses and Monte Carlo simulations for circular slip surfaces produce probabilities of failure of about 10%, 19% and 46%, respectively, all unacceptably high.

• Input random variables are often modeled by the lognormal distribution, which avoids the negative domain. For the case in hand where the coefficient of variation is ≤ 0.21, the computed failure probabilities for correlated lognormals do not differ much from those for correlated normals.

A 75 m long section of the 600 m long trench failed during excavation in August 1970.


Low, B.K., Lacasse, S. and Nadim, F. (2007). "Slope reliability analysis accounting for spatial variation", Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, Taylor & Francis, London, Vol. 1, No. 4, 177‐189.

154.688 γ6 20.1749 19.5 0.975 19150.313 γ7 20.1754 19.5 0.975 19145.938 γ8 20.1753 19.5 0.975 19141.563 γ9 20.1747 19.5 0.975 19137.188 γ10 20.1737 19.5 0.975 19132.813 γ11 20.1721 19.5 0.975 19128.438 γ12 20.1701 19.5 0.975 19124.063 γ13 20.1676 19.5 0.975 19119.688 γ14 20.1647 19.5 0.975 19115.313 γ15 20.1614 19.5 0.975 19

-2-1.5

-1-0.5

00.5

11.5

70 110 150 190

x-Coordinates (m)Des

ign

Poin

t Ind

ex o

f γ

-20

-10

0

10

20

70 90 110 130 150 170 190

Elev

atio

n (m

)

−0.6 m

10050

δ = 1000

105

1

2

( )( )depthratexcc topuu ** , += κ

rate = 2.85 kPa/m

depth = ymid, top – ymid, base

negative α

Coefficient κ for anisotropy:

κ = 0.4 if α ≤ -45° (≈extension)

κ = 1.0 if α ≥ 45° (≈compression)

κ linearly interpolated for −45° ≤ θ ≤ 45°positive α

depth

cu, top = 0.0006x2 - 0.311x + 49.368

010203040

0 50 100 150 200 250x-coordinates

c u, t

op (k

Pa)

-20

-10

0

10

20

70 90 110 130 150 170 190

−0.6 m

Critical: Fs = 1.14

Initial

(m)

(m)

Slope reliability analysis accounting for spatially autocorrelated undrained shear strength and soil unit weight

• Reliability analysis revealed that the slope is less safe when the unit weights near the toe are lower. This implication can be verified by deterministic runs using higher γ values near the toe, with resulting higher factors of safety.

• The design point (of 24 spatially correlated cu values and 24 spatially correlated soil unit weight values) is located automatically in reliability analysis, and reflects parametric sensitivity from case to case in a way specified partial factors cannot.

• The computed reliability index and probability of failure depend on (i) the underlying analytical model, (ii) the number of parameters treated as random variables, and (iii) the statistical inputs including mean values, standard deviations, probability distributions, and correlation structure (including spatial variation and autocorrelation model where pertinent).

Spencer method with search for noncircular slip surface

Fig. 4.2 Undrained shear strength model including anisotropy of a Norwegian slope.

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The overlapping of the failure probability contents for modes 1, 3, 6 and 7, and also for modes 2, 4, 5 and 8 means that it is sufficiently accurate to calculate the bounds for the system failure probability by considering only the two stationary values of reliability index, namely β1 and β2.

System FORM reliability analysis of a soil slope with two equally likely failure modes.

Low, B.K., Zhang J. and Tang, Wilson H. (2011). “Efficient system reliability analysis illustrated for a retaining wall and a soil slope”, Computers and Geotechnics, Elsevier, Vol. 38, No. 2, 196–204.


Fig. 4.3 System reliability analysis for the two critical modes is as accurate as that for all the eight modes

β2=2.893

β1=2.795

β4 β5

β8

β6β3

β7

Clay1

Clay2

X = 0

X > 0


Session 5: Rock slopes and tunnels in rock, i.e. items II(g), II(h) and II(i) of Report.Three examples are shown in the next three pages:

• Page 20: Reliability analysis of Sau Mau Ping slope of Hong Kong;

• Page 21: Reliability analysis of 3‐D tetrahedral wedge mechanism in rock slope;

• Page 22: Tunnel with roof wedge formed by discontinuities, and rock bolt reinforced tunnel.


• The statistical inputs of the 5 random variables follow those in Hoek (2007). For zero reinforcing force T and uncorrelated parameters, the FORM reliability index is β= 1.556, and Pf ≈ 1 ‐ Φ(β) = 6%, in good agreement with the Monte Carlo Pf of 6.4% in Hoek (2007).

• With negative correlations between c and φ and between z and zw/z, as shown in the correlation matrix R, a reinforcing force T of 257 tons (per m length of slope) inclined at θ = 55° is needed to achieve a target reliability index β of 3.0. The most sensitive parameters for the case in hand are the ratio zw/z and the coefficient of horizontal earthquake acceleration α.

• The tension crack depth z and the extent to which it is filled with water (zw/z) are negatively correlated. This means that shallower crack depths tend to be water‐filled more readily (i.e., zw/z ratio will be higher) than deeper crack depths, consistent with the scenario suggested in Hoek (2007) that the water which would fill the tension crack in this Hong Kong slope would come from direct surface run‐off during heavy rains.

• For illustrative purposes, a negative correlation coefficient of ‐0.5 is assumed between z and zw/z. Even though there are no data to quantify this correlation between z and zw/z, it is still useful to explore possible correlations to get a feel for its influence on the reliability index. This is a sensible approach commonly applied in engineering practice for important but not well characterized parameters.

x* Para1 Para2 Para3 Para4 n Correlation matrix RNormal c 8.106 10 2 -0.947 1 -0.5 0 0 0Normal φ 29.653 35 5 -1.069 -0.5 1 0 0 0Normal z 13.831 14 3 -0.056 0 0 1 -0.5 0

Tr_Exp zw /z 0.801 0.5 0 1 1.428 0 0 -0.5 1 0

Tr_Exp α 0.133 0.08 0 0.16 1.527 0 0 0 0 1

H ψf ψp γ γw T θ g(x) β P60 50 35.0 2.6 1 257 55 0.000 3.00

Hoek, E. (2007). Practical rock engineering. http://www.rocscience.com/education/hoeks_corner, Chapter 8 on Factor of safety and probability of failure.Low, B.K. (2007). "Reliability analysis of rock slopes involving correlated nonnormals", International Journal of Rock Mechanics and Mining Sciences, Elsevier Ltd., Vol. 44, Issue 6, 922‐935.Low, B.K. and Phoon, K.K. (2015). “Reliability‐based design and its complementary role to Eurocode 7 design approach”. Computers and Geotechnics, Elsevier, Vol. 65, 30–44.

θ

Tψfψp

WαW

U

V

Tension crack

Failure surface

Assumed water pressure

ZwZ

H

Units: meter, tonne, tonne/m2, tonne/m3.

T

Reliability‐based design of Sau Mau Ping slope of Hong Kong

Fig. 5.1


Low, B.K. (2007). "Reliability analysis of rock slopes involving correlated nonnormals", International Journal of Rock Mechanics and Mining Sciences, Elsevier Ltd., 44(6), 922‐935.

α is the slope face inclination, Ω the upper ground inclination.Slope face dip direction coincides with upper ground dip direction.H and h are related by equation; either can be input.

Tetrahedral wedge

Triangle BDE is horizontal.Lines TS and XR, are horizontal.

β1 and β2 are related to strike directions; δ1 and δ2 are dip angles.

F

b Gs

b Gs

bch

bchs

w w= −⎛

⎝⎜⎜

⎞

⎠⎟⎟ × + −

⎛

⎝⎜⎜

⎞

⎠⎟⎟ × + +a a1

1 11 2

2 22 1

12

23 3γ γ

φ φγ γ

tan tan

in which dimensionless coefficients a1, a2, b1 and b2 are functions of β1, δ1, β2, δ2, α and Ω, and sγ is the specific gravity of rock (γ/γw), c1 and c2 cohesions, Gw1 = 3u1/γwh, Gw2= 3u2/γwh, in which u1 and u2 are average water pressures on joint planes 1 and 2.

D

B

ET Xβ1 β2δ1 δ2

α

Ω

RS

h

H

B′

O

Sliding on both planes, one of the three modes.

α Ω h sγ

70 0 16 2.6

x* µN σN nx Correlation matrixBetaDist β1 4 4 53 71 61.21 62.00 3.263 -0.243 1 0 0 0 0 0 0

BetaDist δ1 4 4 44 56 50.44 50.00 2.179 0.204 0 1 0 0 0 0 0

BetaDist β2 4 4 11 29 19.26 20.00 3.266 -0.226 0 0 1 0 0 0 0

BetaDist δ2 4 4 42 54 48.35 48.00 2.183 0.158 0 0 0 1 0 0 0

BetaDist Gw 4 4 0.14 0.86 0.61 0.51 0.122 0.841 0 0 0 0 1 0 0

BetaDist tanφ 4 4 0.25 1.15 0.59 0.70 0.157 -0.649 0 0 0 0 0 1 0

BetaDist c/γh 4 4 0.04 0.16 0.07 0.10 0.017 -1.548 0 0 0 0 0 0 1

g(x) β Pf

0.00 1.924 0.0272

BetaDist parametersmin max

If tanφ and c/γh neagtivelycorrelated, with ρ67 = ρ76 = -0.5, get reliability index β = 2.214

Reliability analysis of 3‐D wedge in rock slope with uncertain discontinuity orientations

• The uncertainties of discontinuity orientations (β1, δ1, β2, δ2), shear strength of joints (tanφ and c/γh), and water pressure in joints (dimensionless parameter Gw) are modelled by the versatile beta general distributions which can assume non‐symmetrical bounded pdf. Here the beta‐distribution parameters used are (4, 4, mean − 3*StDev, mean + 3*StDev). If desired, other values of lower and upper bounds and mode ≠ mean can be modelled.

• The reliability analysis here assumes the means and standard deviations of tanφ, c/γh and Gw on joint plane 1 are identical to those on joint plane 2. These assumptions are for simplicity, not compulsory.

• Reliability analysis yielded β = 1.924 against sliding on both planes, β = 1.389 against sliding on plane 1, and β > 5 for other modes.

• Although the governing failure mode at mean values is sliding on both planes, the reliability index β against sliding on plane 1 is⎯in the presence of uncertainty in discontinuity orientations (β1, δ1, β2, δ2)⎯ more critical than that against sliding on both planes. This information would not be revealed in a deterministic analysis, or in a reliability analysis that considers only one failure mode.

Fig. 5.2


ααH0 H0S S

N N

W

R

O

h

K0p

p

xθ

(c) A tale of two factors of safety, and reconciliation via reliability analysis

Reliability analysis of tunnels in rocks

• One can design the rockbolt spacings Sz and Sθ and the rockboltlength so as to achieve a target reliability index, for example β = 3.0, corresponding to a probability of failure of about 0.1%. Examples of reliability‐based designs are given in the paper.

• In its current version, EC7 covers little on the characteristic values and partial factors of rock engineering parameters like orientations of discontinuities, in situ stresses, and properties of joints and rock material. RBD is a more flexible approach in dealing with case‐specific uncertainties of input values and can potentially complement EC7 (and LRFD)

(a) Roof wedge in tunnel

(b) Tunnel with rock bolts

Fig. 5.3Low, B.K. and Einstein, H.H. (2013). “Reliability analysis of roof wedges and rockbolt forces in tunnels”. Tunnelling and Underground Space Technology, 38, 1–10.

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Excel-based Subset Simulation • Subset simulation (Au and Beck 2001) is an advanced Monte Carlo

Simulation (MCS) that aims to improve MCS’s computational efficiency, particularly at probability tails, while maintaining its robustness.

• Subset Simulation stems from the idea that a small failure probability can be expressed as a product of larger conditional failure probabilities for some intermediate failure events, thereby converting a rare event (small probability levels) simulation problem into a sequence of more frequent ones. Subset Simulation is performed level by level. The first level is direct MCS, and the subsequent levels utilize Markov chain Monte Carlo to generate conditional samples of interest. Details on Subset simulation are referred to Au and Beck (2001) and Au and Wang (2014).

• An Excel VBA Add-in called “Uncertainty Propagation using Subset Simulation” (UPSS) has been developed and can be obtain from https://sites.google.com/site/upssvba (Au et al. 2010, Au and Wang 2014).

• UPSS divides the reliability analysis or design into three separate processes: (1) deterministic modeling, (2) uncertainty modeling, and (3) uncertainty propagation by Subset simulation. The deterministic modeling is deliberately decoupled from uncertainty modeling and propagation. This allows three separate processes mentioned above to proceed in a parallel fashion. The uncertainty modeling and propagation are performed in a non-intrusive manner, and the robustness of MCS is well maintained. This removes the mathematical hurdles for engineering practitioners when performing reliability analyses or designs.

Au SK and Beck JL (2001). Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 16(4), 263–77. Au SK and Wang Y (2014). Engineering Risk Assessment with Subset Simulation, Wiley, ISBN 978-1118398043, 300p. Au SK Cao Z and Wang Y (2010). Implementing advanced Monte Carlo Simulation under spreadsheet environment. Structural Safety, 32(5), 281-292.

https://sites.google.com/site/upssvba

Session 6:

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TC205/TC304 Excel-FORM-RBD-EC7: Session 6

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Wang Y Cao Z and Au SK (2010). Efficient Monte Carlo Simulation of parameter sensitivity in probabilistic slope stability analysis. Computers and Geotechnics, 37(7-8), 1015-1022. Wang Y Cao Z and Au SK (2011). Practical reliability analysis of slope stability by advanced Monte Carlo Simulations in spreadsheet. Canadian Geotechnical Journal, 48(1), 162-172. Wang Y and Cao Z (2013). Expanded reliability-based design of piles in spatially variable soil using efficient Monte Carlo simulations. Soils and Foundations, 53(6), 820–834. Wang Y and Cao Z (2015). Practical reliability analysis and design by Monte Carlo Simulations in spreadsheet. An invited chapter (Chapter 7) in Risk and Reliability in Geotechnical

Engineering, 301-335, edited by K.K. Phoon and J. Ching, CRC Press.

Subset Simulation Application Example Excel-based Subset simulation has been used in reliability analysis of slope stability (Wang et al. 2010&2011, Wang and Cao 2015) and reliability-based design of foundation (Wang and Cao 2013&2015). A slope stability example is illustrated in this page. Details of the example are referred to Wang et al. (2011).

• Subset simulation significantly improves computational efficiency and resolution, particularly at small probability levels.

• When the spatial variability of soil properties is considered, the critical slip surface varies spatially. Using only one given critical slip surface significantly underestimates failure probability, and it is unconservative. Thus, when the soil property spatial variability is considered, the spatial variability of the critical slip surface should be properly accounted for.

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Potential merits of MCS/Subset Simulation in EC7

Wang Y Schweckendiek T Gong W Zhao T and Phoon KK (2016). Direct probability-based design methods. An invited chapter (Chapter 7) in Reliability of Geotechnical Structures in ISO2394, 193-228, edited by K.K. Phoon and J. V. Retief, CRC Press.

EC7 adopts design formats similar to the traditional allowable stress design (ASD) methods. The factor of safety in ASD methods is replaced by a combination of partial factors in EC7, which are provided after some code calibration processes. Because design engineers are not involved in the calibration processes, many assumptions and simplifications adopted in the calibration processes are frequently unknown to the design engineers. This situation can lead to potential misuse of the partial factors that are only valid for the assumptions and simplifications adopted in the calibration processes. Design engineers may feel uncomfortable to accept these “black box” calibration processes blindly. In addition, design engineers have little flexibility in changing any of these assumptions/simplifications or making their own judgment because recalibrations are necessary when any assumption or simplification is changed. MCS/Subset Simulation has potential merits in the aforementioned aspects. Because MCS/Subset Simulation can be treated as repeated computer (Excel) executions of the traditional ASD calculations, good geotechnical sense and sound engineering judgment that have been accumulated over many years of ASD practice are well maintained during the development of the deterministic (ASD) model for MCS/Subset simulation. The MCS/Subset simulation – based design can be conceptualized as a systematic sensitivity study which is common in geotechnical practice and familiar to design engineers. MCS/Subset simulation is particularly beneficial in the following situations (Wang et al. 2016):

• When the target failure probability needed in the design is different from the target failure probability pre-specified in EC7. • When the exact value of failure probability is needed in engineering applications, such as quantitative risk assessment and

risk based decision making. • When the load and resistance are correlated. For example, the load and resistance for earth retaining structures and slopes are

usually originated from the same sources (e.g., effective stress of soil) and correlated with each other. It is therefore difficult to decide whether the effective stress of soil or earth pressure should be regarded as a load or resistance.

• When dealing with geometric uncertainties, such as orientation of joints in rock engineering. The geometric uncertainties cannot be easily considered by conventional partial factors.

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Probabilistic models for geotechnical data

KK Phoon, J Ching & Y Wang

Introduction

This section provides simple guidelines on how to fit geotechnical data (soil parameters and model factors) to practical probabilistic models.

Geotechnical parameters

1. Single random variable – normal 2. Single random variable – Johnson (lognormal is a special case) 3. Random vector – normal 4. Random vector – Johnson 5. EXCEL Add-in for Bayesian Equivalent Sample Toolkit (BEST) 6. Statistical guidelines

Calculation models

7. Model factors

Geotechnical data can come in the form of 1 column of numbers in an EXCEL spreadsheet, say cone tip resistance with depth. Model factors for one response (defined as the ratio of measured to calculated response) also appear as one column of numbers. The number of rows occupied by the numbers is called the “sample size”. We call this column “univariate data”.

Univariate data can be fitted by a single random variable. This random variable is fully defined once we identify the distribution type (e.g., normal, lognormal) and we evaluate the parameters of the distribution (e.g. mean, standard deviation).

Geotechnical data can come in the form of several columns of numbers in an EXCEL spreadsheet, say cone tip resistance, sleeve friction, and pore pressure with depth. Each row refers to several measurements taken at the same location and at the same depth. Model factors for multiple responses (say deflection and bending moment of a cantilever wall) will appear as two column of numbers. We call this group of columns “multivariate data”.

Multivariate data can be fitted by a random vector. This random vector is fully defined once we identify the joint distribution type (the most common is multivariate normal) and we evaluate the parameters of the distribution (e.g. mean, standard deviation, and correlation matrix). For an EXCEL spreadsheet containing the cone tip resistance, sleeve friction, and pore pressure, the required joint distribution is a 3-dimensional generalization of the more widely known probability density functions in basic texts. This generalization is simple in concept but difficult to actualize in practice, because the columns are related to each other in interesting ways (called a “dependency” structure).

More advanced aspects are not covered to keep this guideline accessible to the general reader. Nonetheless, one should be mindful of the following:

1. Are my data “random”? Data must be “homogeneous” or “stationary”. In simple terms, this means that the data must appear “random” without exhibiting an obvious trend with depth or other auxiliary variables.

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A practical approach is to remove the trend when it exists. There are more subtle kind of non-randomness beyond presence of an average trend. There are formal hypothesis tests to address this.

2. Are my data normal or lognormal?

A histogram is a rather intuitive way to present the range and frequency characteristics of the data. However, lack of fit of a theoretical distribution to a histogram does not imply that the distribution is inadequate, particularly for small sample size.

A practical approach is to compare the empirical and theoretical cumulative distribution function, rather than the histogram and probability density function. There are more formal “goodness of fit” tests to address this aspect in a robust manner.

3. Do I have sufficient data?

The mean, standard deviation, correlation matrix, or other parameters can be estimated from a finite sample size using a variety of methods. Each set of data produces a single value for the mean, standard deviation, and correlation matrix. These point estimates produced by data (called statistics) are affected by “statistical uncertainty”. Statistical uncertainty can be presented as a 95% confidence interval (95% CI). The classical interpretation is that there is a 95% chance of capturing the true value of the mean (or other statistics) within this interval. A small sample size produces a wide 95% CI and in the extreme, the statistics are not meaningful.

The key idea is that statistical uncertainty allows the engineer to answer the practical question “do I have sufficient data?” in a concrete way. This question is critical in geotechnical engineering, because our sample sizes are typically small.

There are general statistical methods to address this, but more work is needed to develop practical tools or softwares for engineers.

4. Can I combine my site investigation data to reduce uncertainties?

The reader is referred to the discussion topic on “Transformation uncertainties & multivariate soil data” for more details. The central idea can be explained using Figure 1 below:

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Figure 1. Relationship between effective stress friction angle and SPT blowcount.

In the absence of site-specific data but in the presence of data from comparable sites, the engineer may assess the effective stress friction angle (φ′) to fall between 28o and 51o based on the scatter of “×” markers. However, if site-specific SPT N-values are available and they fall in the vicinity of 25 blows, it is possible to reduce the uncertainty in φ′ because the “ ” markers fall within a more restrictive range of 36o and 46o.

How to perform “updating” in the presence of several tests? This task can be performed systematically and consistently within a powerful Bayesian framework. More work is needed to develop practical tools or softwares for engineers, but calculations are a lot simpler if the multivariate normal distribution is applicable.

The guidelines below focus on “how to use” using EXCEL, rather than the theory. The user may like to refer to Ching & Phoon (2015) for a more complete treatment that includes the advanced aspects highlighted above.

Single random variable – normal

There are good reasons to consider the normal distribution as a default distribution (Phoon 2006). It is possible to fit data to any distribution, but generalization to higher dimensions (which is necessary to cover multiple test measurements in a site investigation program) is more difficult.

The normal distribution is completely defined by the mean and standard deviation, which can be estimated by applying the EXCEL “average” and “stdev.s” functions on one column of data.

Normal data can be simulated using the mean and standard deviation as inputs (refer to Data Analysis > Random Number Generation under the Data tab in EXCEL).

In principle, the normal distribution is not suitable for geotechnical parameters taking values greater than zero (positive-valued parameters such as strength parameters).

Nonetheless, it can be used for positive-valued parameters as an approximate distribution when the COV is “small”.

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One concrete way of checking suitability of normal distribution is to check the design point produced by the First Order Reliability Method (FORM). If the design point is negative for the positive-valued parameter, then the normal distribution is not suitable.

Another rough check for strength parameters is: (1 – target reliability index × COV) > 0. For example:

target reliability index = 3 and COV = 0.1 ⇒ ok

target reliability index = 3 and COV = 0.5 ⇒ not ok

If a normal distribution is not suitable, one should try to select from the Johnson system, in which the ubiquitous lognormal distribution is one member of this system.

Single random variable – Johnson (lognormal is a special case)

Let Y be a soil parameter that cannot be fitted to a normal distribution. The Johnson system allow Y (which is non-normal) to be calculated from a standard normal random variable (X) analytically:

( )X Yn

X Y

X b Y b Ya a

⎛ ⎞− −= κ = κ⎜ ⎟⎝ ⎠

It suffices to note that the Johnson system can generate distributions with a wide range of shapes by choosing the function κ(.), followed by calibration of 4 parameters, (aX, bX, aY, bY). There are only 3 possible functions for κ(.) and they produce the SU, SB, and SL distributions. The SL distribution is produced by κ(.) = ln(.), which is the well-known lognormal distribution. There is a simple procedure to do this (Section 1.5.3, Ching & Phoon 2015). Examples are shown in Figure 2 below.

(a) SU distribution (Baseline is with aX = 1, bX = 0, aY = 1, bY = 0)

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(b) SB distribution (Baseline is with aX = 1, bX = 0, aY = 1, bY = 0)

(c) SL distribution (Baseline is with aX = 1, bX* = 0, bY = 0)

Figure 2. Examples of SU, SB, and SL probability distributions from the Johnson system.

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Random vector – normal

For concreteness, assume that you have an EXCEL spreadsheet containing 3 columns. Column A contains data for the cone tip resistance, column B contains data for the sleeve friction, and column C contains data for pore pressure. We further assume that these 3 measurements were taken at 100 points in the depth direction. Therefore, the data is contained within the block of cells from A1 to C100.

If the data are normally distributed, we can build a 3-dimensional normal random vector which consists of the following collection of random variables (Z1, Z2, Z3). The random variable Z1 is for cone tip resistance and so forth for Z2 and Z3.

It is easy to calculate the mean (μ1) and standard deviation (σ1) for Z1 using the EXCEL “average” and “stdev.s” functions on each column of data. The means and standard deviations for Z2 and Z3 are obtained in the same way.

The key difference between a random variable and a random vector is a “correlation matrix”, containing the correlation between cone tip resistance and sleeve friction (δ12), the correlation between cone tip resistance and pore pressure (δ13), and the correlation between sleeve friction and pore pressure (δ12). You can get this correlation matrix directly from the data block A1:C100 using “Data Analysis > Correlation” under the Data tab in EXCEL.

Z1 Z2 Z3 Z1 1 δ12 δ13 Z2 δ12 1 δ23 Z3 δ13 δ23 1

Once you obtained this correlation matrix, you can refer to the following sections in Ching & Phoon (2015) for simulation (Section 1.4.4) and Bayesian updating (Section 1.4.5).

Random vector – Johnson

Geotechnical data are multivariate and non-normal, including the example containing 3 columns for the cone tip resistance, sleeve friction, and pore pressure. The standard approach to handle this is to transform each column of non-normal data into a column of normal data using the Johnson system as described above.

Once you obtain 3 columns of normal data, you can use the procedure described in the preceding section for simulation and Bayesian updating. Computational details are given in Section 1.6 and 1.7 (Ching & Phoon 2015) and applications to actual soil databases are given in Ching et al. (2016).

EXCEL Add-in for Bayesian Equivalent Sample Toolkit (BEST)

Although an analytical probability distribution function (e.g., normal or Johnson) is frequently used to fit measurement data and represent a random variable, it is also possible to empirically represent a random variable using a large number of numerical samples simulated from the random variable. This is particularly beneficial when the number of measurement data (i.e., sample size) is too small to directly and properly fit the probability density function (e.g., the 95% CI is too large and the statistics are not meaningful). To deal with the issue of small sample size, Bayesian methods may be used to integrate limited measurement data in a specific site with prior knowledge (e.g., engineering experience and judgment, existing data from similar project sites) to provide updated knowledge on

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the soil parameter of interest (e.g., Wang et al 2016a). Because the updated knowledge might be complicated and difficult to express explicitly or analytically, Markov chain Monte Carlo (MCMC) simulation has been used to transform the updated knowledge into a large number of simulated samples of the soil parameter of interest, which collectively represent the soil parameter as a random variable (Wang and Cao 2013). An EXCEL add-in, called Bayesian Equivalent Sample Toolkit (BEST), has been developed for implementing the Bayesian method and MCMC simulation in a spreadsheet platform (Wang et al. 2016b). The BEST Add-in can be obtained without charge from https://sites.google.com/site/yuwangcityu/best/1. Engineering practitioners only need to provide input to the BEST Add-in, such as site-specific measurement data (e.g., several SPT N values) and typical ranges of soil parameters of interest (e.g., effective friction angle of soil) as prior knowledge. Then, the BEST Add-in may be executed to generate a large number of numerical samples of the soil parameters. Subsequently, conventional statistical analysis can be performed on these simulated samples using EXCEL’ built-in functions (e.g., “average” and “stdev.s”). The BEST Add-in can also be used for estimating soil parameters (e.g., undrained shear strength of clay) from “multivariate data” (e.g., SPT, CPT, and Liquidity index data) in a sequential manner using a Bayesian sequential updating method (Cao et al. 2016).

Statistical guidelines

Extensive statistics have been compiled in the literature. These statistics are summarized by Phoon et al. (2016) and Ching et al. (2016).

The coefficient of variation (COV) is defined as the ratio of the standard deviation to the mean. Guidelines for COV for soil and rock parameters are given in Phoon et al. (2016a). It is important to note that COV of a soil or rock parameter can small or large, depending on the site condition, the measurement method, and the transformation model. Resistance factors should be calibrated using the three-tier COV classification scheme shown in Table 1 to provide some room for the engineer to select the resistance factor that suits a particular site and other localized aspects of geotechnical practice (e.g. property estimation procedure) (Phoon et al. 2016b). A single resistance/partial factor ignores site-specific issues and it shares the same issues as the factor of safety approach where the nominal resistance has to be adjusted to handle site-specific considerations in the presence of a relatively constant factor of safety.

Table 1. Three-tier classification scheme of soil property variability for reliability calibration (Source: Table 9.7, Phoon & Kulhawy 2008)

Geotechnical parameter Property variability COV (%) Undrained shear strength Lowa 10 - 30 Mediumb 30 - 50 Highc 50 - 70 Effective stress friction angle Lowa 5 - 10 Mediumb 10 - 15 Highc 15 - 20 Horizontal stress coefficient Lowa 30 - 50 Mediumb 50 - 70 Highc 70 - 90

a - typical of good quality direct lab or field measurements b - typical of indirect correlations with good field data, except for the standard penetration test (SPT) c - typical of indirect correlations with SPT field data and with strictly empirical correlations

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Model factors

The model factor for the capacity of a foundation is commonly defined as the ratio of the measured (or interpreted) capacity (Qm) to the calculated capacity (Qc), i.e. M=Qm/Qc. The value M = 1 implies that calculated capacity matches the measured capacity, which is unlikely for all design scenarios. Intuition would lead us to think that M takes different values depending on the design scenario. This intuitive observation is supported by a large number of model factor studies (Dithinde et al. 2016). Hence, it is reasonable to represent M as a random variable. It is straightforward to apply this simple definition to other responses beyond foundation capacity. For some simplified calculation models, M can depend on input parameters (i.e., M is not random) and additional efforts are required to remove this dependency (Zhang et al. 2015). A comprehensive summary of model factor statistics is presented by Dithinde et al. (2016). Multivariate model factors are not available at present. Key references

1. Cao, Z., Wang, Y., and Li, D. (2016). “Site-specific characterization of soil properties using multiple measurements from different test procedures at different locations – A Bayesian sequential updating approach.” Engineering Geology, 211, 150-161.

2. Ching, J. & Phoon, K. K., “Constructing Multivariate Distribution for Soil Parameters”, Chapter 1, Risk and Reliability in Geotechnical Engineering, Eds. K. K. Phoon & J. Ching, CRC Press, 2015, 3-76.

3. Ching, J. Y., Li, D. Q. & Phoon, K. K., “Statistical characterization of multivariate geotechnical data”, Chapter 4, Reliability of Geotechnical Structures in ISO2394, Eds. K. K. Phoon & J. V. Retief, CRC Press/Balkema, 2016, 89-126.

4. Dithinde, M., Phoon, K. K., Ching, J. Y., Zhang, L. M. & Retief, J. V., “Statistical characterisation of model uncertainty”, Chapter 5, Reliability of Geotechnical Structures in ISO2394, Eds. K. K. Phoon & J. V. Retief, CRC Press/Balkema, 2016, 127-158.

5. Phoon K. K., “Modeling and Simulation of Stochastic Data”, GeoCongress, ASCE, Atlanta, 2006.

6. Phoon, K. K. & Kulhawy, F. H. (2008). Serviceability limit state reliability-based design. Chapter 9, Reliability-Based Design in Geotechnical Engineering: Computations and Applications, Taylor & Francis, UK, 344-383.

7. Phoon, K. K., Prakoso, W. A., Wang, Y., Ching, J. Y., “Uncertainty representation of geotechnical design parameters”, Chapter 3, Reliability of Geotechnical Structures in ISO2394, Eds. K. K. Phoon & J. V. Retief, CRC Press/Balkema, 2016a, 49-87.

8. Phoon, K. K., Retief, J. V., Ching, J., Dithinde, M., Schweckendiek, T., Wang, Y. & Zhang, L. M., “Some Observations on ISO2394:2015 Annex D (Reliability of Geotechnical Structures)”, Structural Safety, 62, 2016b, 24-33.

9. Wang, Y. and Cao, Z. (2013). “Probabilistic characterization of Young's modulus of soil using equivalent samples.” Engineering Geology, 159, 106-118.

10. Wang, Y., Cao, Z., and Li, D. (2016a). “Bayesian perspective on geotechnical variability and site characterization.” Engineering Geology, 203, 117-125.

11. Wang, Y., Akeju, O. V., and Cao, Z. (2016b). “Bayesian Equivalent Sample Toolkit (BEST): an Excel VBA program for probabilistic characterisation of geotechnical properties from limited observation data.” Georisk, DOI: 10.1080/17499518.2016.1180399.

12. Zhang, D. M., Phoon, K. K., Huang, H. W. & Hu, Q. F. (2015). Characterization of model uncertainty for cantilever deflections in undrained clay. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 141(1), 04014088.

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TC205/TC304 Excel‐FORM‐RBD‐EC734

Merits and limitations of FORM reliability analysis and RBD as illustrated in the 11 examples in the first five sessions of “EXCEL‐based direct reliability analysis and its potential role to complement Eurocodes”

Reliability‐based design (RBD) can provide additional insights to EC7 design or LRFD design when the statistical information (mean values, standard deviations, correlations, probability distributions) of the key parameters affecting design can be estimated and one or more of the following circumstances apply:

• When partial factors have yet to be proposed by EC7 to cover uncertainties of less common parameters, for example in situ stress coefficient K in underground excavations in rocks, dip directions and dip angles of rock discontinuity planes, and other miscellaneous parameters in soil or rock engineering.

• When output has different sensitivity to an input parameter depending on the engineering problems, e.g. sensitivity of shear strength parameters in bearing capacity, retaining walls, slope stability etc. may be different.

• When there is action‐resistance duality in a parameter, for example the vertical load in bearing capacity when horizontal load is also acting (Session 3).

• When input parameters are by their physical nature either positively or negatively correlated.

• When spatially autocorrelated soil properties need to be modelled.

• When there is a target reliability index or probability of failure for the design in hand. The target reliability index or probability of failure can be set explicitly higher or lower in RBD depending on the importance of the structure or consequence of failure.

• When uncertainty in unit weight of soil needs to be modelled.

The design point in RBD is the most probable failure combination of parametric values for a target reliability index, and reflects case‐specific uncertainties, sensitivities, probability distributions and parametric correlations in a way that prescribed partial factors and “conservative” characteristic values cannot.

LIMITATIONS: Statistical inputs are approximate and often involve judgment, due to insufficient data. Also, may have overlooked some factors (e.g. human factors). Hence the probability of failure based on is not exact. (Despite imprecise statistical inputs, RBD may still achieve its aim of “sufficiently safe designs”).

( )βΦ−≈1fP

Conclusions

TC205/TC304 Excel‐FORM‐RBD‐EC7 35

References for Sessions 1 to 5:Frank R, Bauduin C, Driscoll R, Kavvadas M, Krebs Ovesen N, Orr T, and Schuppener , B. (2005). Designers’ guide to EN 1997‐1 Eurocode 7: geotechnical design

– general rules. Thomas Telford.Hoek, E. (2007). Hoek E. Practical rock engineering. http://www.rocscience.com/education/hoeks_corner, Chapter 8 on Factor of safety and probability of

failure.Low, B.K. (2005). "Reliability‐based design applied to retaining walls." Geotechnique, Vol. 55, No. 1, 63‐75. Low, B.K. (2007). "Reliability analysis of rock slopes involving correlated nonnormals", International Journal of Rock Mechanics and Mining Sciences, Elsevier

Ltd., Vol. 44, Issue 6, 922‐935.Low, B.K. (2015). Chapter 9: Reliability‐based design: Practical procedures, geotechnical examples, and insights, (pages 355‐393) of the book Risk and Reliability

in Geotechnical Engineering, CRC Press, Taylor & Francis group, 624 pages, edited by Kok‐Kwang Phoon, Jianye Ching.Low, B.K. and Duncan, J.M., 2013. "Testing bias and parametric uncertainty in analyses of a slope failure in San Francisco Bay mud". Proceedings of Geo‐

Congress 2013, ASCE, March 3‐6, San Diego, 937‐951.Low, B.K. and Einstein, H.H. (2013). “Reliability analysis of roof wedges and rockbolt forces in tunnels”. Tunnelling and Underground Space Technology, 38, 1–

10.Low, B.K. and Wilson H. Tang (2004). "Reliability analysis using object‐oriented constrained optimization." Structural Safety, Elsevier Science Ltd., Amsterdam,

Vol. 26, No. 1, 69‐89.Low, B.K. and Wilson H. Tang (2007). "Efficient spreadsheet algorithm for first‐order reliability method." Journal of Engineering Mechanics, ASCE, Vol. 133, No.

12, 1378‐1387.Low, B.K. and Phoon, K.K. (2015). "Reliability‐based design and its complementary role to Eurocode 7 design approach". Computers and Geotechnics, Elsevier,

Vol. 65, 30‐44.Low, B.K., Lacasse, S. and Nadim, F. (2007). "Slope reliability analysis accounting for spatial variation", Georisk: Assessment and Management of Risk for

Engineered Systems and Geohazards, Taylor & Francis, London, Vol. 1, No. 4, 177‐189.Low, B.K., Teh, C.I, and Wilson H. Tang (2001). "Stochastic nonlinear p‐y analysis of laterally loaded piles." Proc 8th ICOSSAR , Newport Beach, California,, 8

pages. Low, B.K., Zhang J. and Tang, Wilson H. (2011). “Efficient system reliability analysis illustrated for a retaining wall and a soil slope”, Computers and Geotechnics,

Elsevier, Vol. 38, No. 2, 196–204.

Also, 8 references are listed in pages 23-25 (Session 6) by Y Wang,

and 12 references are listed in page 33 (Session 7, pages 26-33) by KK Phoon, J Ching & Y Wang

draft report of a joint tc205/tc304 discussion group on...

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