Probabilistic geotechnical engineering analysis based on first order reliability method

Limit equilibrium engineering analysis needs the incorporation of probabilistic approaches for the determination of soil strength statistical measures for deterministic and probabilistic analyses. For the commonly used Mohr–Coulomb strength model, the uncertainty and the characteristic values of cohesion (c) and angle of shearing resistance (φ) determination is not straight­­forward. This paper applies the first order reliability method (FORM) to estimate these values from the direct shear and the typical triaxial tests. The method is verified with test data. Furthermore, the FORM is applied to the planar failure limit equilibrium problem to determine the statistical measures of the safety margin (SM) and safety factor (SF). It is observed that the critical slip surface for the best estimate of the mean (SMm, SFm), for a 5% probability of exceedance (SMp=5%, SFp=5%) and for the characteristic value (SMk, SFk) do not coincide. It is interesting that the maximum probability of having a SM<0 or SF<1 does not correspond to the minimum best estimate of the SM or SF. FORM can be a very useful tool for complete probabilistic analyses. Further­more, probabilistic approaches applied to soil properties estimation can set a framework for the selection of their characteristic values for deterministic analyses.


INTRODUCTION
he limit equilibrium analysis for the design of geotechnical works requires the application of reliability theory principles for: a) the determination of soil properties in deterministic analyses and b) the uncertainty estimation of the safety margin (SM) or safety factor (SF) in probabilistic analyses. Modern codes of practice (e.g. Eurocode 7 -EC7) provide the alternative of both types of analysis, which may include one of the following methods: a) Monte Carlo, b) Point Estimate, and c) First Order Reliability. The application of First Order Reliability Method (FORM) for geotechnical engineering limit states analyses (e.g. [1,2]), a method for the estimation of error propagation, has been commonly T applied to the uncertainty estimation of the experimental results of laboratory tests (e.g. GUM: 1995, see [3]) and can be applied directly wherever a closed form of analytical solution exists. Under the framework of EC7, the reliability analysis (uncertainty calculation) for a limit equilibrium problem can be performed with respect to the safety margin (SM), which is expected to have a value of SM≥0.0 for a certain level of confidence. That requires the knowledge of the uncertainty of the parameters that affect the value of SM. These parameters normally include the value and the uncertainty of the external and internal loads (permanent and mobile) and of the strength constants, as well as the spatial variability and uncertainty of the model (see Fig.1 based on Kulhawy [4]). One of the most critical components on the overall SM uncertainty is the strength uncertainty both due to spatial variability and model uncertainty. On the other hand, a deterministic analysis requires the best estimate of the loading conditions and the materials properties, which correspond to a specific level of confidence.  Therefore, for the Mohr -Coulomb strength parameters, cohesion (c) and angle of shearing resistance (φ), which are used in limit state analysis, an estimation of their mean value and their corresponding variation (or uncertainty) is required. These measures may be calculated either by direct application of statistical methods (e.g. for the direct shear test) or by an error propagation method (e.g. FORM for the typical triaxial test). The present work explores on the application of the FORM for the statistical evaluation of the strength parameters and for the slope stability analytical solution of a wedge failure mechanism. Issues with respect to the design and characteristic strength are also discussed, as well as the capability to apply the FORM into a general limit equilibrium slope stability problem.

STATISTICAL MEASURES OF SOIL PROPERTIES
deterministic analysis requires the knowledge of the best estimates of its individual components, i.e. loading conditions and material properties, for a specific confidence level. These correspond to the characteristic values of actions (Fk) and of soil parameters (Xk) defined in EC7 (ΕΝ-1997-1). Focusing on soil parameters, for any specific parameter, X, that affects the development of the limit state condition, its characteristic value Χk is defined as a cautious estimate of the mean value, (i.e. of the best estimate) of the mean, Χ m (see ΕΝ-1997-1). The selection of this characteristic value has to be representative of the volume involved in the considered failure mechanism and it can depend on the type of the failure mechanism (e.g. local vs generalized failure).
When the sample size, n, is large enough to apply statistical methods, the characteristic value corresponds to a worse value governing the occurrence of the soil parameter with a calculated probability not greater than 5% (it is 90% confidence interval, see ΕΝ-1997-1), which for a single variable model is given by Eq.(1).
where k is the confidence level coefficient for a given probability distribution, Sd,X is the sample standard deviation and V the variation coefficient. For a specific sample with unknown standard deviation, the S d,X is the corrected standard deviation, which is related to the corrected -unbiased sample variance (s 2 ) according to Eq.(2).
A The use of the corrected variance (sX 2 =var(X), Eq. (2)) instead the uncorrected sample variance (σX 2 ), implies that there is little confidence that σ X 2 is a close estimate of the population variance, σ 2 . Had it been σ X 2 σ 2 , the sample would closely follow a normal distribution, which is not the case in geotechnical engineering investigations as sample size is usually small, despite that population may follow a normal distribution. Therefore, the direct use of the statistical methods may not be applicable, because they may lead to non-representative values for the soil mass (see [2]). In order to account for this difficulty on the statistical error, direct values (e.g. [2,5]) or semi-empirical methods (e.g. [5,6]) have been proposed for the estimation of the variation coefficient V. The V coefficient can also be used to include other types of errors, such as the error uncertainty due to ground spatial variability, the measurement error and the transformation uncertainty of the empirical equations application ( [6]). However, a good knowledge of the statistical background can provide a better understanding on the selection of the cautious estimate (see [5,7]), or even to apply Bayesian statistical methods (e.g. [2,8]), not only in cases of a small sample but also of complex uncertainties.
Concerning the characteristic values used in deterministic analysis, due to the different types of uncertainties involved (e.g. inherent soil variability, sampling disturbance, Fig.1), engineering judgement is also recommended for their selection (e.g. [9]). Such judgement should evaluate the relative importance of the following uncertainties (see also [6]):  Sampling quality (sampling type and sample condition).  The extent of the in-situ and laboratory investigation (the number and spatial distribution of samples and in-situ and laboratory tests).  The quality of the laboratory tests (accreditation and uncertainty of the laboratory measurements)  The spatial variability of the parameters and samples distribution with respect to the extent of the considered mechanism of geotechnical soil model. In addition, the following should also be taken into account: a) existing experience and data on similar soil units (including their uncertainty) and b) the failure mechanism with respect to the geotechnical profile (e.g. generalized vs local failure, short term vs long term conditions, small -large strains). For a single variable model (e.g. undrained strength, Su) and a small sample size (n) of laboratory data with unknown standard deviation of the population (the usual case for geotechnical engineering) and assuming a normally distributed population, the resulting estimated distribution follows the Student t-distribution. The estimated characteristic value (X k ), which corresponds to a probability P(Xk<μ)=1-p=1-α/2 (i.e. in a certain percentage of the cases the expected value of the true mean, μ, is greater than Xk), is then given by Eq.(3), in which SEX is the sample standard error given by Eq.(4).
where tp,n-1 is the n-1 degrees of freedom student distribution confidence parameter for the one-sided 1-p lower confidence limit of the true mean (μ). From Eqs.(1,3) we get the coefficient k=t p,n-1 /n 0.5 . Schneider [10] proposed the approximate relationship Χ k =X m -0.5S d,X , which corresponds to p=5% and n=14. As already mentioned, Eqs. (3,4) correspond to the single variable model (e.g. strength determined from the unconfined compression test, in which q u =2S u , where S u undrained shear strength). The Mohr -Coulomb failure criterion used in limit state analyses, is a two variables linear model that includes two constants (cohesion, c, and angle of shearing resistance, φ) and two variables. The for the cases of: a) the direct shear test is given by Eq.(5), in which σn is the imposed normal stress (the nonrandom or independent variable) and τ the measured -observed shear stress (the random or dependent variable) and b) the typical triaxial test is given by Eq.(6), in which σ 3 is the imposed cell pressure -radial stress (the nonrandom or independent variable) and σ1 the measured -observed vertical stress (the random or dependent variable).
τ=c+σ n tanφ= a+b·σ n (5) With regards to the experiments, if xi is the imposed value and yi is the corresponding observed value of a sample size n, the linear model is given by Eq. (7), where ε i is the error and a and b the model constants. Eq.(7) gives a best estimate of the mean of y for a given x. The measurement error ε i is the deviation of the y i from its best (deterministic) estimate y=a+x i b (Fig.2). In simple linear regression, the best estimate of the constants may be derived from minimizing the error square sum. Assuming that: a) xi is an accurate observation, b) each xi is an independent observation, c) the error εi has a constant variation for each x i and d) the uncertainties of the y i observations are equivalent (otherwise weight coefficients are required), the best mean estimators for the linear regression coefficients are given by Eqs. (8,9). Observation xi can be a predetermined imposed loading (e.g. the σn in direct shear and the σ3 in the typical triaxial), while observation yi a measured reaction (e.g. the τ in direct shear and the σ 1 in the typical triaxial).
In Eqs. (8,9) (x i , y i ) are the data measurements of the two dimensional sample, , y x are their mean values, S d,x and S d,y are the sample standard deviation of x and y measurements (Eqs. (10,11)) and rxy the Pearson sample correlation coefficient (given by Eq. (12)). The r xy is sensitive only to a linear relationship between two variables (|r xy |≤1, when r xy =1 the correlation is a perfect direct, i.e. increasing). Moreover, an unbiased estimate of the variance of y(x) with n-2 degrees of freedom is given by Eq. (13). The standard error estimators SEb and SEa of the b and a regression coefficients are given by Εqs.(14,15), respectively. Some applications in civil and geotechnical engineering of the two variables linear model have been presented by Baecher & Christian [1], Pohl [8] and Kottegoda & Rosso [11], as for instance the case of a variation with depth. A classic example is the increasing undrained shear strength with depth. The application of this model in the Mohr -Coulomb strength failure criterion has some individual characteristics that will be presented later.

STATISTICAL MEASURES FOR THE DIRECT SHEAR TEST -TWO VARIABLES MODEL
he direct shear test gives a direct determination of c and tanφ of the linear Mohr -Coulomb criterion (Eq.(5)), with σn=x and τ=y. For this occasion: a) the best estimates of the mean are cm=am and (tanφ)m=bm given by Eqs. (9,8), respectively (with x i =σ ni and y i =τ i being the data measurements) and b) the standards error estimators SE (tanφ) =SE b and SE c =SE a are given by Εqs.(14,15). Concerning the confidence intervals of the linear regression coefficients estimators, the standard method relies on the normality assumption, which is justified if either: a) the errors in the regression are normally distributed (this leads to a t-statistic) or b) the number of observations n is sufficiently large (in this case the estimator is approximately normally distributed). Applying a statistical t-test, the linear regression random variables follow a student's t-distribution with n-2 degrees of freedom where c and tanφ represent the true mean values (or population mean values). The t-test includes the assumptions that the sample is representative of a specific soil unit, the observations are independent, while the variation of x=σ n depends only on the uncertainty of the laboratory measurement. The σ n variation has a negligible influence on the total uncertainty. Ignoring the influence of sampling disturbance and spatial variability, the characteristic values of cohesion, c k , and angle of shearing resistance, (tanφ) k are given by Eqs.(16,17) and their standard errors by Eqs.(14,15), respectively. The best estimates and their standard errors can be used for probabilistic analyses, by incorporating the standard errors as quantitative measures of the corresponding uncertainties (u c =SE tanφ =S d,tanφ /n 0.5 and u tanφ =SE tanφ =S d,tanφ /n 0.5 ).
Eqs.(16,17) may be used to give the characteristic Mohr -Coulomb failure envelope: τ=ck+σn(tanφ)k. Alternatively, an estimate of the characteristic failure envelope (see also [1,11]) can be obtained by incorporating the shear stress estimates for a specific probability (given by Eq. (18)). The resulting from Eq.(18) failure envelope is non-linear and needs to be approximated by a linear regression to get the characteristic values of the Mohr -Coulomb failure criterion parameters.  The typical test procedure of applying on the three different specimens of each soil sample the σn = 100, 300 and 500 kPa normal stresses respectively, may raise a question concerning the independence of the observations. An alternative is to consider a single variable model for τ on each one of the three σ n values (in our case n=5 samples) and then apply a t-test on the observed τ values, which in our example leads to Table 2. Applying a linear regression on σn, τp=5% pairs we get the following characteristic values: c k3 =14.02 kPa and (tanφ) k3 = 0.38056 (characteristic 3). This approach, overcomes the issues concerning the independence of observation and gives similar results to characteristic 1 (Fig.3), however, it does not give a standard error of the parameters. For this specific case, the maximum standard error on τ lies close to the previously determined standard error of the cohesion.   Fig.4, due to the higher uncertainties.

STATISTICAL MEASURES FOR THE TYPICAL TRIAXIAL COMPRESSION TEST BY FORM
n the typical triaxial compression test the Mohr -Coulomb failure criterion parameters are determined indirectly. For this test, a constant horizontal radial stress (the cell pressure) σr=σc=σ3 is applied to a cylindrical specimen, while the reaction of the axial stress is measured (Δσa, σ1=σ3+Δσa). We can consider that the cell pressure is an accurate observation (i.e. the non -random). The Mohr -Coulomb failure criterion in terms of principal stresses is given by Eq.(6), in which the statistical measures of constants a and b can be determined directly from the two variables models, while the best estimates of the mean for c and tanφ constants are calculated indirectly by the transformation of Eqs. (19,20), respectively.
A way to calculate the uncertainties of c and tanφ is to apply the FORM, which method makes use of the second moment statistics (the mean and the standard deviation) of the random variables and assumes a linearized form of their performance function (e.g. z=g(X 1 ,…, X n )) at the mean values of the random variables and independency between all variables. Truncating at the linear terms the Taylor expansion of the performance function about the mean, it is possible to obtain the first order approximation of the variance (σz 2 ) of the true mean (μ z ) of z. Assuming uncorrelated non -random variables X 1 , …, X n , the approximation of the variance is given by Eq.(21), an equation commonly used to estimate the uncertainties by error propagation for laboratory tests results.
I where: Next, the above equations are applied to laboratory results from the Herakleion marl (see Table 3, data from [12]), with 27 specimens and 9 samples (3 specimens per sample). Treating each specimen separately (i.e. sample size n=27), the best estimates of a and b constants of Eq.(6) are determined from the linear regression (Eqs. ( 8,9)), which gives a m =229. Sample     Table 3.
Treating each sample separately we get the Mohr -Coulomb failure envelope constants of Table 4 (sample size n=9), which are represented in Fig.6 by the thin coloured lines, in the classic τ -σ n Mohr -Coulomb diagram. Again the remaining characteristic envelopes of Fig.6 are explained next. Alternatively, for the characteristic envelope we can apply a t-student distribution into the y(x) estimate (i.e. the predicted σ 1 for given σ 3 , given by Eq. (27)) for a p=5% probability (i.e. t p,n-2 =1.70814) and n-2 dof. In Fig. 5 An alternative for the characteristic envelope is to compute first the characteristic values of a and b constants by applying a t-student distribution (i.e. a k =a m -t p,n-2 SE a and b k =b m -t p,n-2 SE b ) for p=5% and n-2 dof and then apply Eqs. (20,19) to compute the Mohr -Coulomb characteristic constants ck and (tanφ)k. This approach, which does not take into account the error propagation, leads to ak=139.86 kPa and bk=2.58981 constants of Eq.(6) (Fig.5, characteristic 3) and to ck3=43.4 kPa and (tanφ) k3 =0.493949 constants of the Mohr -Coulomb failure criterion (Fig.6, characteristic 3).
Finally, as it was applied for the direct shear test, a common approach in engineering practice for the estimation of all statistical measures of c and tanφ is first to derive the Mohr -Coulomb constants (c, tanφ) for each sample separately (see Table 4) and then apply a statistical t-test on each constant independently for n/3-1 dof (where n the complete number of specimens and n/3 the number of specific locations of soil sampling   Table 4 ck=cm-tp,n-2SEc, tan(φ)k= tan(φ)m-tp,n-2SEtan(φ) 29.82 0.50990  Fig.7) is almost the same with the previously calculated. Fig.7 compares the character istic value from this approach (characteristic 4) with the characteristic 1 from the FORM. The alternate approach is very conservative because it gives a higher SE than the FORM. All the above results are summarized into Tables 5 for the  mean values and the corresponding uncertainties and Table 6 for the characteristic values.

APPLICATION OF THE FORM TO A SIMPLE PLANAR FAILURE PROBLEM
urther extending the FORM application to engineering problems calculations, a limit state analysis may be performed in terms of either the safety margin SM=R-E (R is resistance and Ε action as defined in EC7) in terms of the safety factor SF=R/E. More specifically, in FORM applications the SM application is preferable compared to the FS because the actions Ε in the denominator of the SF enhances the non-linearity effects in the error propagation. For the safety margin the reliability index is β=SM m /S d,SM , where SM m is the best estimate of the mean and S d,SM the standard deviation, while by definition it is βSM=1/VSM, where VSM the variation coefficient. Since resistances and actions describe different types of random variables, they are expected to be uncorrelated, and their covariance, ρ R,D , can be considered zero. Moreover, the variation of permanent loads is generally small compared and should not greatly affect the SM or SF value. The FORM is applied herein to a planar failure problem, which can be adapted to any type of failure surface (e.g. [13]) of limit equilibrium problems (e.g. method of slices). For the planar wedge failure problem considered herein Fig.8 shows the geometry. W is the weight of the wedge, H is the height of the slope, β is the angle of the slope to the horizontal, θ is the angle of the plane of failure with respect to horizontal and N and T are the normal and shear reaction forces on the plane of failure. The safety margin is then determined from the equilibrium equations that lead to Eq.(28). On the other hand the SF is given by Eq.(29). Applying the FORM on Eq.(28) the standard deviation of the SM is given by Eqs 30 to 33, in which u c , u tanφ and u γ are the uncertainties of c, tan(φ) and γ respectively. Likewise, applying the FORM on Eq.(29) the standard deviation of the SF is given by Eqs.(34-37).
c o st a n s i n sin 2 sin A slope geometry example of β=60 ο and Η=25m and the material properties from Tables 5 and 6 Table 5 for soil strength properties have been used.    Tables 9 and 10). The differences for the problem examined are of the order of 4% for the SM with respect to the min(SM p=5% ) value and of the order of 1 to 2% for the SF with respect to the min(SF p=5% ).
Calculations based on min SM for p=5%    Table 8 shows that for the probabilistic analyses it is preferable to determine the critical surface that corresponds to the minimum calculated SM or SF for a probability of exceedance 5% (or 90% confidence level, i.e. minSM p=5% or minSF p=5% ), instead of calculating the critical surface of the minimum mean value SMm or SFm first and then the corresponding SMp=5% or SFp=5%. This happens because there is no linear relationship between SM m (or SF m ) and u SM (or u SF ) for a monotonically increasing or reducing failure plane angle. Concerning the influence of the individual uncertainties Fig.9 shows their influence on the minimum SM and Fig.10 on the probability of having an SM<0. Fig.11 shows their influence on the minimum SF and Fig.12 on the probability of having an SF<1. It is apparent that for this specific problem, the most influential factor on the probabilistic SM or SF is the uncertainty of cohesion. This is important because cohesion generally has a greater uncertainty from the angle of shearing resistance. The dependence SM and SF with the various uncertainties present the same trends.
The above results will now be compared with the results from the deterministic analysis, for which the application of Eurocode 7 has been considered and more specifically the design analysis 3 (DA-3). DA-3 has become the national choice of many countries for the case of overall stability of natural and cut slopes. According to this, for the drained conditions, the following components are used for deterministic analysis: a) the design strength cd=ck/(γΜγm), (tanφ)d=(tanφ)k/(γΜγm) with γ m =1.0 (irrespectively of the type of analysis) and γ Μ =1.25, b) the permanent actions G d =γ G G k , with γ G =1.0 and c) the mobile design loads are Qd=γQQk, with γQ=0 or 1.3 for favourable or unfavourable loads respectively. Table 11 shows the results for the deterministic analyses, in which the four characteristic strength values of Table 6 have been used, factored by γ Μ =1.25 for the design strength values. The minimum safety margin (SM) and safety factor (SF) Figure 9: Influence of each uncertainty coefficient on the value of SM=SMm-kuSM.   and their corresponding critical slip surface (θ cr ) are presented. The material unit weight has been taken γ soil,m =22kN/m3 with uncertainty uγ=2kN/m 3 . According to the calculations, only characteristic 2 gives acceptable safety margin (SM>0) and safety factor (SF>1.00) and. The critical failure plane determined by SM and SF do not coincide. Characteristic values 1 and 4 were derived from the statistical measures used in the above analyses. It is obvious that probabilistic and deterministic analyses can give the opposite result. In our case the deterministic analysis was more conservative, which has to do with the selection of the partial safety factors.

CONCLUSIONS
Reliability methods can be incorporated for the determination of soil strength statistical measures and for probabilistic limit equilibrium analyses. Regarding soil strength, the determination of the uncertainties of the Mohr -Coulomb failure criterion constants c and tanφ from results of the typical triaxial test can be calculated through an error propagation technique, such as the FORM, applied to an appropriate performance function. This function relates the measured (σ1) and applied (σ3) quantities to the material constants (c, φ) of the M-C failure criterion by a nonlinear relationship. The FORM was applied to triaxial data and the results from different approaches for calculating the best estimate of the mean, the uncertainties and the characteristic values of the M-C constants were compared. For the approaches considered, the direct application of the FORM gave a lower uncertainty, since the consideration of each single specimen as independent increased the sample size. The uncertainties together with the corresponding best estimates were also used to estimate the characteristic failure envelope. Regarding the characteristic failure, envelope four different approaches were presented with different results. The FORM gave the more reliable results. In any case, engineering judgement on the results is necessary. The results were applied to a simple planar failure problem, which allows for a direct comparison of the various statistical measures of the SM and SF, since a single failure surface is considered. It was observed that the maximum probability of having a SM<0 or SF<1 does not correspond to the minimum best estimate of the SM or SF. This is influenced by the non -linear relationship of SM and uSM (or SF and uSF) function and needs further exploring. The sensitivity analysis showed that the only influential uncertainty was that of cohesion, which generally happens to have the greater variability with regards to other soil constants. A sensitivity analysis is recommended in probabilistic analyses in order to determine the influence of each uncertainty separately. The deterministic analyses gave in general more conservative results and only one gave acceptable SM and SF values. These analyses greatly depend on the selection of the partial safety factor selection that determines the characteristic value. The partial factors are indirectly related to an empirical probability of exceedance and the confidence of material properties and method of analysis. On the contrary, all probabilistic analyses gave acceptable SM and SF values, which means that probabilistic analyses could be used for an appropriate selection of partial safety factors. Therefore, the application of statistical methods can also set a framework for the selection of the characteristic mechanical properties, for deterministic analyses.