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FORM SORM and spatial modeling in geotechnical engineering B K Low School of Civil and Environmental Engineering Nanyang Technological University Singapore a r t i c l ei n f o Article history Available online xxxx Keywords Reliability index FORM SORM Spatial autocorrelations Geotechnical engineering a b s t r a c t An intuitive ellipsoidal perspective is described together with three spreadsheet automated constrained optimizational FORM procedures and a SORM approach The three FORM procedures are then compared in the context of geotechnical examples of a confi ned soil element a rock slope and an embankment on soft ground with spatially autocorrelated undrained shear strength in the soft clay foundation the per formance function of which is based on a reformulated Spencer method with search for reliability based critical noncircular slip surface Two methods of modeling spatial autocorrelations are presented and the merits and limitations of the three constrained optimizational FORM procedures are studied The comple mentary roles and interconnections among the three constrained optimizational FORM procedures and SORM approach are emphasized Comparisons are also made with Monte Carlo simulations 2013 Elsevier Ltd All rights reserved 1 Introduction The Hasofer Lind 13 index for cases with correlated normal random variables and the fi rst order reliability method FORM for cases with correlated nonnormals are well explained in Ditlevsen 10 Shinozuka 39 Ang and Tang 1 Melchers 34 Haldar and Mahadevan 12 and Baecher and Christian 2 for example The potential inadequacies of the FORM in some cases have been recognized and more refi ned alternatives proposed in Chen and Lind 7 Der Kiureghian et al 9 Wu and Wirsching 47 and Zhao and Ono 49 among others On the other hand the usefulness and accuracy of the FORM in most applications are well recognized for instance by Rackwitz 38 The focus of this paper is on spreadsheet based procedures for FORM which extends the Hasofer Lind index for correlated nor mals into the nonnormal realm SORM on the foundation of FORM results system FORM and reliability analysis accounting for spa tially autocorrelated soil properties Specifi cally a simple geome chanics example is fi rst examined to illustrate spreadsheet based SORM analysis on the foundation of FORM reliability index and FORM design point This is followed by a rock slope with correlated nonnormal random variables solved using the u space approach for comparison with the Low and Tang 29 n space approach Finally spatially autocorrelated shear strength is modeled in the reliability analysis of an embankment on soft ground The advanta ges and limitations of three FORM computational approaches namely constrained optimization with respect to the original randomvariables thenormalizedbutunrotatednvector and the normalized and rotated u vector respectively are investigated Spatial autocorrelation also termed spatial variability arises in geological material by virtue of its formation by natural processes acting over unimaginably long time millions of years This en dows geomaterial with some unique statistical features e g spa tial autocorrelation not commonly found in structural material manufactured under strict quality control For example by the nat ure of the slow precipitation over many seasons of fi ne grained soil particles under gravity in water in nearly horizontal layers two points in close vertical proximity to one another are likely to be more positively correlated likely to have similar undrained shear strength cuvalues for example than two points further apart in the vertical direction System FORM and SORM are extensions of FORM The classical computational approach of FORM in normalized and rotated u space is elegant but shrouded in mathematical details An intu itive perspective and two spreadsheet automated FORM computa tional approaches were provided in Low and Tang 28 29 with the aim to facilitate understanding The two approaches in the x space and n space respectively are summarized in the next section together with a third alternative of spreadsheet constrained opti mization in the u space They are meant to complement the elegant classical u space FORM approach The Low and Tang 29 n space approach easily reverts to the u space required for SORM compu tation using a simple equation The spreadsheet based reliability approaches can be applied to implicit functions or stand alone numerical packages via the response surface methods RSM for example RSM as a bridge betweenspreadsheet basedgeotechnicalFORManalysisand stand alone numerical packages is illustrated by the following 0167 4730 see front matter 2013 Elsevier Ltd All rights reserved http dx doi org 10 1016 j strusafe 2013 08 008 Tel 65 67905270 E mail addresses bklow alum mit edu cbklow ntu edu sg Structural Safety xxx 2013 xxx xxx Contents lists available at ScienceDirect Structural Safety journal homepage Please cite this article in press as Low BK FORM SORM and spatial modeling in geotechnical engineering Struct Saf 2013 http dx doi org 10 1016 j strusafe 2013 08 008 i Xu and Low 48 conducted FORM analysis on second degree polynomial response surface function constructed from fi nite element analysis of embankment stability ii Chan and Low 6 constructed second degree polynomial response surface based on fi nite element analysis of a later ally loaded pile and conducted FORM and SORM analyses iii L and Low 32 conducted FORM and SORM analyses on the movement of a horseshoe shaped highway tunnel in which the response surface function was based on numerical anal yses using the code FLAC 2 Intuitive perspective and effi cient spreadsheet approaches for FORM and SORM The matrix formulation 45 10 of the Hasofer and Lind 13 index b is b min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi x l TC 1 x l q 1a or equivalently b min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi xi li ri T R 1 xi li ri s 1b where x is a vector representing the set of random variables xi lthe vector of mean valuesli C the covariance matrix R the correlation matrix rithe standard deviations and F the failure domain Low and Tang 27 used Eq 1b instead of Eq 1a because the correla tion matrix R is easier to set up and conveys the correlation struc ture more explicitly than the covariance matrix C The point denoted by the xi values which minimize Eq 1 and satisfi es x 2 F is the design point This is the point of tangency of an expand ing dispersion ellipsoid with the LSS which separates safe combina tions of parametric values from unsafe combinations Fig 1 The one standard deviation 1 r dispersion ellipse and the b ellipse in Fig 1 are tilted by virtue of cohesion c and friction angle being negatively correlated The quadratic form in Eq 1 appears also in the negative exponent of the established probability density function of the multivariate normal distribution As a multivariate normal dispersion ellipsoid expands from the mean value point its expanding surfaces are contours of decreasing probability values Hence to obtain b by Eq 1 means maximizing the value of the multivariate normal probability density function and is graphically equivalent to fi nding the smallest ellipsoid tangent to the LSS at the most probable failure point the design point This intuitive and vi sual understanding of the design point is consistent with the more mathematical approach in Shinozuka 39 in which all variables were standardized and the limit state equation was written in terms of standardized variables In FORM one can rewrite Eq 1b as follows Low and Tang 28 and regard the computation of b as that of fi nding the small est equivalent hyperellipsoid centred at the equivalent normal mean value pointlNand with equivalent normal standard devia tionsrN that is tangent to the limit state surface LSS b min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi xi lN i rN i T R 1 xi lN i rN i s 2 wherelN i andrN i can be calculated by the Rackwitz and Fiessler 37 transformation Hence for correlated nonnormals the ellipsoid per spective still applies in the original coordinate system except that the nonnormal distributions are replaced by an equivalent normal ellipsoid centered not at the original mean values of the nonnormal distributions but at the equivalent normal meanlN Eq 2 and the Rackwitz Fiessler equations were used in the spreadsheet automated constrained optimization computational approach of FORM in Low and Tang 28 An alternative to the 2004 FORM procedure is given in Low and Tang 29 which uses the following equation for the reliability index b b min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi nTR 1n p 3 The computational approaches of Eqs 1b 2 and 3 and associated ellipsoidal perspective are complementary to the classical u space computational approach and may help overcome the conceptual and language barriers which Whitman 46 rightly noted The two spreadsheet based computational approaches of FORM are compared in Fig 2 Either method can be used as an alternative to the classical u space FORM procedure A third alternative is also shown in Fig 2 for which the Microsoft Excel s built in constrained optimization routine Solver is invoked to automatically vary the u vector so that b and the design point are obtained This requires only adding one u column and expressing the unrotated n vector in terms of u where u is the uncorrelated standard equivalent nor mal vector in the rotated space of the classical mathematical ap proach of FORM The vectors n and u can be obtained from one another n Lu and u L 1n as follows e g 31 b min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi nTR 1n p min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi nT LU 1n q min x2F ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi L 1n T L 1n q 4a i e b min x2F ffi ffi ffi ffi ffi ffi ffi ffiffi uTu p whereu L 1nandn Lu 4b Fig 1 Illustration of the reliability index b in the plane when c and are negatively correlated Fig 2 Comparison of the two FORM computational approaches of Low and Tang 2004 2007 and the additional u to n to x approach illustrated in this paper 2B K Low Structural Safety xxx 2013 xxx xxx Please cite this article in press as Low BK FORM SORM and spatial modeling in geotechnical engineering Struct Saf 2013 http dx doi org 10 1016 j strusafe 2013 08 008 in which L is the lower triangular matrix of R When the random variables are uncorrelated u n by Eq 4 because then L 1 L I the identity matrix The probability of failure can be estimated approximately as follows Pf 1 U b U b 5 where is the cumulative distribution CDF of the standard normal variate Eq 5 is exact when the LSS is planar and the parameters fol low normal distributions Inaccuracies in Pfestimation may arise when the LSS is signifi cantly nonlinear More refi ned alternatives have been proposed for example the second order reliability method SORM by Fiessler et al 11 Tvedt 40 Tvedt 41 Tvedt 42 Breitung 3 Hohenbichler and Rackwitz 15 Koyluoglu and Nielsen 19 Cai and Elishakoff 4 Hong 16 and Zhao and Ono 49 SORM analysis requires the FORM b value and design point val ues as inputs and therefore is an extension dependent on FORM results Hence the SORM results are displayed alongside the FORM results in the examples to follow In general the SORM attempts to assess the curvatures of the LSS near the FORM design point in the dimensionless and rotated u space The failure probability is calculated from the FORM reli ability index b and estimated principal curvatures of the LSS using established SORM equations The remaining sections of this paper address the following issues 1 Improving the accuracy of FORM Pf Eq 5 by SORM analysis 2 Investigations on the merits and limitations of the three spreadsheet automated FORM computational approaches of Fig 2 3 Presenting two procedures for modeling spatially autocorre lated undrained shear strengths 4 Comparisons with Monte Carlo simulations 3 Extending from FORM to SORM a simple geomechanics example Existing SORM formulas use FORM b and the curvatures at the FORM design point as inputs Pf SORM f bFORM Curvatures at the FORM design point 6 These formulas have been used in Chan and Low 5 which pre sented a practical and effi cient approach of implementing SORM using an approximating paraboloid 9 fi tted to the LSS in the neigh borhood of the FORM design point Complex mathematical opera tionsassociatedwithCholeskyfactorization Gram Schmidt orthogonalization and inverse transformation are relegated to rela tively simple short function codes in the Microsoft Excel spread sheet platform To illustrate how conducting SORM after FORM can improve the accuracy of Pf the simple geomechanics example of Huang and Griffi ths 17 is analyzed here using seven SORM formulas instead of Breitung formula only in Huang and Griffi ths 17 The mean valuesl standard deviationsrand correlation matrix R are as as sumed in Huang and Griffi ths 17 The performance function is g c0 0 r01f r01 r03N 0 2c0 ffi ffi ffi ffi ffi ffi ffi N 0 q r01 0 N 0 1 sin 0 1 sin 0 7a 7b The above equation is mathematically equivalent to Eq 11 of Huang and Griffi ths 17 when g x 0 It can be seen from Fig 3 that SORM Pfbased on the average of seven SORM formulas agrees very well with Monte Carlo Pf If the average of six formulas is used excluding the Breitung formula SORM Pfis practically identical to the Pffrom Monte Carlo simulations It must be mentioned that Breitung s formula gives practically identical results as those of the other six formulas and Monte Carlo simulations when b 2 0 orm 0 3 Breitung s accuracy for practical high reliability prob lems was justly noted in Haldar and Mahadevan 12 The Low and Tang 29 n space computational approach Fig 2 for FORM was used in Fig 3 Had the u space computational ap proach been used the results will be identical because the random variables are uncorrelated Hence the u space approach in spread sheet is illustrated next for correlated nonnormals 4 SORM analysis after FORM and Monte Carlo simulations of a rock slope 4 1 Constrained optimizational FORM approach with respect to the u vector Unlike the previous case which involves two uncorrelated lognormals Fig 4 shows the FORM and SORM analyses of a two dimensional rock slope with fi ve correlated random variables two of which obey the highly asymmetric truncated exponentials The deterministic formulations are in Hoek 14 the FORM analysis invokes Excel Solver to automatically change the u vector initially zeros so as to obtain the reliability index b and the design point x via the n vector and the u vector The SORM analysis uses the Chan and Low 5 Excel spreadsheet approach Five Monte Carlo simulations each with 500 000 trials yielded Pfvalues within the range 2 24 2 28 The FORM Pfis 2 96 higher than the Monte Carlo average Pfof about 2 25 In contrast the average SORM Pfon the basis of FORM b and four estimated components of curvature at the FORM design point is 2 18 as shown if the eight discrete points selected for curvature estimation correspond to k 2 coarser grid and 2 05 if k 1 The Breitung result of 2 28 is the closest to the Monte Carlo Pffor this case where the value of b is nearer to the practical higher design range of b There is no unique SORM Pfvalue It depends on the method used for estimating the curvatures at the design point and on the formula used to compute Pfbased on FORM b and the Fig 3 Compression of a confi ned plane strain element results of FORM and SORM analyses for coeffi cient of variationv 1 and plots of probabilities of failure versus coeffi cient of variation based on FORM SORM and Monte Carlo simulations B K Low Structural Safety xxx 2013 xxx xxx3 Please cite this article in press as Low BK FORM SORM and spatial modeling in geotechnical engineering Struct Saf 2013 http dx doi org 10 1016 j strusafe 2013 08 008 curvatures at the FORM design point Nevertheless in the practical high reliability range Pf 0 and the unsafe do main where g x 0 at the mean value point then perform FORM reliability analysis by increasing T until b btarget iii The reliability based design of the embedment depth of an anchored sheet pile wall in Low 23 provides another example of the need to distinguish negative b from positive b values 5 Slope reliability analysis accounting for spatial variation The geological processes of soil formation impart spatial auto correlation to most soil properties Two methods used by the author in his reliability analysis accounting for 1 D spatial variabil ity are i the method of autocorrelated slices and ii the method of interpolated autocorrelations These are described below focus ing on the method of interpolated autocorrelations to model spa tial autocorrelations in an embankment on soft ground analysed using a reformulated Spencer method in the Excel spreadsheet platform for the purpose of investigating the merits and weakness of the three approaches of Fig 2 It is straightforward to extend both methods to 2 D spatial var iability in limit equilibrium analysis as in Ji et al 18 The method of interpolated autocorrelations can also be applied to 2 D fi nite element method 5 1 Method of autocorrelated slices This method was used in Low et al 30 s limit equilibrium analysis of a clay slope in Norway Fig 6 The horizontal variation Fig 4 FORM based on automatically changing the u vector in the Microsoft Excel platform and SORM estimation of