1. EXCEL-based direct reliability analysis and its potential role to complement Eurocodes B. K. Low Nanyang Technological University (NTU) Singapore

2. Joint TC205/TC304 Report on “Discussion of statistical/reliability methods for Eurocodes”
Chapter 4: EXCEL-Based Direct Reliability Analysis and Its Potential Role to Complement Eurocodes
Lead discusser: Bak Kong Low
Discussers (alphabetical order):
Gregory Baecher, Richard Bathurst, Malcolm Bolton, Philip Boothroyd, Zuyu Chen, Jianye Ching, Peter Day, Nico De Koker, Mike Duncan, Herbert Einstein, Robert Gilbert, D.V. Griffiths, Karim Karam, Suzanne Lacasse, Sónia Marques, Lars Olsson, Trevor Orr, Kok-Kwang Phoon, Brian Simpson, Yu Wang, Tien Wu

3. Intuitive perspective and efficient FORM procedure in a ubiquitous platform
Low & Tang 2004 method (Structural Safety):
Low & Tang 2007 method (J. Engrg. Mech., ASCE):

4. Expanding dispersion ellipsoid in original space
When distributions are non-Gaussian, think in terms of equivalent hyperellipsoids
Mean-value point
Design point: X1,D and X2,D
Characteristic values of loads X1,k and X2,k

5. General concepts of ultimate limit state design in Eurocode 7:
(Low & Phoon, 2015)
(if “=“, then “design point”)
Diminished resistance (ck / c, tank / ) > Amplified loadings
Based on characteristic values and partial factors for loading parameters.
Characteristic values
Partial factors
“Conservative”, for example, 5 percentile for strength parameters, 95 percentile for loading parameters (Implicit probability approach in Eurocode 7)

6. FORM FEM, FDM System FORM Importance sampling SORM RSM
The insights and merits of RBD were illustrated for a gravity retaining wall, an anchored sheet pile wall, a footing with inclined and eccentric loadings, a laterally loaded pile, soil slopes, 2D and 3D rock slopes, tunnels in rocks, and a strut with complex supports.
FORM
System FORM
Importance sampling
SORM
RSM
FEM, FDM

7. FORM analysis and RBD via FORM are very useful in the following ways:
Giving warnings when the computed Pf are unacceptably high;
Aiming at sufficiently safe designs at a target reliability index (and low nominal Pf);
Comparative assessment of the relative reliability of different designs;
Incorporating parametric correlations and spatial auto-correlations in design;
Complement EC7 design for parameters not yet covered in the design code;
Complement EC7 design when the sensitivities of parameters vary from case to case;
Complement EC7 design when reality warrants correlation among parameters;
Complement EC7 design when a parameter possesses stabilizing- destabilising duality (i.e. favourable-unfavourable duality, or action- resistance duality)
Complement partial factors design approaches when uncertainty in unit weight of soil needs to be modelled. 

8. Limitations of RBD-via-FORM
RBD-via-FORM presents greater language and conceptual barriers to practitioners than EC7 or LRFD
Imprecise statistical inputs. Statistical inputs are approximate and often involve judgment, due to limited data.
Neglected factors (“known unknowns”, “unknown unknowns”, …): e.g., human factors, natural hazards, failure modes overlooked, …
Model error due to idealized mechanical model.
Hence the probability of failure based on RBD is not exact. The Pf from RBD is at best approximate (and sometimes way off), and hence the Pf must be regarded as nominal rather than precise.

9. Further concluding thoughts:
That the Pf associated with a target reliability index in a RBD is more indicative and nominal than real should not deter the Geotech profession from appreciating the merits of RBD and its potential complementary role to design approaches like EC7 and LRFD.
The same limitations with respect to approximate inputs, idealizations and non-exhaustive factors also apply to the outputs of deterministic analysis to some extent (e.g. displacement prediction).
One is reminded of Terzaghi’s (and other geotech pioneers’) pragmatic approach of aiming at designs such that unsatisfactory performance is not likely, instead of aiming at designs which would behave precisely (e.g. footing settlement of exactly 25mm). It is in the same spirit that RBD aims to achieve sufficiently safe design, not at a precise probability of failure.

10. Information and insights
Suggestion:
Design outcome based on EC7 or LRFD
Conduct FORM reliability analysis to the EC7/LRFD design, using realistic statistical inputs and correlations
Information and insights
Sensitivity as revealed by FORM design point
Detect potential action-resistance duality (favorable/ unfavorable duality)
Indicative reliability index and Pf
Investigate effects of correlations
The above info/insights aimed at are unlikely to be too sensitive to the statistical inputs and correlations