Probabilistic framework for sensitivity and reliability analyses
Explicit treatment of uncertainties using probabilistic methods has been receiving attention recently, both in the justification of partial factors in codes of practice as well as in design and assessment. EN 1997-1 explicitly states reliability-based methods as one of the options to verify limit states of geotechnical structures alongside the partial factor method, prescriptive rules, testing, and the observational method. Guidelines for reliability-based verification of limit states in design and assessment of geotechnical structures within the safety and reliability concepts of the new Eurocodes, were published in November 2024. The ZSoil team contributed to the elaboration of three examples for these guidelines included in Annex B of this document.
ZSoil 2025 enables users to perform sensitivity, reliability and Bayesian analyses with the help of a new intuitive graphical user interface embedded in the ZSoil post-processor. All probabilistic algorithms are powered by UQLab, a robust framework for uncertainty quantification developed over the past decade by Prof. Bruno Sudret and his team at ETH Zürich. Further details of the included options are given below.
New graphical user interface embedded in the ZSoil post-processor
Definition of random variables
Input variables such as material parameters, water levels, load functions, or existence functions can now be defined as probability density functions. Supported distributions include Gaussian, Lognormal, Gumbel and Uniform with the added capability to introduce dependancies between different variables. This allows users to define one variable as probabilistic (eg. E50), with dependant variables (eg. Eoed, Eur & E0) updated according to the probabilistic value for each sample.
Intercorrelation
To account for correlations between random variables (intercorrelations), copulas can be introduced. In ZSoil v2025 a gaussian copula, characterized by a linear correlation matrix (Pearson correlation matrix), is used. (Lataniotis et al., 2024).
Correlations are considered for Reliability and Bayesian Analysis.
Sampling
The creation of samples of the random variables representing distributions is called sampling. The different distributions are obtained by drawing sample vectors with a uniform distribution between 0-1 for each entry. A transform operation is then applied to the vector in order to get the desired distribution.
The following sampling methods are implemented for the underlying uniform distribution:
• MC (Monte Carlo)
• LHS (Latin Hypercube sampling)
• Sobol
• Halton
Quantities of Interest
Definition of the Quantities of Interest (QoI), such as maximum bending moment (from a list of beam elements), a maximum settlement (from a list of nodes), a safety factor, etc. The process is fully generic, allowing users to define any result produced by the finite element model.
Metamodel
Metamodels are necessary for approximating the finite element model response (specifically the QoI) rapidly, enabling Monte-Carlo simulations to be run to complete Sensitivity, Reliability and Bayesian analyses. In ZSoil, there are 3 approximation methods available for generating metamodels: Kriging, Polynomial Chaos Expansion (PCE) and Polynomial Chaos Kriging (PCK). Note that the intercorrelations defined previously are not considered for building the experimental design and therefore the metamodel. For more detailed information on the respectitve method, please refer to the UQLab manuals for Kriging (Lataniotis et al., 2024), PCE (Marelli et al., 2024) and PCK (Schobi et al., 2022).
Sensitivity Analysis
The sensitivity of a QoI to the variability of a random variable can be assessed in ZSoil v25 using a Sensitivity Analysis. The following methods are available: Sobol indices, Linear Correlation and Standard regression coefficients (SRC).
Reliability Analysis
The generated metamodel can now be used for running Monte-Carlo siulations to obtain distributions of the QoI. The uncertainty associated with the input variables is propagated to the defined QoI, allowing probabilities of failure to be calculated for defined limit states associated with each QoI.
Bayesian Analysis
The ability to refine predictions using Bayesian analysis during project execution or construction, ensures more accurate and adaptive modelling based on real-world conditions. It involves incorporating in-situ measurements such as data from inclinometers and piezometers, to update model predictions, potentially enabling optimisation of the design during construction. In other words, it is the mathematical formulation of the Observational Method.
Comparison of prior and posterior predictions
This short video provides an overview of the entire probabilistic framework process that is demonstrated through a straightforward slope stability analysis example.
Moving loads and masses
The concept focuses on providing users with the flexibility to define loads and added masses freely along specified 2D or 3D paths, and independently from the finite element mesh. This flexibility enables static and dynamic analyses of bridges, roads and train lines when subjected to traffic loads such as high-speed trains or trucks.
Moving loads can be defined along any trajectory —whether a polyline or spline, in global, local coordinates — with the option to specify constant or variable velocities.
Movement paths can be defined on:
• Geometrical objects or edges
• Beams, trusses and continuum 2D
• Membranes, shells and continuum 3D
• Point and the direction and distance
• Splines including smoothing
Moving loads for the static analysis and the added masses (dynamics) can be applied on:
• Points
• Lines
• Surfaces
Users will also appreciate that with this new way of defining loads which is mesh independant means load definition is much simpler, faster and flexible as the mesh no longer needs to mathc the geometry of your load, whether is be a point/node, beam/line, or surface load.
Definition of the point load in global or local coordinates systems
Definition of the moving load including two loading points following a movement path
Dynamic analysis including a movement path for point load and added mass