4. Basis of uncerainty modelling¶
4.1. Basic variables¶
The calculation model for each limit state considered should contain a specified set of basic variables, i.e. physical quantities which characterize actions and environmental influences, material and soil properties and geometrical quantities. The model should also contain model parameters which characterize the model itself and which are treated as basic variables (compare 4.2). Finally there are also parameters which describe the requirements (e.g. serviceability constraints) and which may be treated as basic variables. The basic variables (in
the wide sence given above) are assumed to carry the entire input information to the calculation model.
The basic variables may be random variables (indlucing the special case deterministic variables) or stochastic processes or random fields. Each basic variable is defined by a number of parameters such as mean, standard deviation, parameters determining the correlation structure etc.
4.2. Types of uncertainty¶
Uncertainties from all essential sources must be evaluated and integrated in a basic variable model. Types of uncertainty to be taken into account are:
intrinsic physical or mechanical uncertainty
statistical uncertainty, when the design decisions are based on a small sample of observations or when there are other similar conditions
model uncertainties (see 5.6).
Within given classes of structural design problems the types of probability distributions of the basic variables should be standardized. These standardizations are defined in the parts 2 and 3 of the probabilistic model code.
4.3. Definition of populations¶
The random quantities within a reliability analysis should always be related to a meaningfull and consistent set of populations. The description of the random quantities should correspond to this set and the resulting failure probability is only valid for the same set.
The basis for the definition of a population is in most cases the physical background of the variable. Factors which may define the population are:
the nature and origin of a random quantity
the spatial conditions (e.g. the geographical region considered)
the temporal conditions (e.g. the intended time of use of the structure considered)
The choice of a population is to some extent a free choice of the designer. It may depend on the objective of the analysis, the amount and nature of the available data and the amount of work that can be afforded.
In connection with theoretical treatment of data and with the evaluation of observations it is often convenient to divide the largest population into sub-populations which in turn are further divided in smaller sub-populations etc. Then it is possible to study and distinguish variability within a population and variability between different populations.
In an analysis for a specific structure it may be efficient to define a population as small as possible as far as use, shape and location of the structure are concerned (microzonation). When the results are used for design in a national or international code, it may be necessary or convenient to put the sub-populations together to the large population again in order not to get too complicated rules (randomizing). This means that the variability within the population is increased.
4.4. Hierarchy of uncertainty models¶
This section contains a convenient and recommended mathematical description in general terms of a hierarchical model which can be used for different kinds of actions and materials. The details of this model have to be stated more precisely for each specific variable. The model is associated with a hierarchical set of subpopulations.
The hierarchical model assumes that a random quantity \(\vec{X}\) can be written as a function of several variables, each one representing a specific type of variability:
The \(Y\) represent various origins, time scales of fluctuation or spatial scales of fluctuation.
For instance \(Y_i\) may represent the building to building variation, \(Y_{ij}\) the floor to floor variation in building \(i\) and \(Y_{ijk}\) the point to point variation on floor \(j\) in building \(i\).
In a similar way, \(Y_i\) may represent the constant in time variability, \(Y_{ij}\) a slowly fluctuating time process and \(Y_{ijk}\) a fast fluctuating time process.