For specific countries such as Switzerland, a coefficient of variation for the yield stress of
0.05 seems reasonable.
7.4 Probabilistic Modelling of Model Uncertainties
Probabilistic models for uncertain load and resistance characteristics may in principle be
formulated at any level of approximation within the range of a purely scientific mathematical
description of the physical phenomena governing the problem at hand (micro-level) and a
purely empirical description based on observations and tests (macro-level).
In engineering analysis the physical modelling is, however, normally performed at an
intermediate level sometimes referred to as the meso-level. Reliability analysis will, therefore,
in general be based on a physical understanding of the problem but due to various
simplifications and approximations it will always to some extent be empirical. This essentially
means that if experimental results of e.g. the ultimate capacity of a portal steel frame are
compared to predictions obtained through a physical modelling, omitting the effect of non-
linearity then there will be a lack of fit. The lack of fit introduces a so-called model
uncertainty, which is associated with the level of approximation applied in the physical
formulation of the problem. It is important that the model uncertainty is fully appreciated and
taken into account in the uncertainty modelling.
The uncertainty associated with a particular model may be obtained by comparing
experimental results xexp with the values predicted by the model xmod given the experiment
conditions. Defining the model uncertainty Mas a factor to be multiplied on the value
predicted by the applied model Xmod in order to achieve the desired uncertain load or
resistance characteristic X, i.e.:
XM Xmod (7.36)the model uncertainty M may be assessed through observations of E where:
mod
expx
xE (7.37)
Model uncertainties defined in this way have mean value equal to 1 if they are unbiased.
Typical coefficients of variations deviations for good models may be in the order of
magnitude of 2-5 % whereas models such as e.g. the shear capacity of concrete structural
members the coefficients of variations is in the range of 10–20 %.
When the model uncertainty M is defined as in Equation (7.36) it is convenient to model the
probability distribution function fM( )E by a Lognormal distribution, whereby if the uncertain
load or resistance variable at hand Xmod is also modelled Lognormal distributed the product
i.e. X is also Lognormal distributed.