**Fulvio Tonon, University of Texas at Austin****Alberto Bernardini, Università di Padova**

In the design and evaluation of civil engineering structures, the enormous capacity of the available numerical models to simulate their behavior is frequently hampered by great uncertainty. For example, one cannot precisely predict expected loading conditions, material constitutive relations and their degradation in time, human errors in engineering procedures, or construction means and methods and subsequent operation and management of these structures. These uncertainties seldom can be described by mapping probabilistic input variables through a deterministic model to determine the precise expectation of output parameters.

Extensive research work in the past 30-40 years, from different academic sources and with different aims, has tried to overcome the boundaries of the classical theory of probability. Research has combined both set and random uncertainty and it can be folded within the general idea of convex sets of probability distributions, which describe the input data and hence yield lower/upper bounds on expectations of output parameters.

The tutorial motivates the use of these more general theories in civil engineering, lays out the theoretical background by emphasizing the connections between random sets, interval probabilities, p-boxes and the more general theory of imprecise probabilities. Applications are provided to the prediction of seismic vulnerability of buildings with a couple of illustrative case histories: one in Northern Italy, and one in Central Italy struck by the destructive "L'Aquila earthquake" in 2009.

- Examples of uncertain information in civil engineering and its representation with random sets and imprecise probability.
- Extreme distributions of random sets and calculation of expectation bounds for monotone and general functions.
- White distribution and its relationship with selectors and extreme distributions.
- Extreme distributions of interval probabilities (2-monotone) and p-boxes (infinite-monotone).
- Inclusion of a random set; inclusion of a non-consonant random set into a consonant one.
- Mappings of random sets through point valued and multi-valued mappings.

- Building damage vs. macro-seismic intensities and its formulation with random sets; vulnerability classes.
- Upper and lower vulnerability curves for the vulnerability classes.
- Application 1: Damage scenarios from the 1991 INSTAT inventory in the Vittorio Veneto Area.
- Application 2: Damage scenarios from the 2001 INSTAT inventory in the Abruzzi Region.

**Abstract:**

The purpose of this tutorial is to highlight the role of uncertainty modeling in civil engineering and to demonstrate how and why IP methods can be used.

A simple, but illustrative example from geotechnics is provided by the elastically bedded beam. The uncertainty of the material and geotechnical properties as well as their effect on the response of the beam are addressed. Starting with a purely probabilistic aproach, random set methods are discussed, and finally hybrid methods are presented. A short glimpse at sensitivity analysis using IP methods completes the tutorial, which also adresses decision making in engineering.

- Modeling uncertainty in engineering
- An example from geotechnics: a case study
- Decision making in engineering

Send any remarks to isipta11@uibk.ac.at.