Damage Analysis I: Probit Functions and Probability Distributions

This chapter is a general introduction to damage analysis using mainly probit functions. The chapter first introduces two distinct concepts for the application of distributions to damage assessment. If distributions are used to compute the damage effects directly from hazard source and geometry information, the approach is called direct. The indirect approach, which is more flexible and modular, is to first compute the physical hazard (field, potential) and then to use distributions or further engineering or simulation damage models to finally compute the damage. The present textbook uses both approaches, with a strong focus on the indirect stepwise and systematic approach. Nevertheless, due to lack of hazard source data and often also scenario data (e.g. types of objects at risk) as well as in case of lack of resources, it is often first choice to use direct damage assessment methods. Direct damage assessment models are also basic benchmarks for damage assessment. The chapter shows how distributions are used for the indirect approach rather than using engineering and simulative approaches, e.g. for the description of the response of structures. For instance it shows how the radial distribution of the maximum overpressure is used as an input for lognormal damage assessment distributions. The use of the lognormal distribution (probit) is often motivated by its suitability for representing empirical data distributions with semi-infinite arguments. In contrast, the chapter derives the lognormal approach from a standard binary choice approach which uses the normal distribution. It is shown how it can be transformed to a corresponding lognormal distribution. Formally, the two main application options of the lognormal distributions are cases where the damage increases with increasing argument and where the damage decreases with increasing argument. For both cases application examples are given and the expressions are derived. The chapter gives further examples for the use of lognormal functions for other types of damage. It also gives examples for other types of distributions, in particular the Weibull and logit distributions. In addition, examples for free parameterizations are given. Finally, the chapter applies different types of damage models to the fragment hazard.