## 4 Computational and Mathematical Aspects
### Damage States and Structural Performance Levels  
As is common practice today, performance-based seismic design is initiated with an interplay between demands and appropriate performance objectives. The task of the engineer is to then develop a design capable of meeting these objectives. Performance objectives are expressed as an acceptable level  
Figure 13: Final weights and bias values of the optimum BPNN model 1-3-5-1 (the values are presented in Table 3).  
of damage, typically categorized as one of several performance levels, such as immediate occupancy, life safety or collapse prevention, given that ground acceleration of specified severity is experienced.  
In the past, the practice of meeting performance-based objectives was already included in design practice, but in a rather informal, simplistic and non-standardized way. Some engineers would characterize performance as life-safety or not; others would assign ratings ranging from poor to good. This qualitative approach adopted for performance prediction was appropriate given the limited capability of seismic-resistant design technology to deliver building designs capable of quantifiable performance.  
In the approach presented herein, we consider three structural performance levels in a similar way to the Federal Emergency Management Agency [196]: (a) heavy damage; (b) moderate damage; and (c) insignificant damage. The performance levels and the associated damage states are defined by the values of DI, as shown in Table 4 and Figure 14. In particular, a value of DI less than 15% can be interpreted as insignificant damage; from 15% to less than 25%, as moderate damage; and greater than or equal to 25% as heavy damage. Of course, other approaches could be used, according to the recent European Codes [197], based on a more engineered (and more detailed) estimation of damage.  
It is worth noting that, for the definition of damage States (DS) ranges, expert opinions are needed. When no experimental data are available and analysis of the behavior is not feasible, one or more knowledgeable individuals (experts) can offer an opinion as to values of SL defining the damage states (DS) ranges, based either on experience or judgment. Furthermore, these SL must be periodically updated based on experimental and analytical data, when such data are available.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Damage State** & **Insignificant Damage** & **Moderate Damage** & **Heavy Damage** \\ \hline \multirow{4}{*}{Element} & Minor cracking of veneres. & Extensive cracking. & Extensive cracking. \\  & Minor spalling in veneres at a & Noticeable in-plane offsets of & cause and veneer may \\  & few corner openings. No & masonry and minor & peel off. Noticeable \\  & observable out-of-plane offsets. & out-of-plane offsets. & in-plane and \\ \hline \multirow{2}{*}{[SL]} & 0.05 \(\leq\) DI \(<\)0.15\% & 0.15 \(\leq\) DI \(<\)0.25 & 0.25\% \(\leq\) DI \(<\)1.00 \\  & Immediate occupancy & Life safety & Collapse prevention \\ \hline \hline \end{tabular}
\end{table}
Table 4: Proposed Structural Performance Levels for un-reinforced masonry.  
Figure 14: Definition of Limit States (LS) thresholds and Damage States (DS) ranges.