## 2 Motion-Based Design of Structures under Uncertainty Conditions
### Motion-Based Design of Structures under Stochastic Conditions  
In order to generalize the implementation of the motion-based design method to scenarios with stochastic conditions, it is necessary to consider during the design process the uncertainty associated with the variability of both the external actions and the modal properties of the structure. For this purpose, two types of methods are normally employed [33]: (i) probabilistic methods; and (ii) fuzzy logic methods. Between these two methods, a probabilistic approach was considered herein because engineering practitioners are more used to dealing with probability concepts than with fuzzy logic problems. Concretely, a structural reliability method [39] was adapted herein to deal with the aforementioned uncertainty. According to this method, the vibration serviceability limit state can be expressed as a probabilistic density function, \(\text{g}_{unc}(\theta)\), which is defined in terms of the capacity of the structure, \(\text{C}_{s}\), and the demand of the external actions, \(D_{a}(\theta)\) (where both terms are random variables characterized by their probability density function). Thus, the vibration serviceability limit state can be defined as follows:  
\[\text{g}_{unc}(\theta)=\left\{ \begin{array}{c}C_{s}-D_{a}(\theta)\text{ \ \ \ \ \ \ \ \ \ \ if }\text{g}_{unc}(\theta)\text{ is assumed normally distributed}\\ \frac{\text{C}_{s}}{D_{a}(\theta)}\text{\ \ \ \ \ \ \ \ \ \ \ if }\text{g}_{unc}(\theta)\text{ is assumed }\text{ log - normally distributed}\end{array} \right. \tag{3}\]The above relation (Equation (3)) allows the computation of the probability of failure of the structural system, \(p_{f}(\theta)\), to the vibration serviceability limit state. This probability of failure, \(p_{f}(\theta)\), may be determined as follows:  
\[p_{f}(\theta)=\left\{ \begin{array}{rl}\text{Prob}[g_{unc}(\theta)<0]&\text{if $g_{unc}(\theta)$ is assumed normally distributed}\\ \text{Prob}[g_{unc}(\theta)<1]&\text{f $g_{unc}(\theta)$is assumed $\log- normally distributed}\end{array} \right. \tag{4}\]  
On the other hand, as it is shown in Figure 1, it is possible to characterize the probability of failure, \(p_{f}(\theta)\), via an equivalent index, the so-called reliability index, \(\beta_{s}(\theta)\). The relation between the probability of failure, \(p_{f}(\theta)\), and the reliability index, \(\beta_{s}(\theta)\), may be expressed as follows:  
\[p_{f}(\theta)= \begin{cases}\text{F}_{g_{unc}}(0)=\Phi\Big{(}\frac{\mu_{g_{unc }}(\theta)}{\sigma_{g_{unc}}(\theta)}\Big{)}=\Phi(-\beta_{s}(\theta))&\text{ normally distributed}\\ \text{F}_{g_{unc}}(1)=\Phi\Bigg{(}\frac{\ln\mu_{C_{s}}/\mu_{D_{s}}(\theta)}{ \sqrt{\rho_{h_{h_{C}}}^{2}+\sigma_{h_{h_{D}}(\theta)}^{2}}}\Bigg{)}=\Phi(- \beta_{s}(\theta))&\text{log}-\text{normally distributed}\end{cases}  \tag{5}\]  
where \(\text{F}_{g_{unc}}\) is the cumulative probability distribution function of \(\text{g}_{unc}(\theta)\); \(\mu_{g_{unc}}(\theta)\) and \(\sigma_{g_{unc}}(\theta)\) are respectively the mean and standard deviation of \(\text{g}_{unc}(\theta)\); \(\Phi\) is the standard normal cumulative distribution function; \(\mu_{C_{s}}\) and \(\mu_{D_{s}}(\theta)\) are respectively the mean of the probabilistic distribution function of \(C_{s}\) and \(D_{s}(\theta)\); and \(\sigma_{h_{m}C_{s}}\) and \(\sigma_{h_{m}D_{s}(\theta)}\) are respectively the standard deviation of the log-normal distribution of \(C_{s}\) and \(D_{s}(\theta)\). In this manner, the use of the reliability index, \(\beta_{s}(\theta)\), allows the computation of the vibration serviceability limit state under uncertainty conditions to be simplified. Hence, this design requirement is met if the reliability index, \(\beta_{s}(\theta)\), is greater than the allowable reliability index, \(\beta_{t}\), established by the designer/manufacturer/owner of the structure.