Assessment of mean \(C_{R}\) ratios
In this section, the influence of forward-directivity effects on \(C_{R}\) for NSCs is explicitly quantified and statistically assessed. This is firstly examined through the comparison of the tendencies for mean \(C_{R}\) ratios under FF and NF excitations. Secondly, a series of hypothesis tests are conducted to examine if the differences observed in the mean demand of \(C_{R}\) under NF records are statistically significant, when contrasted with that under FF records.
Figure 12a shows the mean and dispersion of \(C_{R}\) ratio at the roof of a structure for NF records to FF counterparts, for a given Rnsc of 4.0. On the other hand, Fig. 12b shows solely the mean of \(C_{R}\) ratio for all considered levels of Rnsc. Firstly, it is worth recalling that for two given buildings (i.e., different \(T_{1}\)), and irrespective of the set of records, the range of \(T/T_{1}\) does not coincide and hence is not readily comparable. To overcome this and compute the mean and dispersion, a linear interpolation was used, enabling the comparison of \(C_{R}\) at discrete equidistant values of \(T/T_{1}\) with a delta step of \(0.05\). Secondly, it can be observed that in this comparison where all buildings are included, there are three clear spectral regions. In the short region of \(T/T_{1}\) a clear amplification of the demand of \(C_{R}\) is seen due to the effect of NF records, with an increase of 30–40%, and without a discernible trend with respect to \(R_{nsc}\). The central region, not necessarily demarcated in its lower end by \(T/T_{1}\) of 1.0, thus ranging from 0.5 to 1.7, where similar responses are computed through a flat line around 1.0 for all values of \(R_{nsc}\). Interestingly, in the long spectral region, a reduction of the mean demand of \(C_{R}\) is observed, since values below 1.0 occur, with larger reduction of up to 20% for higher \(R_{nsc}\); in other words, the mean demand of \(C_{R}\) is higher due to the effect of FF records. The latter behaviour has been previously observed in the computation of structural inelastic displacement ratios (e.g., Ruiz-Garcia 2011). Furthermore, Fig. 13 shows the same type of results but comparing the mean ratio of \(C_{R}\) between records with forward-directivity effect and the set of FF records. The more significant effect of NFP on the mean demand of \(C_{R}\) is evident when contrasted with the FF case. This effect divides the behaviour of \(C_{R}\) ratio into two spectral regions, enlarging the short-mid span of \(T/T_{1}\) where the ratio is higher than 1.0, approximately up to \(T/T_{1} \approx 2\), and exhibiting up to twice as much \(C_{R}\) mean demand because of records with forward-directivity effect. Moreover, in this region, it is clear that \(C_{R}\) ratio increases with \(R_{nsc}\). In the case of \(T/T_{1} > 2\), the same reversibility is observed, where \(C_{R}\) ratio is below 1.0, and tends to saturate around 0.9 regardless of the level of.
To understand further the behaviour of CR ratio and taking advantage of the large family of buildings considered in this study, 3 bins of 18 buildings are defined based on their fundamental period of vibration. The results are shown in Fig. 14 for each bin of group corresponding to, 0.39 s ≤ 1 ≤ 0.94 s, 0.95 s ≤ 1 ≤ 1.11 s, and 1.14 s ≤ 1 ≤ 1.87 s, and in Fig. 12a–c, respectively. The first observation is that for the three bins, the same global trend is seen for the mean value of CR ratio. For < 1.0, the ratio increases as the normalised period decreases and its absolute value increases with. When > 1.0 the ratio flattens out for all levels of and extends below 1.0 – as noted before – for levels of larger than about 2, the latter being valid for bins 2 and 3. There are however clear differences between the various bins. For example, the first bin of short-period buildings (i.e., stiffer systems) shows a mean CR whose ordinates are larger than 1.0 across the whole span of normalised periods, with the mean CR demand due to NFP being always higher across all NSCs. All in all, it was observed that the mean values of CR ratio oscillates from about 2.3 to 0.8 across. Therefore, it is useful to assess when these differences are statistically significant, a characteristic that is uniquely dependent on the sample size, distribution, and level of dispersion of CR datasets under consideration. To this end, hypothesis tests are carried out as described below.
Hypothesis tests on mean C
R response
In this sub-section, a series of hypothesis tests are conducted to assess if the differences observed in the mean demand of CR due to the action of FF and NF records, are statistically significant. These effects were quantified and discussed above through the computation of the mean CR ratio. If pulse effects present in NF records have no influence, the median demand of CR should be similar to the median demand imposed by a record set without pulse effects, which in this study corresponds to the FF records set. Among the statistical tests available (e.g., t-test and Wilcoxon-test), the Z-score method proposed by Zhou et el. (1997) was chosen for comparing the means of two independent log-normally distributed samples. This is a likelihood test that requires knowledge of the parametric distributions of the data. If both samples of interest are distributed as log-normal, the logarithm of the outcomes (i.e., data points) are normally distributed, such that:
$$\log X_{i} \sim N\left( {\mu_{1} ,\sigma_{1}^{2} } \right),\;\log Y_{i} \sim N\left( {\mu_{2} ,\sigma_{2}^{2} } \right)$$
(3)
and whose corresponding means are M1 and M2 respectively. Hence, the null hypothesis of both mean responses being virtually equal is represented as:
$$H_{0} :M_{1} = M_{2}$$
(4)
To accept or reject the null hypothesis, a significance level \(\alpha\) is defined, which is the probability of rejecting it (i.e., wrongly), given that the null hypothesis was in fact correct. In this case, the estimators for \(\mu_{1}\) and \(\mu_{2}\) are defined as follows:
$$\hat{\mu }_{1} = \frac{1}{{n_{1} }}\mathop \sum \limits_{i = 1}^{{n_{1} }} \log X_{i} ,\;\hat{\mu }_{2} = \frac{1}{{n_{2} }}\mathop \sum \limits_{i = 1}^{{n_{2} }} \log Y_{i}$$
(5)
and the unbiased estimators of \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) are defined by:
$$S_{1}^{2} = \frac{1}{{n_{1} - 1}}\mathop \sum \limits_{i = 1}^{{n_{1} }} \left( {\log X_{i} - \hat{\mu }_{1} } \right)^{2} ,\;S_{2}^{2} = \frac{1}{{n_{2} - 1}}\mathop \sum \limits_{i = 1}^{{n_{2} }} \left( {\log Y_{i} - \hat{\mu }_{2} } \right)^{2}$$
(6)
Since the null hypothesis \(H_{0} :M_{1} = M_{2}\) is equivalent to testing \(\mu_{1} + \left( {1/2} \right)\sigma_{1}^{2} = \mu_{2} + \left( {1/2} \right)\sigma_{2}^{2}\), due to the innate relationship between a normal and a log-normal distribution. Thus, a test can be derived from \(\log \hat{M}_{1} - \log \hat{M}_{2}\), where \(\log \hat{M}_{k} = \hat{\mu }_{k} + S_{k}^{2} /2\). Given that \(\hat{\mu }_{k}\) and \(S_{k}^{2}\) are independent, \(\hat{\mu }_{k} \sim N\left( {\mu _{k} ,\sigma _{k}^{2} /n_{k} } \right)\) and \(\left( {n_{k} - 1} \right)S_{k}^{2} /\sigma _{k}^{2} \sim {\rm X}^{2}\) with \((n_{k} - 1)\) degrees of freedom, this results in:
$${\text{var}} \left( {\hat{\mu }_{2} + \left( \frac{1}{2} \right)S_{2}^{2} - \hat{\mu }_{1} - \left( \frac{1}{2} \right)S_{1}^{2} } \right) = \frac{{\sigma_{1}^{2} }}{{n_{1} }} + \frac{{\sigma_{2}^{4} }}{{2\left( {n_{2} - 1} \right)}} + \frac{{\sigma_{2}^{2} }}{{n_{2} }} + \frac{{\sigma_{1}^{4} }}{{2\left( {n_{1} - 1} \right)}}$$
(7)
and by replacing \(\sigma_{k}^{2}\) by the unbiased estimator \(S_{k}^{2}\) (see Eq. 6), the Z-score test is obtained, whose distribution is approximately normal under \(H_{0}\) when \(n_{1}\) and \(n_{2}\) are large, as follows:
$$Z = \frac{{\hat{\mu }_{2} - \hat{\mu }_{1} + \left( {1/2} \right)\left( {S_{2}^{2} - S_{1}^{2} } \right)}}{{\sqrt {\frac{{S_{1}^{2} }}{{n_{1} }} + \frac{{S_{2}^{2} }}{{n_{2} }}} + \left( {1/2} \right)\left( {\frac{{S_{1}^{4} }}{{n_{1} - 1}} + \frac{{S_{2}^{4} }}{{n_{2} - 1}}} \right)}}$$
(8)
Given that the null hypothesis and the Z-score are defined, the statistical tests can be conducted. Naturally, the next step is to define the CR datasets to be tested. The CR response of the NSCs on the roof of each building was selected to perform the tests, for a given structural period T of each SDOF NSC (i.e., 80 oscillators) as shown in Fig. 15. Importantly, the normalised version of T (i.e., T/T1) that was shown to be better at characterising the behaviour of CR might be indistinctly used here unaffecting the results of the test, as is the same for both NSCs roof responses. Subsequently, a significance level equal to “” is considered, which for a normal distribution corresponds to 0.0455 or approximately 5%. In other words, to accept the null hypothesis that postulates that a pair of set means are virtually identical, the Z-score has to be lower than 2.0. If this is higher than 2.0, the null hypothesis is rejected, meaning that at the 5% significance level, the difference between the means is statistically significant, leading to the conclusion that NF with or without directivity effect is significant. It is important to restate that for a given level of relative strength demand, both at building level (i.e., Rnsc = 1,2) and NSC level (i.e., Rnsc = 1…6), the ground motion records of the 2 sets of records under consideration are scaled based on the yield to induce the same Sa level.
Figure 16 shows the resulting Z-score of the statistical test conducted on the same case shown in Fig. 15, which is presented versus the structural period of the NSC. In this particular case, it is clear that in the short spectral region (shaded in the figure) where T < T1, the null hypothesis is rejected in most NSC cases. The same is observed in the longer spectral region, for between 2.5 s and 3.5 s. These observations confirm the important T-dependant behaviour, particularly the forced dynamic response in the short-spectral region (i.e., T < T1), hence the need to separately assess the behaviour of the inelastic displacement ratio across spectral regions. However, the rate of cases where the null hypothesis is rejected, amongst the full range of 80 structural periods, can be used to gain insight into the statistically significant influence of NF excitations with or without velocity pulses, over the entire set of buildings and relative strength levels of demand. For instance, the case shown in Fig. 16 has a global null hypothesis rejection rate of 47.5%, derived from the total number of rejections across T, that reaches 38 in this case (filled out in the figure), over the total amount of structural systems, which amounts to 80. Similarly, a local null hypothesis rejection rate of the short period can be estimated, as the ratio of rejections in the shaded region (which varies per each building) to the total number of NSCs below T1, that in this particular case amounts to 80%.
In the following, results of null hypothesis rejection rates are presented, among the FF set and NF with and without directivity effects. The first conducted test is between the FF record set and the NF set. The results are shown in Fig. 17 in terms of the global rate, for every structural building (i.e., 54 steel frames) sorted by their fundamental period T1 and every levels of lateral demand imposed on the NSCs mounted only on the roof of the buildings. The first observation is the virtually zero value of the global rejection rate for the elastic case of NSCs, when Rnsc = 1, across all buildings, confirming the practically identical response of CR under both sets of records. These correspond to the black dots at the bottom of the plot for the zero ordinate. Secondly, the overall global rate is always lower than 50%, implying that the statistically significant difference in the mean of the CR demand is neither constant nor governing along the whole range of NSC vibration periods T (see Fig. 16), for a particular building at a given level of NSC lateral demand. Thirdly, the global rate seems to increase modestly for higher levels of Rnsc, indicating that CR central tendencies are significantly different, from a statistical viewpoint. This is coupled with the fact that as Rnsc increases, the dispersion of both lognormal distributions examined augment as well.
It is noticeable that the behaviour of the global rate is not uniform either across the structural periods or Rnsc—see for example the case of buildings with periods within 1.20 s to 1.50 s, where the rate of null hypothesis rejection is considerably higher than the group of more flexible systems immediately next to it on the right-hand side (i.e., 1.50 s < T1 < 1.90 s). Considering the tests conducted between the FF and the NFP set, shown in Fig. 18, similar trends are observed but with an increase in the statistically significant differences among the means of CR between the sets, as expected. Hence the same observations noted above apply in this case. By focussing only to the short spectral region by means of the local rate, it can be seen in Fig. 19 that the difference between the means of CR is remarkably larger, even reaching 100% in the case of buildings with short periods. In other words, the differences among CR means are statistically significant for all NSC elements whose period of vibration T are shorter than the building fundamental period T1. Overall, the differences in the mean CR are statistically significant, across all the range of primary structures especially in the case of short period systems, and are evident even for the lowest level of relative lateral strength imposed on the NSCs (i.e., Rnsc = 2).