Displacement Ratios per Ductility Level

In this section, the overall results are discussed for the displacement ratios given by the two response spectra definitions (RotD50 and RotD100) and their variability at each ductility level as a function of effective period (See Figure 5.8a-g). Each point shown in the figures corresponds to the mean of 65 RC columns, meaning that per figure, 3,250 structures are shown for each definition.

Given that limit states can be based on the capacity of the system to deform, it is worth looking at how the response ratios change as the ductility level increases. Figure 5.8a depicts the displacement ratios given by the median and maximum spectra definition across the observed range of periods (1s to 5s) at a ductility level equal to 1. The RotD50 displacement ratios are shown to remain approximately around a value of 1.25, which is very close to the geometric mean value for RotD50 across all periods of 1.27. In contrast, the averaged maximum displacements for RotD100 seem to coincide with expected response values, for an ideal ratio of one. As shown, the coefficients of variation for both definitions are very small across the period range (< 0.1), nevertheless, on average the variation for the RotD50 ratios is approximately seven times higher than for RotD100 ratios. From this, it can be affirmed that designing structures to a ductility level equal to 1 should be done employing the maximum response spectrum definition (RotD100) because the structure will consistently achieve the intended design performance. However, if a structure is designed to a 𝜇 = 1 using the median response spectrum (RotD50), on average the response will exceed the target by 27%.

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Figure 5.8: Displacement ratios and coefficient of variation per ductility level as a function of period for the median (RotD50) and maximum (RotD100) response spectra definitions.

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Displacement Ratios and corresponding variation as a function of period for μ = 1

Displacement Ratios and corresponding variation as a function of period for μ = 1.5

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Displacement Ratios and corresponding variation as a function of period for μ = 3

Displacement Ratios and corresponding variation as a function of period for μ = 4

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98 For 𝜇 = 1.5 (see Figure 5.8b) the variability in the displacement ratios for both definitions increases as expected given that the structures respond inelastically. Furthermore, there is variability in the ratios as a function of period, unlike results at 𝜇 = 1. At this level, the averaged RotD100 displacements can be said be slightly non-conservative (Δ_𝑚𝑎𝑥 > Δ_𝑡) as they fluctuate above a ratio of one. Similarly, the averaged RotD50 displacement ratios show an increase throughout the observed period range with respect to the previous ductility level (𝜇 = 1). The geometric mean of the displacement ratios across periods is equal to 1.3 and 1.03, for RotD50 and RotD100 respectively, which is similar to the values observed at a displacement ductility of 1 (1.27 and 1). However, to coefficients of variation have increased compared to the results at a displacement ductility of 1.

For higher ductility levels (2 ≤ 𝜇 ≤ 8), shown in Figure 5.8b through Figure 5.8g, the displacement ratios for both definitions start to continuously decrease while their COVs increase. This decrease between ductility levels is more prominent for the RotD100 displacement ratios than for the RotD50 displacement ratios. For the six different ductility levels (𝜇 = 2,3,4,6,8) the ratios start to show trends in the period range. At shorter periods (𝑇𝑒𝑓𝑓 < 1.75𝑠) the RotD100

displacement ratios display a negative slope with lower values than for the rest of the period range. A brief increase in the RotD100 ratios occurs for effective periods around 2 seconds, which is followed by declining portion that remains constant for 2.25𝑠 < 𝑇𝑒𝑓𝑓 < 3.25𝑠. At longer periods

(3.25𝑠 < 𝑇_𝑒𝑓𝑓 > 5𝑠) the ratios start to increase once more. Furthermore, at 𝜇 > 4, the variability

at shorter periods remains higher than for the rest of the period range. It is inferred that the trends observed at shorter periods are due an underestimation of the equivalent viscous damping for structures with 𝑇_𝑒𝑓𝑓 approximating one second. Dwairi et al. (2007) proposed a model for

99 there was some minor dependence at periods between 1 and 2, it was ignored in their model for simplicity. Conversely, at higher levels of ductility, it may be that the equivalent viscous damping is being slightly overestimated, as the RotD100 results drop below a ratio of 1. This, among other factors that affect variability in the results are discussed in section 5.4. Nevertheless, it is worth noting that overall, the coefficients of variation are considered low throughout all ductility levels for the period range observed (COV < 0.4), which grants confidence to the results obtained.

Figure 5.9: Geometric mean and coefficient of variation of the displacement ratios as a function of ductility level for the two response spectra definitions, RotD50 and RotD100.

Figure 5.9 presents the geometric means and coefficients of variation as a function of ductility level for both response spectra definitions, RotD50 and RotD100. This confirms the previous observations noting that displacement ratios for both definitions decrease as the ductility levels increases with an initial increase at ductility equal to 1.5. Similarly, it can be observed that the highest variability is focused at ductility levels below 4, after which it appears to stabilize. Nonetheless, the period dependency of the results is lost in Figure 5.9.

Another way in which the results can be evaluated is by using cumulative distribution functions (CDF). The empirical distribution of the data, separated by response spectra definition,

100 for each ductility level is shown in Figure 5.10. The arrow points towards increases in ductility level, which is also illustrated by the lightening of the hue.

Distribution across ductility levels for RotD50 Distribution across ductility levels for RotD100

Figure 5.10: Empirical data distribution per ductility level for displacement ratios given by each response spectra definition, RotD50 and RotD100. Each curve corresponds to displacement ratios of 3,250 RC

columns.

Distribution across ductility levels for RotD50 Distribution across ductility levels for RotD100

Figure 5.11: Normal distribution per ductility level for displacement ratios given by each response spectra definition, RotD50 and RotD100. Each curve corresponds to displacement ratios of 3,250 RC columns.

As seen in Figure 5.10, the displacement ratios for both definitions fall under a normal distribution. The RotD100 CDF for 𝜇 = 1, agrees with previous observations showing that nearly all structures exhibit a ratio equal to one. This highlights that there is very small variability for the maximum definition of the elastic response spectrum (𝜉 = 5%). As mentioned in Section 5.1, some of this variability may be due to the significant figures for the very small amplitude spectra

101 designs.. In contrast, the RotD50 CDF at 𝜇 = 1 shows that for this ductility level the maximum response exceeded the target response, which is true for all results throughout all ductility levels observed (22,750 RC columns).

Furthermore, it can be seen that for both definitions the spread of the results between ductility levels starts to decrease as the ductility level increases. This is denoted by the curves at high ductility levels (4 ≤ 𝜇 ≤ 8) being very close to each other, which is more acute for the RotD50 displacement ratios. The smooth cumulative distribution of the displacement ratios for the maximum and median response spectra definitions are presented in Figure 5.11.

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