introduction - university of manchester · web view* corresponding author: hector guerrero, school...
TRANSCRIPT
Residual displacement demands of conventional and dual oscillators subjected to earthquake ground motions characteristic of the soft
soils of Mexico City
Héctor Guerreroa,*, Jorge Ruiz-Garcíab and Tianjian Jia
a School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Pariser Building, Manchester, M13 9PL, UK
b Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo Edificio C, Planta Baja, Cd. Universitaria, 58040 Morelia, México
Abstract
This paper presents a statistical evaluation of residual displacement (RDs) demands in conventional and dual single-degree-of-freedom (SDOF) oscillators subjected to a set of 220 earthquake ground motions recorded at soft soil sites of Mexico City. A dual SDOF oscillator represents a highly-dissipating energy system (e.g. buckling-restrained braces, BRBs) acting in parallel with a conventional system (e.g. flexible moment-resisting frames). To provide a context, RDs were normalised with respect to the corresponding: a) maximum transient displacements, and b) elastic spectral displacement. The effects of post-yield stiffness ratio (as affected by P- effects), normalised period of vibration with respect to the predominant period of the ground motion, the type of hysteretic response, maximum displacement ductility, lateral strength ratio, type of transition (from elastic-to-plastic response), and damping ratio on normalised RDs were examined for the conventional oscillators. In addition, the effects of the stiffness ratio, lateral strength ratio, displacement ductility, post-yielding stiffness ratio, and type of hysteretic response of the primary and secondary parts of dual SDOF oscillators on normalised RDs were also evaluated. From the results of this investigation, it was observed that for dual SDOF systems the amplitude of normalised RDs is small when the primary part remains elastic. On the contrary, if the primary part exhibits inelastic response, RDs might increase significantly and the post-yielding stiffness ratio of the secondary part plays a key role for constraining them (i.e. while a positive value reduces RDs significantly; a negative value is highly detrimental). Also, it was found that the type of hysteretic response of the primary part of a dual system has a significant effect on RDs (e.g. it was found that a primary part with self-centring capacity, acting in parallel with a highly dissipation system (e.g. BRBs) as secondary part, is very effective for diminishing RDs). A discussion section is also offered to highlight the new findings of this study and differences on normalised RD demands between soft and stiff soils.
Keywords: residual displacements; dual systems; soft-soil earthquake ground motions; post-yielding stiffness ratio; Buckling-Restrained Braces (BRBs).
* Corresponding author: Hector Guerrero, School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Pariser Building C16, Manchester, UK, M13 9PL, [email protected]
1. IntroductionThe importance of evaluating residual displacement, or residual drift, demands as a
performance parameter to decide whether a building should be repaired or demolished after an earthquake event has been recognized in the up-to-date guidelines for the seismic performance assessment of buildings in the US [1]. Furthermore, Ramirez and Miranda [2] highlighted that economic losses due to earthquakes can be underestimated if the occurrence of excessive post-mainshock residual drifts leading to demolition are neglected in an earthquake loss assessment.
Several studies have focused in assessing the factors affecting the amplitude of residual displacements (RDs) of conventional single-degree-of-freedom (SDOF) oscillators (e.g. [3-6]), and some of them also proposed empirical expressions to estimate RDs for simple structures that can be represented as SDOF systems. For instance, MacRae and Kawashima [3] conducted a parametric study in 2%-damped SDOF bilinear oscillators subjected to 11 ground motions recorded in stiff, medium, and soft soils. The authors found that RDs are highly dependent on the post-yielding stiffness ratio, r (i.e. a negative r produced very high RDs while a positive r produced small RDs). They also reported that the type of ground motion, period of vibration, and peak target ductility did not have a significant effect on RDs. From their results, they proposed a method to estimate RDs for simple structures with different values of r, a damping ratio of 2%, and a target ductility demand, , equal to 4. Years later, Christopoulos et al. [4] conducted a study to evaluate RDs, measured as the ratio of residual-to-maximum displacement demands, of four equivalent 5%-damped SDOF oscillators representative of four reinforced concrete buildings subjected to 20 spectrum-matched earthquake ground motions. Three hysteretic models were used in their study namely: 1) elastoplastic, representative of steel framed structures, 2) Takeda [7], representative of reinforced concrete framed structures, and 3) Flag-Shape [8, 9], representative of structures with self-centring capacity. Also, post-yielding stiffness ratios between -0.10 and 0.10 were evaluated. The authors found that the type of hysteretic model, post-yielding stiffness ratio, and seismic intensity affect both RDs and their dispersions significantly. Particularly, they concluded that structures with self-centring capacity significantly constrained RDs with respect to structures having elastoplastic and Takeda-type global hysteretic behaviours, while similar displacement ductility demands were observed in the three hysteretic models. After that, Ruiz-García and Miranda [5, 10] proposed an equation to estimate a dimensionless parameter called residual displacement ratio, Cr, which allows the direct estimation of RDs of elastoplastic SDOF oscillators from elastic spectral displacement ordinates. The fitted parameters for the introduced equation were obtained from regression analysis using the statistical response of 5%-damped elastoplastic SDOF oscillators subjected to 240 ground motions recorded in firm sites of California, USA. According with the equation proposed by these authors, Cr was a function of the period of vibration (T), the lateral strength ratio (Ry), and the type of soil. A similar study was carried out by Ruiz-García and Miranda [10] to compute Cr
ratios from the response of elastoplastic and stiffness-degrading SDOF systems subjected to 40 near-fault earthquake ground motions having forward-directivity effects. They concluded that the spectral shape of Cr is influenced by the pulse period, Tp, identified in the ground velocity waveform, the peak ground velocity, and the level of unloading stiffness in the stiffness-degrading hysteretic models. Very recently, Liossatou and Fardis [6, 11] conducted an extensive statistical study on residual displacements, described by Cr and residual/maximum ratios, of SDOF systems using three types of hysteretic models representative of reinforced concrete (RC) structures, namely: good quality, fair quality, and poor quality RC construction. The results were compared to the results from bilinear models without stiffness and strength degradation. The
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most significant conclusion of the study was that RC oscillators present smaller Cr and residual/maximum ratios than the bilinear oscillators. The RDs were from larger to smaller in the following order: bilinear, RC of good quality, RC of fair quality and RC of poor quality. They also reported that velocity pulses of the ground motions increase both, residual and peak inelastic displacements in similar proportion, and they confirmed that the amplitude of Cr ratios depends on Tp. Other studies developed in refs. [12-14] proposed simple empirical equations to estimate maximum displacements from residual displacement demands. Certainly, these equations can also be used to obtain residual displacements from maximum displacement demands. Although useful, these studies were conducted for soil classes A to D, D, and B, respectively; which significantly differ from the soil conditions found on Mexico City (objective of this study).
After searching the literature, it has been observed that RDs are dependent on the post-yielding stiffness ratio, the period of vibration, the lateral strength ratio, the level of ductility demand, the type of soil, and the type of hysteretic behaviour [3-5, 10, 11]. However, there is still very limited information about the effects of the aforementioned parameters on RDs triggered in structural systems subjected to earthquake ground motions recorded in very soft soil sites (i.e. narrow-band motions characterised by long predominant periods, low frequency content, and long duration) such as those found in Mexico City, and only the work of Bojórquez and Ruiz-García [15] examined residual drifts demands. In their study, the authors developed peak and residual inter-story drift demand hazard curves for four typical steel moment-resisting frames designed according to the 2004 Mexico City Building Code [16] when subjected to 30 records gathered in accelerographic stations located in the lake-bed zone of Mexico City. They found that the amplitude of residual drift demands strongly depended on the members’ post-yield stiffness ratio (e.g. increasing post-yield stiffness ratio from 1% to 3% decreased residual drift demands as the seismic intensity increased). They also reported that if the case-study frames would experience maximum inter-storey drifts around 3%, they also could exhibit residual inter-story drifts of approximately 1%, which may lead them to demolition.
Dual structural systems consisting of highly-dissipating energy systems (e.g. buckling-restrained braced frames) acting in parallel with a conventional system (e.g. flexible moment-resisting frames) such as those proposed in refs. [17-19] are very attractive in highly active earthquake-prone regions such as Mexico City. However, they might be subjected to large residual displacement demands due to the wide hysteretic features of the energy-dissipating elements [20, 21]. As an example, Kiggins and Uang [21] concluded that BRBFs acting as lateral-load resisting system might experience large residual displacement demands, and that they should incorporate moment-resisting frames acting as a restoring system to be designed as a dual system. This observation was confirmed in subsequent studies carried out by Pettinga et al. [22], Terán-Gilmore et al. [23], as well as Sahoo and Chao [24]. Therefore, the amplitude of residual displacement demands of such dual structural systems in buildings built on soft soil conditions should be investigated before their practical implementation.
The objective of this paper is to present main results of a statistical study focused on evaluating residual displacement demands, in terms of the residual-to-maximum displacement ratios and Cr ratios, for simple structures representative of conventional and dual structural systems, subjected to a relatively large set of 220 accelerograms recorded in Mexico City. Since practical applications may require residual displacement demands in absolute terms, they can easily be obtained by multiplying the results presented in this paper by the corresponding maximum displacement demands (which may be calculated from [25]) or the elastic spectral displacement ordinates (which are commonly available to designers), as appropriate. It is worth
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to highlight that, in this study, the response of conventional systems refers to the response of SDOF oscillators, while the response of dual systems is related to the response of coupled primary and secondary SDOF systems acting in parallel. The latter systems are aimed at representing highly-dissipating energy systems (e.g. buckling-restrained braced frames) acting in parallel with a conventional system (e.g. flexible moment-resisting frames) such as those proposed in [23]. In this paper and to be consistent with [26], the primary part of the dual system refers to the main structural system (e.g. moment-resisting frames) while the secondary part refers to the energy dissipation system.
Although this work introduces new findings on the subject, it is constrained to the response of SDOF oscillators. However, it should be recognised that residual displacements and residual inter-storey drifts in multiple-degree-of freedom systems (e.g. multi-storey buildings) depend on additional variables such as higher-mode effects, P- effects, type of sway mechanism, and type of member hysteresis characteristics (e.g. [27, 28]) that are not accounted for in SDOF oscillators. These effects can be approximately taken into account by scaling the RDs of SDOF oscillators (e.g. those presented in this study) using amplification factors, such as those discussed in refs. [27, 28] for moment-resisting frames. A study to estimate amplification factors in multi-storey structures equipped with BRBs is the subject of a further study and the results will be available in the future. Another consideration, that the reader must be aware of, is that soil-structure interaction (SSI) effects have been neglected in this study and the results presented should be interpreted in that light. Certainly, stiff structures located in soft soils tend to be affected by these effects. The issue will be addressed in a future paper.
2. Earthquake ground motions and measures of residual displacements To obtain measures of residual displacement demands, a set of 220 earthquake ground
motions recorded in the lake-bed zone of Mexico City (i.e. soil type III, as defined by the Mexico City Building Code [16]) were used in this study. The records were gathered from far-field events (having more than 300 km of epicentral distance), occurred between 1985 and 2012 in the subduction zone of the Pacific Coast of Mexico, with seismic wave magnitudes between 6.3 and 8.1. The predominant period of each earthquake ground motion, Tg, was defined as the period at which the 5%-damped pseudo-acceleration spectrum reaches its peak, which ranges between 1.3 and 3 seconds for the records included in this dataset. More details of the ground motion set can be found in ref. [29].
To facilitate comparison with results from prior studies, the residual displacement demands are presented as normalised residual displacements with respect to the maximum displacement demands (i.e. residual/maximum displacement ratio), or as Cr ratios in this paper. Two statistical measures were computed, while the central tendency is measured by the sample mean (calculated by averaging the result data), the dispersion was computed from the coefficient of variation.
3. Conventional OscillatorsIn this section, the effects of different parameters on central tendency and dispersion of
residual displacements demands for conventional SDOF oscillators are evaluated when subjected to narrow-band earthquake ground motions gathered in soft soil sites of Mexico City. The considered parameters are the period of vibration normalised by the predominant period of the ground motion, the post-yielding stiffness ratio (as affected by P- effects), the type of global hysteretic response, the level of maximum displacement ductility, the strength reduction factor,
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the damping ratio, and the type of transition from elastic-to-plastic response. For convenience, all the calculations were conducted using the authors’ subroutines, which were calibrated using the computer software Degtra [30]. It is also important to highlight that a zero-acceleration segment of 40 seconds was added at the end of each ground motion to measure the residual displacements after stabilisation of the response.
3.1 Normalised period of vibration, T/Tg
Prior studies on the estimation of displacement demands from oscillators subjected to narrow-band earthquake ground motions have noted that a better characterisation of demands is found when the period of vibration of the system, T, is normalised with respect to the predominant period of the ground motion, Tg [31]. Therefore, the main results of normalised RDs are shown as a function of the T/Tg ratio. For instance, Figure 1a shows the spectral variation of mean normalised RDs from conventional SDOF oscillators having a range of T/Tg ratios between 0.2 and 3, damping ratio of 5%, target maximum ductility demand of 2, and assuming zero post-yielding stiffness ratio (i.e. r=0%). In general, it can be seen that normalised RDs are not constant over the whole spectral region, while three spectral regions can be identified. In the first region, corresponding to T/Tg ratio smaller than about 0.7, the normalised RDs tend to increase as the normalized period decreases. In the second spectral region, corresponding to T/Tg ratios between 0.7 and 1.5, normalised RDs decrease which is particularly true for periods of vibration close to Tg. A third region can be identified, for T/Tg ratios greater than about 1.5, where the normalised RDs tend to be follow a constant trend. It should be noted that the spectral variation of normalised RDs for soft soil sites is very different from that shown for earthquake ground motions recorded in firm soil sites, where the normalized RDs do not significantly changes with variations of the period of vibration as illustrated in ref. [5]. Regarding the dispersion the coefficient of variation reached values between 0.5 and 0.6, but no clear tendency is appreciated as shown in Figure 1b.
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a) b)Figure 1. Effects of T/Tg ratio on normalised residual displacements: a) mean, and b) coefficient of
variation
3.2 Effect of post-yielding stiffness ratioPost-yielding stiffness ratio, r, is one of the most important parameters affecting the amplitude
of RDs [3, 4, 8]. As noted in [32], P- effects might affect the hysteretic characteristics of inelastic systems and, particularly, the value of r in the force-displacement relationship. When P- effects are important, the value of r can be significantly reduced up to negative values. Therefore, P- effects are approximately taken into account through the values of r in this study.
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For instance, Figure 2 shows the lateral displacement time-history response of conventional oscillators subjected to the East-West component of the record gathered at SCT station during the September 19, 1985, Michoacan earthquake. Three values of r were modelled in each oscillator (r=-5%, 0%, and 5%). The period of vibration, viscous damping ratio, and yield strength coefficient Cy (i.e. yield lateral strength normalized with respect to the weight), of the oscillators were chosen as 0.5s (considered as representative of a short-period structure), 5%, and 0.16, respectively. From the figure, it can be observed that r affects significantly the amplitude of RDs, which is consistent with previous studies (e.g. [3, 22]).
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r = -5%r = 0%r = 5%
Parameters: T=0.5s, x=5%, Cy=0.16
Figure 2. Response of SDOF systems with three different post-yielding stiffness ratios subjected to the SCT-EW earthquake ground motion recorded during the 1985 Michoacan earthquake.
To further illustrate the effect of the post-yield stiffness ratio in the first spectral region identified in Figure 1a, the response of conventional bilinear oscillators was estimated considering a target maximum ductility of 2 and periods of vibration, T, equal to 0.25Tg, 0.5Tg, and 0.75Tg. For example, if Tg is about 2 seconds, the periods of the oscillators correspond to 0.5, 1.0, and 1.5 seconds, respectively. A damping ratio of 5% was considered for the three cases. Figure 3 shows the sample mean and coefficient of variation of the normalised RDs for different post-yielding stiffness ratios. In general, it can be appreciated that the normalised RDs are highly dependent on the level of post-yielding stiffness ratio (i.e. the higher the post-yielding stiffness, the smaller the normalised RDs). Especially, values of post-yield stiffness ratio higher than about 5% or 10% seem to be convenient to reduce normalised RDs. On the other hand, the coefficient of variation increased significantly as the post-yielding stiffness ratio increased, reaching values higher than 0.6.
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a) b)Figure 3. Normalised residual displacements for conventional oscillators: a) mean, and b) coefficient of
variation.
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3.3 Effect of hysteretic responseThe response of SDOF oscillators was studied using three different hysteretic models shown
in Figure 4: a) elastoplastic (representative of steel framed structures), b) Takeda [7] (representative of well-detailed reinforced concrete framed structures), and c) Flag-shape [8](representative of structures with self-centring capacity). For the Takeda model (Figure 4b), the parameters =0.0 and =0.4 (as defined in the figure) were considered, which are commonly accepted values for reinforced concrete framed structures [31]. For the flag-shaped model (Figure 4c), =0.7 (also defined in the figure) was assumed as a representative value of self-centring systems [4].
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a) Elastoplastic b) Takeda [7] c) Flag-shaped [8]Figure 4. Types of hysteretic response.
3.3.1 Oscillators with elastic-perfectly plastic behaviourIn this subsection, the three hysteretic models of Figure 4 were analysed considering zero
post-yielding stiffness ratio (r=0%). Additionally, a damping ratio of 5% and target maximum ductility demand of 2.0 were considered. Figure 5a shows the effect of the type of hysteretic behaviour on mean normalised RDs, where it can be seen that the type of hysteretic response significantly affects the amplitude of normalised RDs. In general, the elastoplastic hysteretic response leads to normalised RDs about three times larger than those obtained from the Takeda hysteretic response. However, it is appreciated that the Takeda response generally presented slightly higher coefficients of variation than the elastoplastic response in the whole period range as shown in Figure 5b. The coefficients of variation of the flag-shaped response are not shown in the figure because all the residual displacements were zero.
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a) b)Figure 5. Effect of the type of hysteretic response on normalised residual displacements: a) mean, and b)
coefficient of variation.
3.3.2 Oscillators with post-yielding stiffness ratio different from zeroNow, this subsection presents the results for the three hysteretic models of Figure 4 for post-yielding stiffness ratios different from zero (i.e. r=5% and r=-5%). Again, a damping ratio of 5% and target maximum ductility demand of 2 were considered. Figure 6 shows the results for normalised periods between 0.2 s and 3 s. Comparison of figures 5 and 6 shows that the effects of positive and negative post-yielding stiffness ratios are negligible for oscillators with Takeda and Flag-shaped hysteretic behaviours. However, the effects on normalised RDs for oscillators with bilinear behaviour is very significant in both mean values and coefficients of variation. It is seen that a positive r tends to reduce the mean normalised RDs, especially for normalised periods around unity. The opposite is observed for a negative r. Regarding the coefficient of variation, a positive r produces larger values; while a negative r generates significantly smaller coefficients.
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c) d)Figure 6. Effect of post-yielding stiffness ratio and the type of hysteretic response on normalised residual
displacements.
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3.4 Effect of maximum displacement ductility The influence of maximum displacement ductility is examined for conventional oscillators
having elastic-perfectly plastic behaviour (i.e. r=0), followed by a discussion about results of bilinear oscillators with post-yielding stiffness ratio different from zero.
3.4.1 Oscillators with elastic-perfectly plastic behaviourFigure 7a shows the mean normalised residual displacements for oscillators having five
levels of maximum displacement ductility (=1.5, 2, 3, 4, and 6) with r=0%. As it can be observed, mean normalised RDs for all levels of keep the spectral trend illustrated in Figure1a; however, the amplitude of mean normalised RDs increases as the level of increases. On the other hand, coefficients of variation corresponding to different levels of follow, in general, a similar trend, with values between 0.5 and 0.6.
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a) b)Figure 7. Effect of maximum displacement ductility in normalised residual displacements for elastoplastic
oscillators: a) mean, and b) coefficient of variation
3.4.2 Oscillators with post-yielding stiffness ratio different from zeroIn order to illustrate the variations of the normalised RDs with respect to when r is different
from zero, one oscillator with normalized period of vibration of 0.25Tg, damping ratio of 5% and target ductility demands of 2, 4, and 6 was analysed under the same set of soft soil records. This period was selected as representative of short-period structures, on which normalised RDs tend to be large (as identified from Figure 7). Figure 8a shows that, opposite to the case of r=0% (see previous section), the normalised RDs do not always increase as the ductility demand increases. In fact, for post-yielding stiffness ratios greater than 2.5% the normalised RDs decreased as the ductility demand increased, while the opposite was true for r < 2.5%. It is worth to note that the dispersion increased significantly as the value of r increased as shown in Figure 8b. In order to analyse the observed effects of r for other periods of vibration, mean and dispersions of normalised RD demands were calculated for r=5%. They are shown in Figure 9 for a normalised periods between 0.2 and 3.0, and ductility levels of 2, 4 and 6. It is seen that the higher the ductility level, the smaller the normalised residual displacements in the full period range analysed. Other two interesting observations, worthy to be highlighted, are: 1) the normalised RDs tend to be very small for normalised periods around unity; and 2) the coefficients of variation are larger than those of Figure 7b; which is consistent with Figure 8b. The results of
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Figures 8 and 9 highlight the importance of the post-yielding stiffness ratio on residual displacements because r is capable of inverting the effects of the ductility, , on normalised RDs. That is, while normalised RDs increase with the displacement ductility level for r<2.5%, they decrease with increasing displacement ductility for r>2.5%.
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a) b)Figure 8. Effect of displacement ductility and post-yielding stiffness ratio on normalised residual
displacements: a) mean, and b) coefficient of variation.
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a) b)Figure 9. Effect of displacement ductility and post-yielding stiffness ratio on normalised RDs for different
periods of vibration: a) mean, and b) coefficient of variation.
3.5 Effects of damping ratioSome structures could include supplemental damping devices, or passive systems that may
increase the damping ratio significantly–such as the case of buckling-restrained braces, BRBs, in framed buildings as found in recent experiments carried out by Guerrero [29]. Therefore, the effect of damping ratio on normalised RDs were also investigated in oscillators with damping ratios of 2%, 5%, 10%, and 20% having elastoplastic behaviour, and target ductility demand of 2. Figure 10a shows the mean of the normalised residual displacements, where it is appreciated that they are only slightly affected by variations in the damping ratio for normalised periods longer than unity (i.e. in this spectral region, the normalised RDs reduce slightly as the damping ratio increases). However, no significant differences are appreciated for normalised periods
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smaller than one. However, when the residual displacement demands are normalised by the mean of the 5%-damped maximum displacements (Figure 10c), it is clearly appreciated that the higher the damping ratio the smaller the residual displacements in the whole period range. This means that incrementing of the damping ratio provides the benefit of reducing both, residual and peak displacement demands. Regarding the dispersion, it can be said that no clear trend with changes of the damping ratio is appreciated, but most of the values of the coefficient of variation are between 0.5 and 0.6.
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c)Figure 10. Effects of damping ratio on normalised residual displacements: a) mean, b) dispersion, and c)
mean of residual displacement normalised by the 5%-damped max. displacements.
3.6 Effects of transition from elastic-to-plastic hysteretic responseGlobal force-displacement response in structures (e.g. buildings) shows smooth transition
between the elastic and plastic response since their elements do not yield at the same time. However, numerical modelling of simple structures is often simplified by assuming a bilinear hysteretic behaviour with a sharp transition from elastic-to-plastic response. Therefore, it is pertinent to evaluate the effect of smooth elastic-to-plastic transition. The effects of the transition from elastic-to-plastic hysteretic response were analysed using a simplified version of the Bouc-
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Wen hysteretic model [33]; which has been widely used for modelling the hysteretic behaviour of the restoring forces for non-linear oscillators [34]. In this model, the non-linear force-deformation relationship is given by the following relationship:
(1)
where t is the time, d is the displacement, r is the post-yielding stiffness ratio, k is the elastic stiffness, and z(t) is a hysteretic parameter, usually called hysteretic displacement, given by the following differential equation:
(2)
where A, , and n are dimensionless parameters that control the shape of the hysteresis loop. n is the parameter that controls the transition from elastic-to-plastic response (a small value, say n=3, provides a very smooth transition, while a large value, say n=100, provides a sharp transition); A, and were considered 1.0, 0.5 and 0.5, respectively, in this study, as proposed in [35].
Three transitions were simulated in the global oscillator’s hysteretic response as illustrated in Figure 11: a) sharp (i.e. bilinear response), b) smooth, and c) very smooth transitions, which correspond to yielding exponents n=100, 7, and 3, respectively. Values of target ductility demand of 2.0 and r=5% were considered for the three cases. It should be noted that the effects of the type of transition are only significant for reduced levels of ductility (e.g. ≤ 2) since these effects on highly nonlinear systems may turn to be negligible.
0.0
0.5
1.0
1.5
0.0 1.0 2.0 3.0 4.0Forc
e / Y
ield
ing
forc
e
Ductility
n=100n=7n=3
Figure 11. Transition from elastic-to-plastic hysteretic response with r=5%
The mean and coefficient of variation of normalised RDs were computed for the three transitions and the spectral trends are shown in Figure 12. From the left figure, it is appreciated that the amplitude of normalised RDs are affected by the type of transition in the whole normalised period range. The very smooth transition led to smaller values than the sharp and the smooth transitions, which is particularly true for normalised periods close to unity, and the differences increases for normalized periods longer than 2. On the other hand, the dispersion was very high in the three cases, with higher values for the smooth and very smooth transitions. Except for normalised periods in the range of about 0.75Tg and 1.5Tg, values of the coefficient of variation were around 0.8.
12
The combined influence of the type of transition and r was further examined in this study. For example, the mean of the normalized RDs of an oscillator with period of vibration of 0.25Tg, damping ratio of 5%, target ductility demand of 2, and post-yielding stiffness ratios between -0.1 and 0.1 is shown in Figure 13a. It is observed that, with exception of post-yielding stiffness ratio close to zero, the normalised residual displacements are highly influenced by the type of transition. In the positive range of r, the smoother the transition type, the smaller the normalised RDs, while the opposite is true in the negative range of r. From Figure 13b, it can be seen that the coefficient of variation is larger for very smooth transition than for the other transitions for positive r, but it tends to decrease for the three types of transition for negative values of r.
0%
10%
20%
30%
0 0.5 1 1.5 2 2.5 3
Res
idua
l / m
ax. d
ispla
cem
ents
T / Tg
n=100n=7n=3
0
0.4
0.8
1.2
1.6
0 0.5 1 1.5 2 2.5 3
Coe
ffic
ient
of v
aria
tion
T / Tg
n=100n=7n=3
a) b)Figure 12. Effect of the type of transition from elastic-to-plastic response on normalised residual
displacements: a) mean, and b) coefficient of variation
0.0
0.2
0.4
0.6
0.8
-0.1 -0.05 0 0.05 0.1
Res
idua
l / m
ax. d
ispla
cem
ents
post-yielding stiffness ratio
0.0
0.2
0.4
0.6
0.8
1.0
-0.1 -0.05 0 0.05 0.1
Coe
ffic
ient
of v
aria
tion
post-yielding stiffness ratio
a) b)Figure 13. Influence of post-yielding stiffness ratio and the type of hysteretic transition on the normalised
residual displacements: a) mean, and b) coefficient of variation
3.7 Effects of lateral strength ratioFor seismic evaluation structures where the displacement ductility is not known, it is
convenient to compute residual displacement demands for SDOF systems with known lateral strength. For this purpose, the lateral strength ratio, Ry, is defined as the ratio of the strength required to maintain elastic a SDOF oscillator to its yielding strength. Therefore, Ry is a measure
13
of the lateral strength of the structure relative to the ground motion intensity [36], which is expressed as follows:
2e e
yy y
V m dR
V V
= =(3)
where Ve is the strength required to maintain an SDOF oscillator elastic, Vy is the yielding strength of the system, m is the mass, is the circular frequency of vibration and de is the corresponding spectral elastic displacement. The convention of Ruiz-García and Miranda [36]for strong systems (Ry ≤ 3.0) and weak systems (Ry > 3.0) is adopted here. Prior studies evaluated the influence of level of lateral strength ratio through the Cr ratio, defined as the ratio of residual displacements on elastoplastic SDOF oscillators to elastic spectral displacement ordinates [5, 6]. Therefore, for comparison purposes, the following subsections show the mean and dispersion of Cr ratios for five different levels of lateral strength ratios (Ry = 1.5, 2, 3, 4, and 6) on very soft soils.
3.7.1 Oscillators with elastic-perfectly plastic behaviourFive percent-damped elastic-perfectly plastic SDOF oscillators (i.e. r=0%) with normalised
periods, T/Tg, between 0.2 and 3 were subjected to the 220 ground motions described previously. Figure 14 shows the mean and dispersion of the Cr ratio for different values of Ry, where it is observed that Cr is very sensitive to both Ry and the T/Tg ratio. In general, it is appreciated that the higher the lateral strength ratio, the higher the value of Cr. Particularly, Cr can reach values higher than unity in the short-period range (i.e. for T/Tg ratios smaller than around 0.75), which means that the residual displacements can be higher than the elastic displacement demands. On the other hand, Cr ratios become small for oscillators with period of vibration close to the predominant period of the earthquake motion (i.e. for T/Tg ratios close to one). For long-period oscillators (i.e. T/Tg ratios larger than around 2), Cr ratios become nearly constant. It should be noted that the overall trend of Cr for soft-soil sites also follow a very similar trend than that observed for the inelastic displacement ratio, CR computed for the same soil conditions [5]. It is also interesting to note that similar trend is observed for Cr ratios computed for SDOF systems subjected to near-fault earthquake ground motions, where the pulse period has significant influence on the spectral shape of Cr [6, 10]. On the other hand, there was no clear tendency in the coefficient of variation of Cr with respect to Ry and T/Tg ratio, with most of the values between 0.7 and 1.0 as shown on Figure 14b.
14
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3
Cr
T / Tg
Ry=6 Ry=4Ry=3 Ry=2Ry=1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3
Coe
ffic
ient
of v
aria
tion
of C
r
T / Tg
Ry=6 Ry=4Ry=3 Ry=2Ry=1.5
a) b)Figure 14. Statistics of residual displacement ratios, Cr, computed with soft-soil earthquake ground
motions: a) mean, and b) coefficient of variation
3.7.2 Oscillators with post-yielding stiffness ratio different from zeroIn order to identify the effects of Ry on Cr when r is different from zero, a 5%-damped
oscillator with a period of vibration of 0.25Tg was analysed for -2.5% ≤ r ≤ 10% and for Ry= 1.5, 2, 3, 4, and 6. Note that the negative value of r was limited to r=-2.5% because very high values of Cr were computed in the negative range. The oscillator was subjected to the same set of 220 ground motions, and the mean and dispersion of Cr are shown in Figure 15. It can be observed that the effects of Ry on Cr are highly affected by r. For r ≤ 0, the higher the values of Ry, the higher the values of Cr. A transition interval is observed between r=0% and 2% - except for Ry=1.5 which needs a longer transition interval (between 0% and 6%). For values of r greater than about 2%, the higher the values of Ry, the smaller were the values of Cr. Regarding the dispersion, no clear tendency was distinguished for the coefficient of variation of Cr. However, for r>0 values around 0.9 were observed. On the other hand, for r≤0 two trends may be appreciated: a) for Ry≥3 the dispersion tends to zero – which may be attributed to the fact that, for almost all the seismic records, residual displacement demands are very large and close to the maximum inelastic displacement demands; and b) for Ry<3 the dispersion tends to be higher – which is attributed to higher uncertainty in the estimation of Cr in that interval.
From the results of Figure 15, it is worth noting that small increments of r have a significant impact on RDs. This observation is in agreement with observations by [5, 15], where increments of r from 1% to 3% produced significant reductions of RDs. In general, it can be said that values of r>5% may be convenient to ensure small amplitudes of residual displacements. Therefore, recommendations proposed by Pettinga et al. [22] to reduce RDs can also be effective in structures subjected to excitations characteristic of the lake-bed zone of Mexico City.
15
0.00
0.50
1.00
1.50
2.00
-2.5% 0.0% 2.5% 5.0% 7.5% 10.0%
Coe
ffic
ient
Cr
Post-yielding stiffness ratio
Ry=6Ry=4Ry=3Ry=2Ry=1.5
0.00
0.30
0.60
0.90
1.20
1.50
1.80
-2.5% 0.0% 2.5% 5.0% 7.5% 10.0%
Coe
ffic
ient
of v
aria
tion
of C
r
Post-yielding stiffness ratio
Ry=6 Ry=4Ry=3 Ry=2Ry=1.5
a) b)Figure 15. Effects of the lateral strength ratio and post-yielding stiffness ratio on Cr: a) mean, and b)
coefficient of variation.
4. Dual OscillatorsDual structural systems composed by highly-dissipating energy systems (e.g. buckling-
restrained braced frames) acting in parallel with a conventional system (e.g. moment-resisting frames) have been proposed in the literature [21-24, 26]. Particularly, and to be consistent with [26], the primary part is related to the main structural system (e.g. moment-resisting frames) while the secondary part is related to highly dissipative systems (e.g. buckling-restrained braces, BRBs). In this way, it is considered that the secondary part (dissipative system) yields at earlier displacements than the primary part. For instance, Figure 16a shows a SDOF dual system in which the primary part contributes with a stiffness k1, a lateral load capacity of Vy1, and a damping coefficient c1; while the secondary part contributes with k2, Vy2, and c2, respectively. Assuming elastoplastic behaviour, both parts act in parallel and have the total lateral load capacity-displacement behaviour (V-d) illustrated in Figure 16b.
As can be anticipated, the dynamic response of dual systems should be influenced by the characteristics of both parts that constitute the system. Particularly, RDs highly depend on where the maximum displacement demand is located of the three zones defined in Figure 16b. Conceptually, RDs are null in zone 1 (defined by the square dot) because both parts of the system present elastic deformation and they have enough restoring capacity to return the system to its un-deformed state. In zone 2 (defined by the circular dot), RDs may exist because plastic deformation is present in the secondary part. Here, the amplitude of RDs depends on the elastic properties of the primary part and the elastoplastic properties of the secondary part. Note that the behaviour of RDs in zone 2 is similar to that of conventional oscillators with a high value of post-yielding stiffness ratio. Finally, in zone 3 (defined by the triangular dot) the RDs tend to be higher than in zone 2 because plastic deformation is present in both parts of the dual system. The amplitude of RDs in zone 3 depends on the elastoplastic properties of both, the primary and secondary parts. For practical applications, it should be noted that in zone 2, RDs can be fully eliminated after removing damaged BRBs, while in zone 3 RDs cannot be fully eliminated when the BRBs are removed because the primary part also presents permanent deformations.
16
m
k1,Vy1
c2
c1
Primary
Secondaryk2,Vy2
V
d
Vy1
k2
dy2
k1
dy1
Vy2
VyT
Primary
Secondary
Total
(2)(1) (3)
a) Parts of a dual SDOF system b) Force-displacement curvesFigure 16. Representation of dual systems.
For convenience and taking advantage of Figure 16b, two variables, which are going to be used in subsequent sections, are defined: 1) the ductility ratio of the primary part, 1=dmax/dy1, where dmax is the peak transient displacement and dy1 is the yielding displacement of the primary part; and 2) the ductility ratio of the secondary part, 2=dmax/dy2, where dy2 is the yielding displacement of the secondary part. It should be noted that if the maximum displacement demand is located in: a) zone 1, thus 1≤1.0 and 2≤1.0; b) zone 2, 1≤1.0 and 2>1.0; and c) zone 3, 1>1.0 and 2>1.0. It is highlighted that, in order to estimate the dynamic response of a SDOF dual oscillator for given target ductility ratios, an iterative process shall be conducted similar to that used for conventional SDOF systems with a given target ductility.
Now, it is considered that the amplitude of RDs in dual systems is also influenced by the period of vibration, predominant period of the ground motion, strength reduction factor, damping ratio, and type of transition (from elastic-to-plastic response), similarly to the conventional systems analysed previously. In addition, other factors need to be further taken into account since they affect RDs in dual systems due to the interactions between both parts. These factors, described in Figure 17, are the strength and stiffness ratios (which represent the contribution of the primary and secondary parts to the strength and stiffness of the system), displacement ductility, post-yielding stiffness ratio, and type of hysteretic response, for both the primary and secondary parts of the dual system.
Post-yielding stiffness ratio
Ductility
Hysteresis response
•In primary part, 1•In secondary part, 2
•Of primary part, r1•Of secondary part, r2
Strength and stiffness ratios, a & b
Contribution of primary and secondary parts
•Of primary part•Of secondary part
Factors studied in this section
Figure 17. Additional factors affecting residual displacements in dual systems
17
4.1 Effect of stiffness and strength ratiosIn this study, the stiffness ratio (a) is defined as the ratio of the stiffness of the secondary part
(k2) to that of the primary part (k1), i.e. a=k2/k1; while the strength ratio (b) is defined as the ratio of the lateral strength at yielding of the secondary part (Vy2) to that of the primary part (Vy1), i.e. b=Vy2/Vy1.
To illustrate the influence of a and b ratios, a 5%-damped dual SDOF oscillator, having elastoplastic behaviour in both parts of the dual system, with normalised period of vibration with respect to the predominant period of the ground motion, T/Tg, equal to 0.25 is studied when subjected to the same 220 ground motions described in previous sections. The target maximum ductility ratios of the primary and secondary parts are set to be 1=1.0 and 2=4.0. It is appreciated that 1=1.0, which means that, as described previously, the maximum displacement demands are located in the limit of zone 2 and the primary part remains elastic while the secondary part presents plastic deformation (see Figure 16b). It should be noted that the ratio 2/1 is equal to four. This ratio can be related to the a/b ratio, assuming that both parts experience a maximum displacement dmax, as follows [29]:
2
1
ab
=(4)
Figure 18a shows the mean of the residual displacements normalised by the maximum displacement demands. Note, in the horizontal axes, that parameter a is proportional to 4b due to the constrain of 1=1 and 2=4. It is appreciated in the figure that the normalised RDs increase as the stiffness and strength ratios increase (i.e. the normalised RDs increase as the contribution of the secondary part becomes larger with respect to that of the first part). Figure 18b shows that the dispersion is high with coefficients of dispersion around 0.8.
Observations of Figure 18a are explained as follows. In one side, the peak displacement demands were located within zone 2 (as defined in Figure 16b) and the total force-displacement relationship is bilinear and similar to that on conventional oscillators. On the other hand, a small contribution of the secondary part results in a large post-yielding stiffness ratio, r, on the total capacity curve, while a large contribution of the secondary part, leads to reduced values of r. In the former case and in agreement with observations on conventional oscillators with large values of r, RD ratios tend to be significantly reduced; while in the latter case, and again, in agreement with observations on conventional oscillators, residual displacement demands tend to be large.
0 4 8 12 16 20 24
0%
10%
20%
30%
0 1 2 3 4 5 6
stiffness ratio (a)
Res
idua
l / M
ax. d
ispl
acem
ent
strength ratio (b)
0 5 10 15 20
00.20.40.60.8
11.2
0 1 2 3 4 5 6
stiffness ratio (a)
Coe
ffic
ient
of v
aria
tion
strength ratio (b)
a) b)Figure 18. Effects of stiffness and strength ratios on normalised residual displacements:
a) mean, and b) coefficient of variation
18
4.2 Effect of ductility ratios of the primary and secondary parts
4.2.1 Ductility ratio of the secondary partThe effects of the ductility ratio of the secondary part, 2, on normalised RDs are assessed in a
5%-damped oscillator having elastoplastic behaviour of both parts of the dual system, with T/ Tg
=0.25 when subjected to the set of 220 ground motions. The value of 2 is varied from 2 to 7 using constant values of a and b (i.e. a=10 and b=1.25). In this way, a/b=8 and 1 ≤1 according to equation (4). Note that 2 varies proportional to 81 (see equation (4)), and that the primary part remains elastic (i.e. the maximum displacement demand is located in zone 2, as defined in Figure 16b).
0.00 0.25 0.50 0.75 1.00
0%
5%
10%
15%
20%
25%
0 2 4 6 8
Ductility of primary part (1)
Res
idua
l / M
ax. d
ispla
cem
ent
Ductility of secondary part (2)
0.00 0.25 0.50 0.75 1.00
0.000.200.400.600.801.001.20
0 2 4 6 8
Ductility of primary part (1)
Coe
ffic
ient
of v
aria
tion
Ductility of secondary part (2)
a) b)Figure 19. Effects of ductility of the secondary part on normalised residual displacements:
a) mean, and b) coefficient of variation.
Figure 19a shows that the ratio of normalised RDs tend to decrease as the ductility ratio of the secondary part increases. This means that the normalised RDs of dual systems would be constrained providing that the moment-resisting frames behave elastically although the energy-dissipating systems, such as BRBs, developed large levels of displacement ductility. This observation is in agreement with the observations from section 3.4, where normalised RDs in conventional oscillators with post-yielding stiffness ratios larger than about 2.5% decrease as ductility demands increase. In addition, Figure 19b shows no clear tendency of the coefficient of variation, with values between 0.65 and 0.85.
4.2.2 Ductility ratio of the primary partIn order to assess the effects of the target ductility of the primary part, 1, on RDs, the same
oscillator described in the previous section is subjected to the same set of records using values of a=10 and b=2.5 (i.e. a/b=4). The value of 1 is varied from 0.5 to 3.0 (i.e. the primary part reaches inelastic deformation as shown in zone 3 of Figure 16b), which imply that values of 2
are varied from 2 to 12 according to equation (2).In Figure 20a, it is observed that once the primary part yields, the normalised residual
displacements tend to significantly increase. This means that allowing plastic deformations in both primary and secondary parts has the detrimental effect of increasing the RD demands. In other words, the effect of 1 is opposite to the effect of 2. Therefore, designers should avoid plastic deformations on the primary part if elastic-perfectly plastic behaviour is expected in both
19
parts of the system. Regarding to dispersion levels, Figure 20b shows that the coefficient of variation reduces once 1 becomes larger than one.
0%
10%
20%
30%
40%
0 1 2 3 4
Res
idua
l / M
ax. d
ispla
cem
ent
Ductility of primarypart (1)
Zone 2 Zone 3
0.00
0.50
1.00
0 1 2 3 4
coef
ficie
nt o
f var
iatio
n
Ductility of primarypart (1)
Zone 2 Zone 3
a) b)Figure 20. Effects of ductility of the primary part on normalised residual displacements:
a) mean, and b) coefficient of variation.
4.3 Effects of post-yielding stiffness ratio of the primary and secondary parts The effects of post-yielding stiffness ratio of the primary part (r1) and the secondary part (r2)
of dual systems are studied in this section. For this purpose, two cases are compared to the previous results shown in Figure 20, which corresponded to zero post-yielding stiffness in the primary and secondary parts (i.e. r1=r2=0). For the first case (Case 1), the post-yielding stiffness ratios are set to r1=5% and r2=0, while for the second case (Case 2) are specified as r1=0 and r2=5%.
Figure 21a shows that the effect of increasing r1 on normalised residual displacements is small, while increasing r2 has a significant effect. This can be explained since the plastic deformation of the secondary part is four times larger than that of the primary part, which leads to higher participation of r2. Regarding the dispersion, Figure 21b shows that it tends to be similar in a certain region (say 1<1.5) and then increases for Case 2 (after 1>1.5).
0%
10%
20%
30%
40%
0 1 2 3 4
Res
idua
l / M
ax. d
ispla
cem
ent
Ductility of primarypart (1)
Zone 2 Zone 3 r1=r2=0
Case 1: r1=5%, r2=0
Case 2: r1=0, r2=5%
0.00
0.50
1.00
0 1 2 3 4
coef
ficie
nt o
f var
iatio
n
Ductility of primarypart (1)
Zone 2 Zone 3
r1=r2=0
Case 1: r1=5%, r2=0
Case 2:r1=0, r2=5%
a) b)Figure 21. Effects of post-yielding stiffness ratios on normalised RDs: a) mean, and b) coefficient of
variation.
20
The effects of negative post-yielding stiffness ratios on normalised residual displacements are also examined with the aid of the same oscillator. Again, two cases are considered (r1=-5% and r2=0 for Case 1; while r1=0% and r2=-5% for Case 2). Results are compared with those illustrated in Figure 20. Figure 22a shows that a negative value of r1 slightly increases the normalised RDs as 1 increases. Unlike the effect of r1, the effect of negative r2 is significant in increasing the normalised RDs. This observation means that energy-dissipating systems cannot exhibit strength degradation leading to negative post-yield stiffness. Figure 20b shows that the effect of r2 on the dispersion was the opposite, i.e. negative r2 generates smaller dispersions than negative r1.
0%
20%
40%
60%
80%
100%
0 1 2 3 4
Res
idua
l / M
ax. d
ispla
cem
ent
Ductility of primarypart (1)
Zone 2 Zone 3
r1=r2=0
r1=-5%, r2=0
r1=0, r2=-5%
0.00
0.50
1.00
0 1 2 3 4co
effic
ient
of v
aria
tion
Ductility of primarypart (1)
Zone 2 Zone 3
r1=r2=0
r1=-5%, r2=0
r1=0, r2=-5%
a) b)Figure 22. Effects of negative post-yielding stiffness ratios on normalised RDs: a) mean,
and b) coefficient of variation.
From the aforementioned results and study-cases, it can be said that the effect of r2 on residual displacements is significant while that of r1 is negligible. Designers should be aware that positive values of r2 are very beneficial while negative values are highly detrimental. If a negative r2 is identified at the beginning of the design process, designers can make an early corrective decision that could help the structure to avoid excessive losses due to RDs.
4.4 Effects of hysteretic response In this section, only the effect of the type of hysteretic response of the primary part on RDs of
dual systems is assessed and that of the secondary part is considered elastoplastic (since this represents well the behaviour of some passive dissipation systems such as BRBs). Three types of hysteretic response are evaluated for the primary part, namely elastoplastic, Takeda [7], and flag-shaped [8] as illustrated in Figure 4. It should be mentioned that the influence of hysteretic response of the primary part is only significant when 1>1 (i.e. when the primary part yields, according to zone 3 in Figure 16b). Otherwise, the response of the primary part would remain elastic and the total response of the dual system would be the same for the three types of hysteretic response. Dual oscillators with normalised periods of vibration between 0.2Tg and 3Tg
and total viscous damping ratio of 5% are subjected to the same set of 220 earthquake ground motions described in previous sections. A strength ratio of b=2 and a stiffness ratio of a=5 are considered. The target ductility ratios of the primary and secondary parts are set to be 1=2 and 2=5. The post-yielding stiffness ratio in both parts was assumed to be zero (i.e. r1=r2=0).
Figure 23a shows that the hysteretic global response of the primary part significantly affects the ordinates of normalised RDs. As can be expected, the highest values are obtained for the
21
elastoplastic response while the smallest correspond to the flag-shaped response. This means that a dual-systems consisting of a self-centring system (e.g. having post-tensioned beams or columns) acting as primary part along with BRBs acting as secondary part will be very efficient at restraining residual displacements. It is worth to note that, contrary to the response observed on conventional oscillators (Figure 5), the normalised RDs of the flag-shaped model were not null due to the contribution of the secondary part. From Figure 23b, it can be seen that the elastoplastic response leads to the smallest coefficient of variation while the Takeda response presented higher values. The coefficient of variation of the flag-shaped response was not presented in the figure because unrealistic dispersions were observed due to mean RDs close to zero.
0%
10%
20%
30%
40%
0 0.5 1 1.5 2 2.5 3
Res
idua
l / M
ax. d
ispla
cem
ent
T / Tg
Bilinear
Takeda
Flag-shaped
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
coef
ficie
nt o
f var
iatio
n
T / Tg
Bilinear
Takeda
a) b)Figure 23. Effect of the type of hysteretic response of the primary part on normalised residual
displacements: a) mean, and b) coefficient of variation.
5. Recommendations to Mitigate Residual DisplacementsBefore providing recommendations to mitigate residual displacements, it is stressed that the
results presented in this paper are presented either in terms of the peak transient displacement or the elastic displacement ordinates. Therefore, the results should be interpreted in that light. If absolute values of residual displacements are required, they can be determined by the multiplication of the normalised RDs presented here and either the peak transient displacements or the elastic spectral displacement ordinates (ESDOs), as it corresponds. It should also be noted that peak transient displacements can also be determined from ESDOs, as recommended by [25]. Therefore, both peak and residual displacement demands can be determined from ESDO; which are commonly available to designers.
Now, from the identified factors affecting normalised residual displacements, the following recommendations are suggested in order to reduce or minimise them. It is recognised that, while some factors can be easily or economically achieved, others may be more difficult.
5.1 Conventional systemsAlthough dual systems may be a better way to control normalised RDs, it is recognised that
their implementation may not always be possible. Therefore, for those cases where dual systems may not be feasible, reduced RDs may be expected by providing:
1. High post-yielding stiffness ratio (say) r > 5% or 10%. According to Pettinga et al. [22], this parameter can be easily increased by: 1) using different steel reinforcement with beneficial features; 2) re-designing geometry and properties of primary elements; and 3)
22
providing a secondary resisting part that should remain elastic (which is equivalent to using a dual system). Sahoo and Chao [24] recommend increasing the stiffness of the non-yielding members (such as columns of a BRBF), which can effectively increase the post-yield stiffness.
2. Self-centring technology. This can be achieved by using posttensioned beam-to-column connections, as proposed by Christopoulos et al. [8] and Priestley et al. [9]. This technology may be more expensive but very effective.
The previous solutions are regarded as very effective to reduce RDs; however, if for any reason the post-yielding stiffness ratio, r, is negative or self-centring technology is not feasible, reductions can be achieved by following the next recommendations.
1. Designing for reduced levels of ductility demands. Although this may result in an expensive structure.
2. Designing for reduced strength reduction factors. Similar to the previous point, the resultant structure may be expensive if this solution is used.
3. Providing supplemental damping. This provides the benefit of reducing both, residual and peak displacements.
4. Negative values of r must be avoided because they increase RDs dramatically.
If, after following the previous recommendations, the RDs are still not acceptable, the use of a dual system is highly advisable.
5.2 Dual systemsDual systems are very effective to reduce or eliminate residual displacements, as long as the
maximum displacement demand remains located within zone 2, as defined in Figure 16b. This means that the main structural system shall remain essentially elastic while all the plastic deformation is concentrated in the dissipation system, such as BRBs. Besides, any residual displacements can be removed by replacing damaged dissipation devices after an earthquake.
On the other hand, if the maximum displacement demand reaches zone 3, permanent deformations may be present in the primary structure and they will remain even after the damaged dissipation devices are removed. Taking this into account, the following recommendations are suggested to reduce RDs in dual structures:
1. Designing for a maximum displacement demand within zone 2, i.e. 1 ≤ 1. Additionally, the smaller the contribution of the secondary part, the better. Values of the strength ratio of b < 1 are preferable. A high ductility demand of the secondary part (2) is also desirable.
2. If, for a reason, the maximum displacement demands do not remain in zone 2 and reach zone 3, a post-yielding stiffness ratio of the secondary part r2 > 5% or 10% should be provided. Again, a high ductility demand of the secondary part (2) is preferable.
3. If the previous suggestions are not feasible, self-centring capacity is advisable to provide flag-shaped response to the primary part. Posttensioned beam-to-column connections, as suggested by Christopoulos et al. [8] and Priestley et al. [9], may be an effective solution.
The previous recommendations, especially the first one, are very effective to reduce RDs. If however, they are not feasible, reductions can also be achieved by considering:
1. Primary part made of reinforced concrete. 2. Providing supplemental damping.3. No negative values of post-yielding stiffness ratio of the secondary part (r2).
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If the RDs are still not acceptable, the design should be modified. Smaller values of b and a, and 1 ≤ 1 may be the wise choice this time.
6. DiscussionComparison of the responses to those on stiff soils. Several factors affecting residual
displacement demands on conventional structures located on stiff soils also affect them on soft soils. One example is the post-yielding stiffness ratio, r, which affected the magnitude of normalised RDs significantly (i.e. a negative r produced large normalised RD demands while a positive value produced small demands). Comparing the magnitude and spectral trend of the normalised RDs reported Figures 7 and 14 of this study to those computed from records gathered at stiff soils published in refs. [5, 8, 9], it can identified that mean normalised RDs are only similar in the long-period range where they tend to be constant. However, significant differences are observed in the short-period range and in periods close to the predominant period of the ground motion, Tg, where normalised RDs tend to reduce significantly for normalised periods (T/Tg) close to unity, which is not observed for stiff soils. Regarding the coefficients of variation, values higher than 0.6 were commonly observed, which agrees with dispersions reported on firm soil sites.
New findings. A few parameters that, to the knowledge of the authors, have not been previously addressed, have been presented in this paper because they provide interesting findings that should be taken into account in the context of Performance-Based Seismic Design and Assessment; so that RD demands are estimated with reliability. These parameters are summarised as follows: For conventional oscillators:
o The type of transition from elastic-to-plastic response. As observed from Figure 12, the type of transition may produce a significant effect on normalised RDs, especially for post-yielding stiffness ratios different than zero and reduced levels of ductility demand.
o Damping ratio. This parameter affects both the peak and residual displacements proportionally.
o Changes on some trends produced by the magnitude of the post-yielding stiffness. This finding is very interesting since contrary to what other studies based on elastic-perfectly plastic oscillators have reported (e.g. [10, 11]), some trends on the normalised RDs change with the magnitude of r. As an example, observation of Figure 7 could mistakenly lead to believe in general that the larger the ductility demand (i.e. the inelastic behaviour) the larger the normalised RDs. However, when figures 8 and 9 are analysed, it is seen that that trend is only valid for r ≤ 0; and that the opposite trend is clearly observed for r > 5% in all the analysed period range.
For dual oscillators:o Effects of the contribution of the secondary part to the stiffness and strength. When a
dual structure is designed so that the primary part remains elastic under seismic loading, a reduced contribution of the dissipative secondary part may be desirable so that the primary part can return the structure to its un-deformed position. This is confirmed in Figure 18.
o Effects of the ductility demands of the primary and secondary parts. These findings are also very significant because they show that, as long as the primary part remains
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elastic, the normalised RDs will be well constrained (i.e. the greater the inelastic behaviour of the secondary part, the smaller the normalised RDs). However, if the primary part reaches inelastic deformation, the normalised RDs may increase dramatically, leading to costly repairs. Therefore, a suggestion to designers is that, when designing dual structures, the primary part should remain elastic so that RD demands remain small and are easy to remove.
o Effects of the post-yielding stiffness ratio of the primary and secondary parts. Within this study, it was seen that the post-yielding stiffness ratio of the primary part, r1, does not have a significant effect on the normalised RDs. However, that of the secondary part, r2, has a significant effect. A recommendation to designers is to pay special attention to the post-yielding stiffness ratio of the secondary part so that losses due to excessive RDs are avoided.
o Effects of the type of hysteretic response of the primary part. Here, it was shown that, even when the secondary part of a dual system may have a bilinear behaviour, a primary part with self-centring capacity is effective to diminish normalised RDs in dual systems. Although normalised RDs are not fully eliminated, they certainly remain small.
Applicability of the results presented in this paper. It is important to stress that all the results presented in this paper are applicable for the soft-soils of Mexico City and sites with similar characteristics. The reader should be aware that these results may not be valid for other types of soils.
7. Summary and ConclusionsResidual displacement (RD) demands from conventional and dual single-degree-of-freedom
oscillators subjected to a relatively large set of earthquake ground motions recorded in the lake-bed zone of Mexico City were statistically evaluated in this study. To provide a context on the amplitude of RDs, they were normalised by: 1) the corresponding maximum transient displacement demands, named as normalised RDs, and 2) the corresponding elastic spectral displacement, designated as Cr ratio [5, 10]. Both measures of normalised RDs were plotted against normalised periods of vibration with respect to the predominant period of the ground motion, T/Tg. The influence of several parameters such as the post-yielding stiffness ratio, r, the displacement ductility, , the type of hysteretic behaviour, the damping ratio, the lateral strength ratio, Ry, and the type of transition, from elastic-to-plastic, in the global hysteretic response were examined. The following conclusions are offered from this study:
For conventional oscillators:1. The most significant parameter affecting normalised RDs was r. Similarly to findings derived
from earthquake ground motions recorded in firm and rock sites [3, 4, 10], RDs decrease as positive r values increase. It was also found that RDs significantly increase as negative values of r increase, which is also consistent with findings noted in [3, 4].
2. Due to the dominant influence of r, the effects of other parameters, such as and Ry were inverted. For instance, for r>2.5% the higher the levels of and Ry, the smaller the normalised RDs; while the opposite observation was true for r<0%.
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3. The amplitude of the parameter Cr is very sensitive to the T/Tg ratios. Oscillators with short T/Tg ratios present very high Cr values (i.e. Cr tends to increase as T/Tg decreases). On the other hand, Cr values become smaller for periods of vibration close to the predominant period of the excitation, Tg, than in other spectral regions. Finally, Cr values tend to be constant for T/Tg ratios larger than about 1.5.
4. The type of hysteretic response affects RDs. The flag-shaped is the most effective to reduce them. Takeda response (representative of concrete structures) presented about a third RDs of those of elastoplastic response (representative of steel structures).
5. The level of damping ratio affects the amplitude of residual and peak displacements almost proportionally. The higher the damping ratio, the smaller the RDs and peak displacement demands; which means that increasing the damping ratio provides the benefit of reducing both residual and peak displacements.
6. The type of transition, from elastic-to-plastic global hysteretic response, affects RDs when r 0 and ≤ 2. RDs may be very different when the transition is very smooth or sharp. RDs are smaller in sharp than in smooth transition if r < 0. The opposite is true if the value of r > 0.
For dual oscillators:1. RDs are small if the primary part (e.g. moment-resisting frames) remains elastic (i.e. 1<1).
On the contrary, if 1>1, RDs increase as 1 increases.2. RDs increase as the stiffness and strength ratios, a and b, increase. This means that small
contributions of the secondary part (e.g. Buckling-Restrained Braces) are preferred to constrain RDs. However, it is recognised that small contribution of the BRBs are not always feasible, since they constitute most of the times the main lateral resisting system. There are currently some approaches where the contribution of the BRBs and the moment resisting system can be decided from the beginning of the design process (e.g. [37]) leaving room to provide more capacity to the moment resisting frame than to the BRBs. This could be the case, for example, of retrofitting existing buildings, where a small contribution of the BRBs is desirable to avoid damage in existing elements and has the additional benefit of reducing RDs.
3. The post-yielding stiffness ratio of the primary part, r1, does not significantly affect RDs. On the contrary, that of the secondary part, r2, has a significant effect. A positive r2 reduces RDs significantly while a negative value is very detrimental.
4. Assuming that the secondary part exhibits elastoplastic hysteretic behaviour, the type of hysteretic response in the primary part has a significant effect on RDs. While the elastoplastic response presents the highest RDs, the flag-shaped response presents very small RDs. Takeda response is somehow in the middle. The RDs observed for the flag-shaped response show that a primary part with self-centring capacity, acting in parallel with a highly dissipation system (e.g. BRBs) as secondary part, is very effective to diminish RDs.
Recommendations to reduce RDs have also been proposed. In general, it can be said that the most effective ways to reduce RDs are: 1) providing a high value of r in conventional oscillators; 2) in dual systems, controlling the displacement demand so that the primary part remains elastic or providing r2 > 5%; and 3) for both conventional and dual systems, providing self-centring capacity to the primary structure. The other parameters analysed in this study can also help to reduce RDs - however they may be less effective.
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Finally, it is highlighted that dispersion was very high in all the studied cases. Coefficients of variation higher than 0.6 were commonly observed, which agrees with previous findings about the dispersion of residual displacement demands for systems in firm soil sites. These values must be taken into account when assessing the probabilistic performance of conventional and dual structures.
AcknowledgementsThe first author acknowledges the sponsorship provided by the National Council for Science and Technology (CONACyT) in Mexico, while the second author expresses his gratitude to the Universidad Michoacana de San Nicolás de Hidalgo.
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