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Inelastic spectra for displacement-based seismic design
Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
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Inelastic spectra for displacement-based seismic design B. Borzi a,1, G.M. Calvi b, A.S. Elnashai a,*, E. Faccioli c, J.J. Bommer a a
Department of Civil and Environmental Engineering, Imperial College, London SW7 2BU, UK Dipartimento di Meccanica Strutturale, Universita degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy c Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy b
Accepted 6 September 2000
Abstract In recognition of the emergence of displacement-based seismic design as a potentially more rational approach than force-based techniques, this paper addresses derivation of inelastic displacement spectra and associated topics. A well-constrained earthquake strong-motion dataset is used to derive inelastic displacement spectra, displacement reduction factors and ductility±damping relationships. These are in a format amenable for use in design and assessment of structures with a wide range of response characteristics. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Attenuation relationship; Displacement spectrum; Displacement-based approach; Ductility; Damping; Displacement modi®cation factor
1. Introduction Force-based seismic design remains, in spite of its shortcomings, the method widely used in codes. However, deformation-based (Calvi and Kingsley [1], Kowalsky et al. [2], Priestley et al. [3]) or deformation-controlled (Panagiotakos and Fardis [4]) procedures have recently been proposed. The new approach utilises displacement spectra expressed as spectral ordinates versus effective period (period at maximum displacement) to quantify the demand imposed on structural systems. The origins of displacement-based design may be traced to work published as early as the 1960s, where comments on the displacements of inelastic systems and their relationship to their elastic counterparts were made (e.g. Muto et al., 1960, as reported by Moehle [5]). However, it was the work of Sozen and his associates (Gulkan and Sozen [6], Shibata and Sozen [7]) that developed the concept of a substitute structure. The substitute structure is a single degree of freedom elastic system, the characteristics of which represent the inelastic system. This concept enables the use of an elastic displacement spectrum in design, while using the displacement capacity of an inelastic system. Various contributions were made towards the development of displacement-based seismic design since the early
work mentioned above. However, it was in the 1990s that formal proposals were made to implement the emerging ideas into a design procedure. The earliest is that by Moehle and his co-workers (Moehle [5], Qi and Moehle [8]). Thereafter, a complete and workable procedure for seismic design of structures that sets aside forces and relies on displacement as a primary design quantity was proposed by Kowalsky et al. [2] for single degree of freedom systems (such as bridge piers). A parallel paper on multi-degree of freedom systems is due to Calvi and Kingsley [1]. The steps comprising the design process for SDOF systems are given below for simplicity:
* Corresponding author. Tel.: 144-20-7594-6058; fax: 144-20-72252716. E-mail address: [email protected] (A.S. Elnashai). 1 Present address: EQE International Ltd, Warrington WA3 6WJ, UK. 0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(00)00075-0
a. A target displacement for the structure is selected, based on the type of structure and the governing limit state. b. Knowing the yield displacement, and the material and structural system, a value of equivalent damping is determined. c. Displacement spectra representative of the seismotectonic environment are used. The inputs are the target displacement and the equivalent damping. The output is an effective period of vibration. d. The structure is dimensioned to give an effective period, taking into account reduced stiffness consistent with the level of deformation, equal to that obtained from the displacement spectra. e. If the effective period is not suf®ciently close to the required period, go to step b. above and repeat until convergence.
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It is clear from the above that whereas displacementbased design is the logical framework for seismic design, since the primary source of seismic energy dissipation is inelastic deformation, it imposes new requirements for veri®able design. Primarily, accurate, representative and parametrically described displacement spectra are essential ingredients that have only very recently become available (Bommer et al. [9], Bommer and Elnashai [10], Tolis and Faccioli [11]). In the study by Bommer et al. [9], it was shown that a well-balanced catalogue for acceleration studies may be unsuitable for displacement purposes. The ordinates of acceleration response spectra are insensitive to the processing applied to the accelerograms, whereas the in¯uence of the ®lter parameters on the displacement spectra is pronounced. In this paper attenuation coef®cients for inelastic displacement constant ductility spectra are calculated. Attenuation coef®cients for residual displacements were also taken into account. This is of great importance in the assessment of displacement-based design situations, hence the spectra are presented as ratio of maximum response displacement. By means of a comparison between elastic and inelastic displacement spectral ordinates a reduction coef®cient taking into account the effects of real inelastic behaviour in a simpli®ed method of design based on displacement is calculated. For this coef®cient, herein denoted h -factor, regression coef®cients were evaluated (Borzi et al. [12], Borzi [13]) for all magnitudes, distances and site conditions. Further, average values obtained constitute a valid reference for new code implementations for simpli®ed methods based on displacements. 2. Input motion 2.1. Strong-motion dataset selection and processing The dataset employed for the de®nition of inelastic constant ductility spectra was presented by Bommer et al. [9] and used for the derivation of frequency-dependent attenuation equations for ordinates of displacement response spectra. All records were ®ltered individually at Imperial College (London, UK) and used to derive displacement spectra for different levels of damping, from 5 to 30%. Each record was individually processed using an elliptical ®lter and a long-period cut-off that resulted in a physically reasonable displacement time-history (Bommer and Elnashai [10]). This ®ltering process removes permanent displacements, caused by ground deformation or fault slip, in order not to mix these phenomena with transitory ground vibrations. The dataset has been adapted from that employed by Ambraseys et al. [14,15] to derive attenuation relationships for ordinates of acceleration response spectra. This is a high-quality dataset in terms of both accelerograms that have been individually corrected, and information regarding the recording stations and earthquake character-
Fig. 1. Distribution of records for: (a) rock; (b) stiff; and (c) soft soil.
istics. The original dataset of Ambraseys et al. [14,15] has been modi®ed in the following ways: 1. records for small magnitude earthquakes have been removed; 2. some site classi®cations have been corrected in the light of new information; 3. two new strong-motion records have been added; 4. all of the original records have been ®ltered with individually selected cut-offs. The necessity of eliminating all the records generated by earthquakes of magnitude less than Ms 5.5 is due to the particular interest in long-period radiation. In Fig. 1 the distribution of records comprising the dataset with regard to magnitude, distance and site classi®cation are
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
demonstrated to emphasise credentials of this set of natural records. While the source distance and the surface-wave magnitude are available for all the accelerograms, for three records the local site geology is unknown. For the remaining 180, the percentages of distribution in the three site groupings of rock, stiff and soft soil are 25.0, 51.1 and 23.9%, respectively. These percentages are close to the distribution of the original dataset, being 25.5, 54.3 and 20.2%, respectively. For two records only one component of the motion is known. Therefore a total of 364 accelerograms were processed. It is acknowledged that site intensity could be used for the selection of strong-motion datasets. However, this has its own problems, and is not considered to be superior to the selection based on magnitude, distance and site condition. 2.2. Attenuation model and regression analysis Studies concerning the evaluation of seismic hazard related to earthquakes utilise predictive models commonly referred to as attenuation relationships. These models generally express values of strong-motion parameters as a function of source characteristics, propagation path and local site geology. In order to de®ne elastic spectra, a common approach to perform a hazard analysis is to de®ne the hazard in terms of the peak ground acceleration (PGA), which anchors the zero-period ordinate for a standard spectral shape. A criticism of this approach is that the resulting spectra do not correspond to the same seismic hazard for all periods (i.e. non-uniform hazard spectra). To obtain spectra characterised according to the same seismic hazard for every spectral ordinate (uniform hazard spectra) the de®nition of period-dependent attenuation relationships are proposed. The attenuation model used in this work is that of Ambraseys et al. [14,15] employed to de®ne elastic acceleration spectra. The formulation of the attenuation relationship is: log y C1 1 C2 Ms 1 C4 log r 1 Ca Sa 1 Cs Ss 1 sP 1 where y is the strong-motion parameter to be predicted, Ms is the surface wave magnitude and: q 2 r d 2 1 h20 in which d is the shortest distance from the station to the surface projection of the fault rupture in km and h0 is a regression constant. The h0 coef®cient takes into account that the fault projection is not necessarily the source of the peak motion, but it does not represent explicitly the effect of the focal depth on the motion. The coef®cients C1, C2, C4, Ca, Cs and h0 are determined by regression. In this attenuation model three soil conditions are distinguished by the average shear wave velocity. When the shear wave velocity is higher than 750 m/s the soil is classi®ed as rock. Soft soil has a shear wave velocity is less than 360 m/s, whilst stiff soil is assumed in the intermediate
49
range of shear wave velocity. In the regression model Sa takes the value of 1 for soft soil conditions and 0 otherwise, while Ss takes the value of 1 for stiff soil conditions and 0 otherwise. Finally, P is a parameter that is multiplied by the standard deviation s and takes the value of 0 when the mean value of log(y) is calculated and 1 for the 84-percentile value of log(y). 3. Structural models 3.1. Elastic perfectly-plastic model In order to determine the in¯uence of magnitude, distance and soil condition on inelastic response spectra, attenuation relationships have been de®ned using an elastic perfectlyplastic (EPP) response model. The EPP model was employed since it is the simplest form of inelastic forceresistance as well as being the basis for early relationships between seismic motion and response modi®cation factors. Moreover, by virtue of its two parameters de®nition: level of force-resistance and stiffness, few structural characteristics are included, hence the in¯uence of strong-motion records may be better visualised. The stiffness corresponds to the period of vibration for which the spectral ordinate has to be calculated and the resistance is derived iteratively. In this work inelastic constant ductility spectra were obtained. Therefore the resistance of the system corresponds to the resistance for which the system has a required ductility equal to the target ductility. The ensuing inelastic spectra would re¯ect solely the characteristics of the input motion. 3.2. Hysteretic hardening±softening model In order to investigate the in¯uence of the response characteristics of structures on inelastic displacement spectra, a hysteretic hardening-softening model (HHS) was used (Ozcebe and Saatcioglu [16]). The structural model is characterised by the de®nition of a primary curve, unloading and reloading rules. The primary curve for a hysteretic force± displacement relationship is de®ned as the envelope curve under cyclic load reversals. For non-degrading models the primary curve is taken as the response curve under monotonic loading. In this model the primary curve is used to de®ne the limits for member strength. Two points on the primary curve have to be de®ned. It is essential to de®ne cracking and yield loads (Vcr and Vy) and the corresponding displacements (D cr and D y), as shown in Fig. 2. If, for example, this model was used to describe the hysteretic behaviour of reinforced concrete members, the cracking load would correspond to the spreading of cracks in the concrete and the yielding load would be related with the load at which the strain in bars is equal to the yield strain of steel or some other criterion can be selected by the user. Unloading and reloading branches of the HHS model have been established through a statistical analysis of experimental data
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Fig. 2. HHS model for structural members.
(Saatcioglu and co-workers [17,18]). The load reversal rules are brie¯y described below. Structural members exhibit stiffness degradation under cyclic loading. When the number of cycles or the amplitude of inelastic deformation increases, the system becomes softer. Furthermore, the hysteretic behaviour is affected by pinching. The axial load is an important parameter in predicting pinching effects (due to the onset of crack closure). The slope of reloading branches increases beyond the crack load. The slopes of the lines connecting the origin to the cracking point (K1 in Fig. 2) and the yield point to the cracking point in the opposite quadrant (K2 in Fig. 2) are used to de®ne the unloading branches under cyclic loads. The latter slope depends on deformation and force levels attained at the beginning of unloading. Experimental results indicate that if unloading starts between the cracking and the yield load, and the yield load has not been exceeded in the relevant quadrant, then unloading stiffness is bounded by K1 and K2. In this model a linear variation between these limits was proposed as a function of displacement ductility. If the unloading load exceeds the yield load, the unloading curve changes the slope to a value close to the cracking load.
is not representative of the structural behaviour. A low damping value of 1% was included. This viscous damping is representative of the non-hysteretic dissipation, since hysteretic damping is already included. Sensitivity analyses were undertaken and have indicated that further precision of this assumption is unwarranted. The input parameters for the HHS model described above are the monotonic curve and the relationship between axial compressive force and nominal concentric axial capacity. In order to de®ne the inelastic constant ductility spectra the magnitude of the monotonic curve is not an input parameter. It is de®ned in an iterative way forcing the relationship between maximum and yield displacements to satisfy the target ductility. To obtain the inelastic spectra and displacement modi®cation factor (h ) an approximation of the primary curve with three linear branches has been assumed (Fig. 3). Consequently, the input parameters de®ning the shape of the primary curve are: 1. the relationship between the cracking and the yielding load (Vcr/Vy); 2. the relationship between the stiffness before the cracking load and the secant stiffness (Kcr/Ky); 3. the slope of the post yield branch.
4. Procedural considerations In order to de®ne inelastic displacement spectra and displacement modi®cation (h ?, displacement ductilities of 2, 3, 4 and 6 are considered. Ductility levels higher than 6 are not included because they constitute global displacement ductility and structures very rarely have a local ductility supply commensurate with global ductility above 6. The inelastic spectra have been de®ned between 0.05 and 3 s. For the HHS model the initial elastic period can be considered either as the stiffness before Vcr or the secant stiffness. In the current work period corresponding to the secant stiffness was considered. This is because the stiffness before Vcr
Fig. 3. Shape of primary curve used in this work.
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
To select the values of parameters to be employed, extensive analysis of the in¯uence of each parameter on the inelastic spectra was undertaken. The results of a parametric investigation indicate that the parameter with the strongest in¯uence on inelastic spectra is the slope of the post yield branch. Hence ®xed ratios between Vcr and Vy and between Kcr and Ky were considered. Although the aforementioned ratios have a large variability, constant values are assumed, since the results of preliminary parametric investigations using a ®ne mesh of variation show that they do not have a signi®cant in¯uence in terms of the inelastic displacement spectra. From the experimental results of Paulay and Priestley [19], Priestley et al. [20], Calvi and Pinto [21] and Pinto [22], it is reasonable to consider the secant stiffness at the yield point equal to 50% of the stiffness before Vcr; The latter is taken equal to 30% of Vy. The ratio between the cracking and yield loads in¯uences the pinching behaviour that does not occur often for structures with loads higher than approximately 30% of the yielding load Vy. The considered representative slopes of the structural behaviour are: ² ² ² ²
K 3 0 : (elastic perfect plastic behaviour) K3 10%Ky : (hardening behaviour) K3 220%Ky : (softening behaviour) K3 230%Ky : (softening behaviour)
An axial load equal to 10% of the nominal axial load is assumed, since the model does not account for second-order effects. The above characteristics have been veri®ed to cover both new structures with seismic detailing, and existing poorly detailed structures (Borzi et al. [12], Borzi [13]). An iterative procedure was utilised for the de®nition of spectral ordinates corresponding to target ductility. In rare cases it has not been possible to obtain a convergent solution with the HHS model. The percentage of spectral ordinates that have not been considered in the regression analyses are reported in Table 1. It is observed that the number of spectral ordinates to be excluded from parametric analysis for the slope of the third branch equal to 230%Ky and ductility equal to 4, is very high. The attenuation relationship for this combination of parameters was therefore not considered. The above observations (non-convergence) are fully justi®ed by noting that highly degrading systems are inherently of low ductility. This was further investigated by comparison with a different program for inelastic spectra available to the authors (program inspect). The lack of convergence was observed Table 1 Percentage of ordinates excluded from regression analysis K3
m 2 (%)
m 3 (%)
m 4 (%)
m 6 (%)
0 10%Ky 220%Ky 230%Ky
0.16 0.03 1.67 2.45
0.58 0.13 5.12 8.16
1.23 0.27 12.00 30.98
3.16 0.80 ± ±
51
for the same range of degrading stiffness and high ductility, thus con®rming that these situations correspond to structural collapse. 5. Inelastic displacement spectra The cut-off period of every accelerogram is associated with an initial elastic period TI, which corresponds to the secant stiffness at yield. This is less than the cut-off periods employed in order to ®lter the records. For inelastic systems there is doubt as to whether spectral ordinates should correspond to the equivalent elastic period TE rather than the initial elastic period TI, both of which are less than the cut-off period. However, it was observed that even if equivalent elastic periods are somewhat higher than the cut-off period of 3 s, the inelastic spectral ordinates are still valid. This was con®rmed by comparison to selected digital records Borzi [13]. In order to study the in¯uence of input motion parameters on inelastic displacement spectra, attenuation coef®cients were calculated for the EPP model. Fig. 4a±c show the in¯uence of magnitude, distance and soil conditions on inelastic displacement spectra. They con®rm the strong in¯uence of input motion parameters on inelastic displacement spectra, as already demonstrated in previous studies for elastic displacement spectra by Bommer et al. [9] (Fig. 5). To compare the elastic and inelastic spectra, attenuation coef®cients were de®ned for elastic spectra. Fig. 6 shows the in¯uence of ductility on inelastic displacement spectra. In this representation the initial elastic period of vibration corresponding to the secant stiffness at the yield point is assumed. Thus the elastic and inelastic systems with different ductility requirements are characterised by the same initial stiffness. The results con®rm the established observation that elastic and inelastic systems with the same initial stiffness reach similar maximum displacements. However, the difference between elastic and inelastic spectral ordinates calculated in this work is larger than in previous ones. This is due to the fact that the damping value used herein is small (1% of critical). Therefore, the elastic spectral ordinates for periods corresponding to the soil frequency tend to be higher than the inelastic ones. As an example, in the work of Miranda [23] a damping of 5% of critical was employed. In comparing elastic and inelastic displacements, small damping values should be employed, because the damping must represent only the dissipation of energy not related with inelastic behaviour. However the ratio between inelastic and elastic displacements tend to be equal to 1 only in the long period range, as already observed in previous studies (Miranda [23], Gupta and Sashi [24], Whittaker et al. [25], Rahnama and Krawinkler [26]). In the short period range the inelastic demand exceeds the elastic one as shown in Fig. 6b. In this work the ratio between inelastic and elastic displacement demand of a system characterised
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B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 4. In¯uence of: (a) magnitude; (b) distance; and (c) site condition on inelastic displacement spectra evaluated for the EPP model.
Fig. 5. In¯uence of: (a) magnitude; and (b) distance on elastic displacement spectra obtained by Bommer et al. (1996).
by the same initial elastic period of vibration TI is not further investigated, because in the direct displacement-based approach an equivalent elastic period of vibration TE is assumed. The substitute elastic systems used in previous
work and in the current one are shown in Fig. 7a and b, respectively. Attention is drawn to the comparison of displacement reached by the system shown in Fig. 7b, as discussed hereafter.
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
53
Fig. 6. In¯uence ductility on: (a) displacement spectra; and (b) ratio between inelastic and elastic displacement spectra (EPP model, Ms 4; d 10 km; rock site).
Fig. 7. Elastic and inelastic systems compared in: (a) previous work; and in (b) current work.
Fig. 8. In¯uence of: (a) ductility and; (b) hysteretic behaviour on inelastic displacement spectra Ms 6; d 10 km; soft soil).
Inelastic displacement spectra were also obtained for the HHS model. In the case of softening behaviour high levels of ductility were not considered. This must be done because from a practical point of view high ductilities are not significant for softening systems. In the case of post elastic stiff-
ness equal to 220 and 230% the secant stiffness for the yield point ductility up to 4 and 3, respectively, were assumed. Fig. 8 shows the in¯uence of ductility and hysteretic behaviour on inelastic displacement spectra. For these
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B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 9. Standard deviation of the calculated displacement spectra.
graphs an equivalent elastic period of vibration was assumed. This representation renders it possible to observe that there is an appreciable difference between elastic and inelastic spectral ordinates, but both the ductility and the hysteretic characteristics have negligible in¯uence. Consequently, a displacement spectra derived for a certain level of ductility and hysteretic model characterisation will be an acceptable approximation for different values of the latter parameters. The standard deviations s (Eq. (1)) of the logarithm of the calculated elastic and inelastic displacement spectra are reported in Fig. 9. From the aforementioned ®gure it is observed that the standard deviations are almost constant in the whole period range. Furthermore, the uncertainties in terms of elastic and inelastic
displacement spectral ordinates are similar. As a consequence, the calculated ratios between elastic and inelastic spectral ordinates are characterised by a lower level of uncertainty. Veri®cation of the derived spectra is non-trivial, due to the dearth of information in the literature on inelastic displacement spectra. However, the models and methods were veri®ed, alongside the dataset, by comparing a sub-set of results of this study with previously published work for force spectra and response modi®cation factors (R or q). Details are give elsewhere [27], where comparisons have con®rmed the validity of the results obtained for inelastic acceleration spectra by comparison with published ones (Vidic et al. [28], Miranda and Bertero [29], Krawinkler and Nassar [30], Newmark and Hall [31]).
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
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Fig. 10. Relation between TE and TI. Fig. 11. Relation between SD EL(TE) and SD IN(TE).
6. h -factor determination 6.1. De®nition of h -factor The displacement-based design approach herein considered is based on an idealisation of an equivalent elastic SDOF system representing the structure. Two basic characteristics of structure play an important role in determining the response to strong-ground motion: change in period of vibration and energy dissipation capacity. Therefore the maximum inelastic response can be interpreted in terms of linear elastic analysis by means of a hypothetical elastic structure (substitute structure). The substitute structure has to be equivalent to the original system in terms of period of vibration and amount of dissipated energy. This is achieved by de®ning an equivalent or effective period TE and an equivalent damping value j E. The latter may be related to a reduction coef®cient (h ) of the elastic spectral ordinates. It was observed that the most representative period of vibration for the global response is the period corresponding to the secant stiffness at maximum displacement (Gulkan and Sozen [6]). As a consequence, the relationship between the initial elastic period TI and the equivalent elastic period TE is: r m 3 T E TI 1 1 am 2 a as shown in Fig. 10. A modi®cation of the spectral ordinates is necessary to include a measure of the energy dissipation capacity of the structure. The equivalent elastic period TE is used to determine spectra in terms of periods of the substitute structure. The relationship between the elastic and inelastic spectral ordinates is shown in Fig. 11. In order to de®ne a reduction coef®cient to transform the elastic displacement spectral ordinates the following relationship is utilised:
h
SDEL TE SDIN TE
4
where SD EL and SD IN are, respectively, elastic and inelastic
spectral displacement ordinates corresponding to the equivalent elastic period. In the context of displacement-based design, h is the coef®cient equivalent to the behaviour factor, q, in forcebased design. Deformation beyond the elastic range principally has the effect of increasing both the vibration period and the energy dissipation. While both the stiffness degradation and the dissipation of energy are considered in the behaviour factor, the factor h is only a function of dissipation of energy. This is due to the fact that the increase of vibration period is already taken into account in the de®nition of an equivalent elastic period of vibration TE. The coef®cient h may be expressed as a function of damping using for example the relationship given in Eurocode 8: s 21j 5 h 7 where h is equal to 1 when j is equal to 5%. In this work the elastic displacement spectra are for j equal to 1%. Thus the Eq. (5) above must be modi®ed as: sr s 21j 7 21j 6 h 7 3 211 The damping coef®cient may now be expressed as:
j 3h 2 2 2
7
In this study, coef®cients for the de®nition of h as a function of magnitude, distance, soil condition and period are obtained. Mean values for all ductility levels and both response models (EPP and HHS) are also calculated in order to obtain damping values pertaining to displacement-based design of inelastically responding structures. 6.2. Attenuation relationships Attenuation coef®cients for the displacement modi®cation factor (h ) for EPP and HHS models were calculated. Fig. 12 demonstrates the in¯uence of magnitude and distance on h . The in¯uence of the above-listed parameters
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B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 12. In¯uence of: (a) magnitude; and (b) distance on h -factor evaluated for the EPP model.
on h can be neglected. This is a consequence of having the same dependence from input motion parameters that was observed for elastic and inelastic spectra. Displacement modi®cation factor is obtained as the ratio between elastic and inelastic displacement spectra, the in¯uence of input motion parameters on h -factor is therefore eliminated. 6.3. Average values The mean values of h -factor for different periods, magnitudes, distance and soil conditions were evaluated. The input motion parameters employed in this analysis are: ² magnitude in the range [5.5,7] with magnitude step of 0.5; ² distance less than 150 km and distance step 2 km; ² soil characteristics considered in the attenuation model.
spectral ordinates obtained for the HHS model in the case of elastic-perfectly plastic behaviour and the elastic spectral ordinates for damping equal to j E are compared. In this representation the inelastic spectral ordinates are those at equivalent elastic periods TE. When elastic and inelastic response displacements correspond to the same equivalent elastic vibration period TE, different ductility requirements correspond to different maximum displacements. Thus, comparing the inelastic displacements mentioned above with the elastic ones, different equivalent damping values are obtained for various levels of ductility. The equivalent damping value, which accounts for energy dissipation, is obtained for post-yield cycles of a given amplitude through the equation:
jE
EH 4pEEL
8
Not all the combinations of the above parameters have been used, because for earthquakes of low magnitude there is no need to consider long distances. In order to select magnitude and distance pairs of engineering signi®cance, the following limits have been set:
where EH is the energy dissipated in a full cycle of load reversals and EEL the elastic strain energy. Eq. (8) can be written for the EPP model as: 1 jE a 1 2 9 m
² for magnitude less than 5.5 distances over 50 km are excluded; ² for magnitude less than 6 distances over 75 km are excluded; ² for magnitude less than 6.5 distances over 100 km are excluded.
where a is 0.64 for the EPP hysteretic model when all the cycles of load reversals have the same amplitude up to the target ductility. Using the results obtained from the above procedure, values of a can be re-evaluated for a more realistic de®nition of j E. Eq. (9) which relates the equivalent
The mean values and the standard deviation of h obtained above are calculated considering the de®nition of the reduction coef®cient given in Eq. (4). The standard deviations calculated herein represent the dispersion of the average values of h on the period, when magnitude, distance and soil condition have changed. These values are presented in Table 2. Using Eq. (7) damping coef®cients j E corresponding to the same parameters were evaluated. In Fig. 13 the inelastic
Table 2 Average (kxl) and standard deviation (s ) values of h -factor
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
kxl
s
kxl
s
kxl
s
kxl
s
1.83 2.07 1.97 2.27 2.41
0.07 0.13 0.11 0.18 0.21
2.23 2.39 2.25 2.73 3.12
0.13 0.19 0.17 0.31 0.42
2.45 2.55 2.39 3.15 ±
0.18 0.23 0.21 0.44 ±
2.65 2.70 2.51 ± ±
0.23 0.28 0.26 ± ±
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
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Fig. 13. (a) Inelastic displacement spectra for different ductilities; and (b) elastic displacement spectra for equivalent viscous dampings (HHS model for K3 0; Ms 6; d 10 km; soft soil).
ing reach the limit curve, leading to a greater dissipation of energy.
damping value and the ductility factor is assumed for the HHS model too. Therefore, differences in hysteretic behaviour are represented by variations in a . The equivalent damping values j E and the corresponding a coef®cients are reported in Table 3. Assessment of the mean values of h lead to the following observations:
For long period systems the mean values of h -factor obtained above are unconservative, since they are in¯uenced by the high values characteristic of the response modi®cation factor of stiff systems. In order to improve the estimation of h -factor for long period systems, mean values were evaluated comparing the energy of inelastic and elastic displacement spectra, both considered for an equivalent period TE. The displacement modi®cation factor is thus given by the following expression:
1. Lower h is obtained with the EPP model than with the HHS when applied to a perfectly plastic case. This is due to the higher initial stiffness of the HHS model that uses a secant stiffness of 50% of the initial stiffness. Therefore, for the same maximum displacement the ductility demand for the EPP case is twice that of HHS case, provided the same initial stiffness is used. 2. In the hardening case lower h was obtained than in the elasto-plastic case, because the equivalent system corresponding to hardening is stiffer than that corresponding to the elasto-plastic case. 3. In the softening case the average h value is higher than in both the EPP and the hardening cases. This is due to the high values of the displacement modi®cation factor for stiff systems with softening behaviour. In terms of dissipation of energy, high h values correspond to high energy dissipation. Stiff systems tend to have a large number of load reversals. Therefore, for softening systems a large number of cycles of loading and reload-
ZT EMax
h
0 ZTEMax 0
SDEL SD
10
IN
Long period ordinates become of greater importance than short period ordinates, since the area under both the elastic and inelastic spectra in the short period range is only a small percentage of the total area. The results of this investigation for the range of ductilities taken into account are given in Tables 4 and 5. The mean values of h found from Eq. (10) are lower than those reported in Table 2. Therefore, the latter values are
Table 3 Equivalent damping j E and a values
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
j E (%)
a
j E (%)
a
j E (%)
a
j E (%)
a
8 11 10 13 15
0.16 0.22 0.19 0.27 0.31
13 15 13 20 27
0.19 0.23 0.20 0.31 0.41
16 18 15 28 ±
0.21 0.23 0.20 0.37 ±
19 20 17 ± ±
0.23 0.24 0.20 ± ±
58
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
The formulation of displacement modi®cation factor proposed is: ( h h if TE # T 11 h 1 1 h 2 1 T=T if TE . T
Table 4 Average (kxl) and standard deviation (s ) values of h -factor calculated with Eq. (10)
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
kxl
s
kxl
s
kxl
s
kxl
s
1.53 1.81 1.78 1.84 1.85
0.11 0.19 0.18 0.21 0.21
1.66 2.04 2.05 1.98 1.93
0.15 0.29 0.28 0.27 0.27
1.72 2.12 2.17 1.98 ±
0.17 0.33 0.33 0.29 ±
1.74 2.21 2.30 ± ±
0.18 0.39 0.40 ± ±
where T is the equivalent elastic period at the border of the increasing branch of the inelastic displacement spectra and h is a constant value which de®nes the displacement modi Considering the inelastic displace®cation factor up to T: ment spectra calculated in this work, an adequate value of T; for all the levels of ductilities and hysteretic behaviours used, seems to be 2 s. Therefore the constant branch of the functions de®ning h is operative for most practical applications. The assumed values of h are the mean values calculated via Eq. (10) and reported in Table 4. As a consequence of this choice the functions are conservative in the short period range. However, this is reasonable because in the short period range the displacements are small and therefore the accuracy of the estimates is not crucial. Nevertheless, h must be more conservative in the short period range because the displacement spectra present an increasing branch. Erroneous evaluation of the period of vibration can lead to an under-estimation of the required displacement capacity. This cannot occur in the long period range that is characterised by practically horizontal line of the displacement spectrum. Fig. 15 shows the approximate curves representing Eq. (11). It is evident that the approximate curves do not vary signi®cantly as a function of ductility and hysteretic behaviour.
conservative as they are not in¯uenced by the behaviour of systems in the short period range. 6.4. Period-dependent response modi®cation functions Period-dependent h functions were evaluated. The average and the standard deviation values of h -factor obtained changing input motion parameters, were calculated for all periods and ductilities considered in this study. The results of these analyses are represented in Fig. 14. The hysteretic behaviour and the ductility have a strong in¯uence on the value of h -factor, but this is evident only in the short period range. For these periods h is higher for softening systems than for perfectly elastic and hardening ones. This is due to the fact that short period systems tend to have an increased number of load reversals. Therefore, for softening system hysteresis cycles reaching the post-elastic branch of the force-displacement primary curve occur more after, thus increasing the energy dissipation. The displacements of both elastic and inelastic systems tend to the ground displacement in the long period range, independently of the hysteretic behaviour. In order to obtain usable values of h in the whole period range, they are expressed as a function of period as follows:
7. Residual inelastic displacements Perfectly feasible designs may undergo large irreversible displacements leading to a permanent off-set. This would impair the function of the structure. It is therefore of practical signi®cance to derive relationships between various inputs and response parameters and the permanent irrecoverable displacement of structures. This has been undertaken and the results are herein discussed. The response characteristics chosen are for softening behaviour with stiffness K3 of 220 and 230%. This is because in hardening and perfectly plastic cases the residual displacements are less signi®cant than in the case of softening behaviour (Kawashima et al. [32], MacRae and Kawashima [33]). The
1. a horizontal portion in the period range limited by the equivalent elastic period in which the inelastic displacement spectra reach the peak value; 2. a decaying branch in the long period range. This function must tend to 1 when the period increases. This is reasonable because both the elastic and inelastic displacements converge to the peak ground displacement if the SDOF system is in®nitely ¯exible.
Table 5 Equivalent damping j E and a values corresponding to h -factor reported in Table 4
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
j E (%)
a
j E (%)
a
j E (%)
a
j E (%)
a
5 8 8 8 8
0.10 0.16 0.15 0.16 0.17
6 10 11 10 9%
0.09 0.16 0.16 0.15 0.14
7 11 12 10 ±
0.09 0.15 0.16 0.13 ±
7 13 14 ± ±
0.08 0.15 0.17 ± ±
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
59
Fig. 14. (a) In¯uence of ductility; and (b) hysteretic behaviour on h -factor.
Fig. 15. (a) In¯uence of ductility; and (b) hysteretic behaviour on period-dependent h -factor functions.
Fig. 16. Ratio between inelastic and residual displacement spectra for softening behaviour Ms 6; d 10 km; soft site).
regression coef®cients for the de®nition of residual displacement spectra were calculated. Fig. 16 shows the ratio between maximum inelastic displacement and residual displacement. The residual displacements are a high percentage of the maximum displacements reached by the systems. In the aforementioned ®gure they are strongly dependent on
the ductility requirements and the post-elastic stiffness, as observed in previous work (Kawashima et al. [32], MacRae and Kawashima [33]). If the designer considers that such permanent displacements are unacceptable, another iteration in the displacement design cycle, described in Section 1 above, is necessary.
60
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
8. Conclusions Seismic assessment and design based on displacements provides a conceptually appealing alternative to conventional force-based design. Since one of the commonly used deformational quantities employed to assess the satisfaction of limit states are displacements, the only question that arises from considering displacement-based design is why was it not adopted earlier. Seismic design displacement spectra are three-parameter demand representations, the three parameters being effective period, relative response displacement and equivalent damping ratio. In three recently published works (Bommer and co-workers [9,10], Tolis and Faccioli [11]) displacement spectra (as a function of magnitude, distance and site condition) and modi®cations to Eurocode 8 spectrum were proposed. However, the derived spectra would not be used unless robust and veri®able relationships between global ductility factor and corresponding equivalent damping are derived. Also, in the latter papers, elastic highly damped spectra were derived to represent inelastic response. The paper introduces a new form of inelastic displacement spectra and displacement reduction factor, as well as residual displacement spectra. The results of non-linear analyses undertaken have been employed for the de®nition of an equivalent elastic system (substitute structure). The equivalence between elastic and inelastic systems is interpreted in terms of period of vibration and dissipation capacity. There is a simple relation between the initial period and equivalent elastic period of vibration, which is a function of required ductility and primary curve shape, while for the energy dissipation a reduction factor h has been proposed. This may be related to equivalent elastic damping, which can conveniently describe the energy dissipation. The displacement modi®cation factor h for displacementbased approaches is the reduction coef®cient corresponding to the behaviour factor (R or q) in force-based methods. The advantage of h over the behaviour factor is that for the de®nition of the former a correct identi®cation of ductility capacity and hysteretic behaviour of the structure is not important, because their in¯uence is already taken into account via an equivalent elastic vibration period. On the other hand, for the behaviour factor that takes into account the effects of plasticity for a force-based design or assessment method, a strong in¯uence from the ductility has been demonstrated. This is because the behaviour factor considers both the increase in vibration periods and the dissipation of energy due to inelastic behaviour. The evaluation of h is therefore an indispensable ingredient in the displacementbased design approach that was hitherto unavailable. Acknowledgements The writers would like to express their gratitude to Mr G.O. Chlimintzas for his help with the strong-motion
dataset. The regression program was kindly provided by Dr S.K. Sarma, whilst support was given by Mr D. Lee during the implementation of the hysteretic model. All the above are from Imperial College. Funding for the stay of the primary author at Imperial College was provided by the EU network programs ICONS and NODISASTR.
References [1] Calvi GM, Kingsley GR. Displacement-based seismic design of multi-degree-of-freedom bridge structures. Earthquake Engng Struct Dyn 1995;24:1247±66. [2] Kowalsky MJ, Priestley MJN, MacRae GA. Displacement-based design of RC bridge columns in seismic regions. Earthquake Engng Struct Dyn 1995;24:1623±43. [3] Priestley MJN, Seible F, Calvi GM. Seismic design and retro®t of bridges. New York: Wiley, 1996. [4] Panagiotakos TB, Fardis MN. Deformation-controlled earthquakeresistant design of RC buildings. Journal of Earthquake Engineering 1999;3(4):495±518. [5] Moehle JP. Displacement-based design of RC structures subjected to earthquakes. Earthquake Spectra 1992;8(3):403±28. [6] Gulkan P, Sozen M. Inelastic response of reinforced concrete structures to earthquake motions. ACI J 1974;71:604±10. [7] Shibata A, Sozen M. Substitute-structure method for seismic design in R/C. J Struct Div, ASCE 1976;102:1±18. [8] Qi X, Moehle JP. Displacement design approach for reinforced concrete structures subjected to earthquakes. Report No. UCB/ EERC-91/02, 1991, p. 186. [9] Bommer JJ, Elnashai AS, Chlimintzas GO, Lee D. Review and development of response spectra for displacement-based seismic design. ESEE Research Report No. 98-3, ICONS, Imperial College, London, 1998. [10] Bommer JJ, Elnashai AS. Displacement spectra for seismic design. J Earthquake Engng 1999:1±32. [11] Tolis SV, Faccioli E. Displacement design spectra. J Earthquake Engng 1999:107±25. [12] Borzi B, Elnashai AS, Faccioli E, Calvi GM, Bommer JJ. Inelastic spectra and ductility±damping relationships for displacement-based seismic design. ESEE Research Report No. 98-4, ICONS, Imperial College, London, 1998. [13] Borzi B. Design spectra based on inelastic response. PhD thesis, Politecnico di Milano, Italy, 1999. [14] Ambraseys NN, Simpson KA. Prediction of vertical response spectra in Europe. Earthquake Engng Struct Dyn 1996;25:401±12. [15] Ambraseys NN, Simpson KA, Bommer JJ. Prediction of horizontal response spectra in Europe. Earthquake Engng Struct Dyn 1996;25:371±400. [16] Ozcebe G, Saatcioglu M. Hysteretic shear model for reinforced concrete members. J Struct Engng, ASCE 1989;115:133±48. [17] Saatcioglu M, Ozcebe G, Lee BCK. Tests of reinforced concrete columns under uniaxial and beaxial load reversals. Department of Civil Engineering, University of Toronto, Toronto, Canada, 1988. [18] Saatcioglu M, Ozcebe G. Response of reinforced concrete columns to simulated seismic loading. ACI Struct J 1989:3±12. [19] Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry buildings. New York: Wiley, 1992. [20] Priestley MJN, Seible F, Calvi GM. Seismic design and retro®t of bridges. New York: Wiley, 1996. [21] Calvi GM, Pinto PE. Experimental and numerical investigations on the seismic response of bridges and recommendations for code provisions. ECOEST/PREC8 Report No. 4, Laboratorio Nacionale de Engenharia Civil, Portugal, 1996, p. 137.
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61 [22] Pinto AV. Pseudodynamic and shaking table tests on R.C. Bridges. ECOEST & PREC8, 1996, p. 5. [23] Miranda E. Evaluation of site dependent inelastic seismic design spectra. J Struct Engng, ASCE 1993;119:1319±38. [24] Gupta B, Sashi KK. Effect of hysteretic model parameters on inelastic seismic demand. In: Proceedings of the Sixth US National Conference on Earthquake Engineering, Oakland, California, 1998. [25] Whittaker A, Constantinou M, Tsopelas P. Displacement estimates for performance-based seismic design. J Struct Engng, ASCE 1998;124:905±12. [26] Rahnama M, Krawinkler H. Effects of soft soil and hysteretic model on seismic demands. Report No. 108, The John A. Blume Earthquake Engineering Center, Stafford University, Stanford, California, 1993. [27] Borzi B, Elnashai AS. Re®ned force reduction factor for seismic design. J Engng Struct 2000 (in press).
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[28] Vidic T, Fajfar P, Fischinger M. Consistent inelastic design spectra: strength and displacement. Earthquake Engng Struct Dyn 1994;23:507±21. [29] Miranda E, Bertero VV. Evaluation of strength reduction factor for earthquake-resistance design. Earthquake Spectra 1994;10:357±79. [30] Krawinkler H, Nassar AA. Seismic design based on ductility and cumulative damage demand and capacities. In: Fajfar P, Krawinkler H, editors. Nonlinear seismic analysis and design of reinforced concrete buildings. New York: Elsevier, 1992. [31] Newmark NM, Hall WJ. Earthquake spectra and design. EERI monograph series, EERI, Oakland, 1982. [32] Kawashima K, MacRae G, Hoshikuma J, Nagaya K. Residual displacement response spectra. J Struct Engng, ASCE 1998;124:523±50. [33] MacRae G, Kawashima K. Post-earthquake residual displacement of bilinear oscillators. Earthquake Engng Struct Dyn 1997;26:701±16.
www.elsevier.com/locate/soildyn
Inelastic spectra for displacement-based seismic design B. Borzi a,1, G.M. Calvi b, A.S. Elnashai a,*, E. Faccioli c, J.J. Bommer a a
Department of Civil and Environmental Engineering, Imperial College, London SW7 2BU, UK Dipartimento di Meccanica Strutturale, Universita degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy c Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy b
Accepted 6 September 2000
Abstract In recognition of the emergence of displacement-based seismic design as a potentially more rational approach than force-based techniques, this paper addresses derivation of inelastic displacement spectra and associated topics. A well-constrained earthquake strong-motion dataset is used to derive inelastic displacement spectra, displacement reduction factors and ductility±damping relationships. These are in a format amenable for use in design and assessment of structures with a wide range of response characteristics. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Attenuation relationship; Displacement spectrum; Displacement-based approach; Ductility; Damping; Displacement modi®cation factor
1. Introduction Force-based seismic design remains, in spite of its shortcomings, the method widely used in codes. However, deformation-based (Calvi and Kingsley [1], Kowalsky et al. [2], Priestley et al. [3]) or deformation-controlled (Panagiotakos and Fardis [4]) procedures have recently been proposed. The new approach utilises displacement spectra expressed as spectral ordinates versus effective period (period at maximum displacement) to quantify the demand imposed on structural systems. The origins of displacement-based design may be traced to work published as early as the 1960s, where comments on the displacements of inelastic systems and their relationship to their elastic counterparts were made (e.g. Muto et al., 1960, as reported by Moehle [5]). However, it was the work of Sozen and his associates (Gulkan and Sozen [6], Shibata and Sozen [7]) that developed the concept of a substitute structure. The substitute structure is a single degree of freedom elastic system, the characteristics of which represent the inelastic system. This concept enables the use of an elastic displacement spectrum in design, while using the displacement capacity of an inelastic system. Various contributions were made towards the development of displacement-based seismic design since the early
work mentioned above. However, it was in the 1990s that formal proposals were made to implement the emerging ideas into a design procedure. The earliest is that by Moehle and his co-workers (Moehle [5], Qi and Moehle [8]). Thereafter, a complete and workable procedure for seismic design of structures that sets aside forces and relies on displacement as a primary design quantity was proposed by Kowalsky et al. [2] for single degree of freedom systems (such as bridge piers). A parallel paper on multi-degree of freedom systems is due to Calvi and Kingsley [1]. The steps comprising the design process for SDOF systems are given below for simplicity:
* Corresponding author. Tel.: 144-20-7594-6058; fax: 144-20-72252716. E-mail address: [email protected] (A.S. Elnashai). 1 Present address: EQE International Ltd, Warrington WA3 6WJ, UK. 0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(00)00075-0
a. A target displacement for the structure is selected, based on the type of structure and the governing limit state. b. Knowing the yield displacement, and the material and structural system, a value of equivalent damping is determined. c. Displacement spectra representative of the seismotectonic environment are used. The inputs are the target displacement and the equivalent damping. The output is an effective period of vibration. d. The structure is dimensioned to give an effective period, taking into account reduced stiffness consistent with the level of deformation, equal to that obtained from the displacement spectra. e. If the effective period is not suf®ciently close to the required period, go to step b. above and repeat until convergence.
48
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
It is clear from the above that whereas displacementbased design is the logical framework for seismic design, since the primary source of seismic energy dissipation is inelastic deformation, it imposes new requirements for veri®able design. Primarily, accurate, representative and parametrically described displacement spectra are essential ingredients that have only very recently become available (Bommer et al. [9], Bommer and Elnashai [10], Tolis and Faccioli [11]). In the study by Bommer et al. [9], it was shown that a well-balanced catalogue for acceleration studies may be unsuitable for displacement purposes. The ordinates of acceleration response spectra are insensitive to the processing applied to the accelerograms, whereas the in¯uence of the ®lter parameters on the displacement spectra is pronounced. In this paper attenuation coef®cients for inelastic displacement constant ductility spectra are calculated. Attenuation coef®cients for residual displacements were also taken into account. This is of great importance in the assessment of displacement-based design situations, hence the spectra are presented as ratio of maximum response displacement. By means of a comparison between elastic and inelastic displacement spectral ordinates a reduction coef®cient taking into account the effects of real inelastic behaviour in a simpli®ed method of design based on displacement is calculated. For this coef®cient, herein denoted h -factor, regression coef®cients were evaluated (Borzi et al. [12], Borzi [13]) for all magnitudes, distances and site conditions. Further, average values obtained constitute a valid reference for new code implementations for simpli®ed methods based on displacements. 2. Input motion 2.1. Strong-motion dataset selection and processing The dataset employed for the de®nition of inelastic constant ductility spectra was presented by Bommer et al. [9] and used for the derivation of frequency-dependent attenuation equations for ordinates of displacement response spectra. All records were ®ltered individually at Imperial College (London, UK) and used to derive displacement spectra for different levels of damping, from 5 to 30%. Each record was individually processed using an elliptical ®lter and a long-period cut-off that resulted in a physically reasonable displacement time-history (Bommer and Elnashai [10]). This ®ltering process removes permanent displacements, caused by ground deformation or fault slip, in order not to mix these phenomena with transitory ground vibrations. The dataset has been adapted from that employed by Ambraseys et al. [14,15] to derive attenuation relationships for ordinates of acceleration response spectra. This is a high-quality dataset in terms of both accelerograms that have been individually corrected, and information regarding the recording stations and earthquake character-
Fig. 1. Distribution of records for: (a) rock; (b) stiff; and (c) soft soil.
istics. The original dataset of Ambraseys et al. [14,15] has been modi®ed in the following ways: 1. records for small magnitude earthquakes have been removed; 2. some site classi®cations have been corrected in the light of new information; 3. two new strong-motion records have been added; 4. all of the original records have been ®ltered with individually selected cut-offs. The necessity of eliminating all the records generated by earthquakes of magnitude less than Ms 5.5 is due to the particular interest in long-period radiation. In Fig. 1 the distribution of records comprising the dataset with regard to magnitude, distance and site classi®cation are
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
demonstrated to emphasise credentials of this set of natural records. While the source distance and the surface-wave magnitude are available for all the accelerograms, for three records the local site geology is unknown. For the remaining 180, the percentages of distribution in the three site groupings of rock, stiff and soft soil are 25.0, 51.1 and 23.9%, respectively. These percentages are close to the distribution of the original dataset, being 25.5, 54.3 and 20.2%, respectively. For two records only one component of the motion is known. Therefore a total of 364 accelerograms were processed. It is acknowledged that site intensity could be used for the selection of strong-motion datasets. However, this has its own problems, and is not considered to be superior to the selection based on magnitude, distance and site condition. 2.2. Attenuation model and regression analysis Studies concerning the evaluation of seismic hazard related to earthquakes utilise predictive models commonly referred to as attenuation relationships. These models generally express values of strong-motion parameters as a function of source characteristics, propagation path and local site geology. In order to de®ne elastic spectra, a common approach to perform a hazard analysis is to de®ne the hazard in terms of the peak ground acceleration (PGA), which anchors the zero-period ordinate for a standard spectral shape. A criticism of this approach is that the resulting spectra do not correspond to the same seismic hazard for all periods (i.e. non-uniform hazard spectra). To obtain spectra characterised according to the same seismic hazard for every spectral ordinate (uniform hazard spectra) the de®nition of period-dependent attenuation relationships are proposed. The attenuation model used in this work is that of Ambraseys et al. [14,15] employed to de®ne elastic acceleration spectra. The formulation of the attenuation relationship is: log y C1 1 C2 Ms 1 C4 log r 1 Ca Sa 1 Cs Ss 1 sP 1 where y is the strong-motion parameter to be predicted, Ms is the surface wave magnitude and: q 2 r d 2 1 h20 in which d is the shortest distance from the station to the surface projection of the fault rupture in km and h0 is a regression constant. The h0 coef®cient takes into account that the fault projection is not necessarily the source of the peak motion, but it does not represent explicitly the effect of the focal depth on the motion. The coef®cients C1, C2, C4, Ca, Cs and h0 are determined by regression. In this attenuation model three soil conditions are distinguished by the average shear wave velocity. When the shear wave velocity is higher than 750 m/s the soil is classi®ed as rock. Soft soil has a shear wave velocity is less than 360 m/s, whilst stiff soil is assumed in the intermediate
49
range of shear wave velocity. In the regression model Sa takes the value of 1 for soft soil conditions and 0 otherwise, while Ss takes the value of 1 for stiff soil conditions and 0 otherwise. Finally, P is a parameter that is multiplied by the standard deviation s and takes the value of 0 when the mean value of log(y) is calculated and 1 for the 84-percentile value of log(y). 3. Structural models 3.1. Elastic perfectly-plastic model In order to determine the in¯uence of magnitude, distance and soil condition on inelastic response spectra, attenuation relationships have been de®ned using an elastic perfectlyplastic (EPP) response model. The EPP model was employed since it is the simplest form of inelastic forceresistance as well as being the basis for early relationships between seismic motion and response modi®cation factors. Moreover, by virtue of its two parameters de®nition: level of force-resistance and stiffness, few structural characteristics are included, hence the in¯uence of strong-motion records may be better visualised. The stiffness corresponds to the period of vibration for which the spectral ordinate has to be calculated and the resistance is derived iteratively. In this work inelastic constant ductility spectra were obtained. Therefore the resistance of the system corresponds to the resistance for which the system has a required ductility equal to the target ductility. The ensuing inelastic spectra would re¯ect solely the characteristics of the input motion. 3.2. Hysteretic hardening±softening model In order to investigate the in¯uence of the response characteristics of structures on inelastic displacement spectra, a hysteretic hardening-softening model (HHS) was used (Ozcebe and Saatcioglu [16]). The structural model is characterised by the de®nition of a primary curve, unloading and reloading rules. The primary curve for a hysteretic force± displacement relationship is de®ned as the envelope curve under cyclic load reversals. For non-degrading models the primary curve is taken as the response curve under monotonic loading. In this model the primary curve is used to de®ne the limits for member strength. Two points on the primary curve have to be de®ned. It is essential to de®ne cracking and yield loads (Vcr and Vy) and the corresponding displacements (D cr and D y), as shown in Fig. 2. If, for example, this model was used to describe the hysteretic behaviour of reinforced concrete members, the cracking load would correspond to the spreading of cracks in the concrete and the yielding load would be related with the load at which the strain in bars is equal to the yield strain of steel or some other criterion can be selected by the user. Unloading and reloading branches of the HHS model have been established through a statistical analysis of experimental data
50
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 2. HHS model for structural members.
(Saatcioglu and co-workers [17,18]). The load reversal rules are brie¯y described below. Structural members exhibit stiffness degradation under cyclic loading. When the number of cycles or the amplitude of inelastic deformation increases, the system becomes softer. Furthermore, the hysteretic behaviour is affected by pinching. The axial load is an important parameter in predicting pinching effects (due to the onset of crack closure). The slope of reloading branches increases beyond the crack load. The slopes of the lines connecting the origin to the cracking point (K1 in Fig. 2) and the yield point to the cracking point in the opposite quadrant (K2 in Fig. 2) are used to de®ne the unloading branches under cyclic loads. The latter slope depends on deformation and force levels attained at the beginning of unloading. Experimental results indicate that if unloading starts between the cracking and the yield load, and the yield load has not been exceeded in the relevant quadrant, then unloading stiffness is bounded by K1 and K2. In this model a linear variation between these limits was proposed as a function of displacement ductility. If the unloading load exceeds the yield load, the unloading curve changes the slope to a value close to the cracking load.
is not representative of the structural behaviour. A low damping value of 1% was included. This viscous damping is representative of the non-hysteretic dissipation, since hysteretic damping is already included. Sensitivity analyses were undertaken and have indicated that further precision of this assumption is unwarranted. The input parameters for the HHS model described above are the monotonic curve and the relationship between axial compressive force and nominal concentric axial capacity. In order to de®ne the inelastic constant ductility spectra the magnitude of the monotonic curve is not an input parameter. It is de®ned in an iterative way forcing the relationship between maximum and yield displacements to satisfy the target ductility. To obtain the inelastic spectra and displacement modi®cation factor (h ) an approximation of the primary curve with three linear branches has been assumed (Fig. 3). Consequently, the input parameters de®ning the shape of the primary curve are: 1. the relationship between the cracking and the yielding load (Vcr/Vy); 2. the relationship between the stiffness before the cracking load and the secant stiffness (Kcr/Ky); 3. the slope of the post yield branch.
4. Procedural considerations In order to de®ne inelastic displacement spectra and displacement modi®cation (h ?, displacement ductilities of 2, 3, 4 and 6 are considered. Ductility levels higher than 6 are not included because they constitute global displacement ductility and structures very rarely have a local ductility supply commensurate with global ductility above 6. The inelastic spectra have been de®ned between 0.05 and 3 s. For the HHS model the initial elastic period can be considered either as the stiffness before Vcr or the secant stiffness. In the current work period corresponding to the secant stiffness was considered. This is because the stiffness before Vcr
Fig. 3. Shape of primary curve used in this work.
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
To select the values of parameters to be employed, extensive analysis of the in¯uence of each parameter on the inelastic spectra was undertaken. The results of a parametric investigation indicate that the parameter with the strongest in¯uence on inelastic spectra is the slope of the post yield branch. Hence ®xed ratios between Vcr and Vy and between Kcr and Ky were considered. Although the aforementioned ratios have a large variability, constant values are assumed, since the results of preliminary parametric investigations using a ®ne mesh of variation show that they do not have a signi®cant in¯uence in terms of the inelastic displacement spectra. From the experimental results of Paulay and Priestley [19], Priestley et al. [20], Calvi and Pinto [21] and Pinto [22], it is reasonable to consider the secant stiffness at the yield point equal to 50% of the stiffness before Vcr; The latter is taken equal to 30% of Vy. The ratio between the cracking and yield loads in¯uences the pinching behaviour that does not occur often for structures with loads higher than approximately 30% of the yielding load Vy. The considered representative slopes of the structural behaviour are: ² ² ² ²
K 3 0 : (elastic perfect plastic behaviour) K3 10%Ky : (hardening behaviour) K3 220%Ky : (softening behaviour) K3 230%Ky : (softening behaviour)
An axial load equal to 10% of the nominal axial load is assumed, since the model does not account for second-order effects. The above characteristics have been veri®ed to cover both new structures with seismic detailing, and existing poorly detailed structures (Borzi et al. [12], Borzi [13]). An iterative procedure was utilised for the de®nition of spectral ordinates corresponding to target ductility. In rare cases it has not been possible to obtain a convergent solution with the HHS model. The percentage of spectral ordinates that have not been considered in the regression analyses are reported in Table 1. It is observed that the number of spectral ordinates to be excluded from parametric analysis for the slope of the third branch equal to 230%Ky and ductility equal to 4, is very high. The attenuation relationship for this combination of parameters was therefore not considered. The above observations (non-convergence) are fully justi®ed by noting that highly degrading systems are inherently of low ductility. This was further investigated by comparison with a different program for inelastic spectra available to the authors (program inspect). The lack of convergence was observed Table 1 Percentage of ordinates excluded from regression analysis K3
m 2 (%)
m 3 (%)
m 4 (%)
m 6 (%)
0 10%Ky 220%Ky 230%Ky
0.16 0.03 1.67 2.45
0.58 0.13 5.12 8.16
1.23 0.27 12.00 30.98
3.16 0.80 ± ±
51
for the same range of degrading stiffness and high ductility, thus con®rming that these situations correspond to structural collapse. 5. Inelastic displacement spectra The cut-off period of every accelerogram is associated with an initial elastic period TI, which corresponds to the secant stiffness at yield. This is less than the cut-off periods employed in order to ®lter the records. For inelastic systems there is doubt as to whether spectral ordinates should correspond to the equivalent elastic period TE rather than the initial elastic period TI, both of which are less than the cut-off period. However, it was observed that even if equivalent elastic periods are somewhat higher than the cut-off period of 3 s, the inelastic spectral ordinates are still valid. This was con®rmed by comparison to selected digital records Borzi [13]. In order to study the in¯uence of input motion parameters on inelastic displacement spectra, attenuation coef®cients were calculated for the EPP model. Fig. 4a±c show the in¯uence of magnitude, distance and soil conditions on inelastic displacement spectra. They con®rm the strong in¯uence of input motion parameters on inelastic displacement spectra, as already demonstrated in previous studies for elastic displacement spectra by Bommer et al. [9] (Fig. 5). To compare the elastic and inelastic spectra, attenuation coef®cients were de®ned for elastic spectra. Fig. 6 shows the in¯uence of ductility on inelastic displacement spectra. In this representation the initial elastic period of vibration corresponding to the secant stiffness at the yield point is assumed. Thus the elastic and inelastic systems with different ductility requirements are characterised by the same initial stiffness. The results con®rm the established observation that elastic and inelastic systems with the same initial stiffness reach similar maximum displacements. However, the difference between elastic and inelastic spectral ordinates calculated in this work is larger than in previous ones. This is due to the fact that the damping value used herein is small (1% of critical). Therefore, the elastic spectral ordinates for periods corresponding to the soil frequency tend to be higher than the inelastic ones. As an example, in the work of Miranda [23] a damping of 5% of critical was employed. In comparing elastic and inelastic displacements, small damping values should be employed, because the damping must represent only the dissipation of energy not related with inelastic behaviour. However the ratio between inelastic and elastic displacements tend to be equal to 1 only in the long period range, as already observed in previous studies (Miranda [23], Gupta and Sashi [24], Whittaker et al. [25], Rahnama and Krawinkler [26]). In the short period range the inelastic demand exceeds the elastic one as shown in Fig. 6b. In this work the ratio between inelastic and elastic displacement demand of a system characterised
52
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 4. In¯uence of: (a) magnitude; (b) distance; and (c) site condition on inelastic displacement spectra evaluated for the EPP model.
Fig. 5. In¯uence of: (a) magnitude; and (b) distance on elastic displacement spectra obtained by Bommer et al. (1996).
by the same initial elastic period of vibration TI is not further investigated, because in the direct displacement-based approach an equivalent elastic period of vibration TE is assumed. The substitute elastic systems used in previous
work and in the current one are shown in Fig. 7a and b, respectively. Attention is drawn to the comparison of displacement reached by the system shown in Fig. 7b, as discussed hereafter.
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
53
Fig. 6. In¯uence ductility on: (a) displacement spectra; and (b) ratio between inelastic and elastic displacement spectra (EPP model, Ms 4; d 10 km; rock site).
Fig. 7. Elastic and inelastic systems compared in: (a) previous work; and in (b) current work.
Fig. 8. In¯uence of: (a) ductility and; (b) hysteretic behaviour on inelastic displacement spectra Ms 6; d 10 km; soft soil).
Inelastic displacement spectra were also obtained for the HHS model. In the case of softening behaviour high levels of ductility were not considered. This must be done because from a practical point of view high ductilities are not significant for softening systems. In the case of post elastic stiff-
ness equal to 220 and 230% the secant stiffness for the yield point ductility up to 4 and 3, respectively, were assumed. Fig. 8 shows the in¯uence of ductility and hysteretic behaviour on inelastic displacement spectra. For these
54
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 9. Standard deviation of the calculated displacement spectra.
graphs an equivalent elastic period of vibration was assumed. This representation renders it possible to observe that there is an appreciable difference between elastic and inelastic spectral ordinates, but both the ductility and the hysteretic characteristics have negligible in¯uence. Consequently, a displacement spectra derived for a certain level of ductility and hysteretic model characterisation will be an acceptable approximation for different values of the latter parameters. The standard deviations s (Eq. (1)) of the logarithm of the calculated elastic and inelastic displacement spectra are reported in Fig. 9. From the aforementioned ®gure it is observed that the standard deviations are almost constant in the whole period range. Furthermore, the uncertainties in terms of elastic and inelastic
displacement spectral ordinates are similar. As a consequence, the calculated ratios between elastic and inelastic spectral ordinates are characterised by a lower level of uncertainty. Veri®cation of the derived spectra is non-trivial, due to the dearth of information in the literature on inelastic displacement spectra. However, the models and methods were veri®ed, alongside the dataset, by comparing a sub-set of results of this study with previously published work for force spectra and response modi®cation factors (R or q). Details are give elsewhere [27], where comparisons have con®rmed the validity of the results obtained for inelastic acceleration spectra by comparison with published ones (Vidic et al. [28], Miranda and Bertero [29], Krawinkler and Nassar [30], Newmark and Hall [31]).
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
55
Fig. 10. Relation between TE and TI. Fig. 11. Relation between SD EL(TE) and SD IN(TE).
6. h -factor determination 6.1. De®nition of h -factor The displacement-based design approach herein considered is based on an idealisation of an equivalent elastic SDOF system representing the structure. Two basic characteristics of structure play an important role in determining the response to strong-ground motion: change in period of vibration and energy dissipation capacity. Therefore the maximum inelastic response can be interpreted in terms of linear elastic analysis by means of a hypothetical elastic structure (substitute structure). The substitute structure has to be equivalent to the original system in terms of period of vibration and amount of dissipated energy. This is achieved by de®ning an equivalent or effective period TE and an equivalent damping value j E. The latter may be related to a reduction coef®cient (h ) of the elastic spectral ordinates. It was observed that the most representative period of vibration for the global response is the period corresponding to the secant stiffness at maximum displacement (Gulkan and Sozen [6]). As a consequence, the relationship between the initial elastic period TI and the equivalent elastic period TE is: r m 3 T E TI 1 1 am 2 a as shown in Fig. 10. A modi®cation of the spectral ordinates is necessary to include a measure of the energy dissipation capacity of the structure. The equivalent elastic period TE is used to determine spectra in terms of periods of the substitute structure. The relationship between the elastic and inelastic spectral ordinates is shown in Fig. 11. In order to de®ne a reduction coef®cient to transform the elastic displacement spectral ordinates the following relationship is utilised:
h
SDEL TE SDIN TE
4
where SD EL and SD IN are, respectively, elastic and inelastic
spectral displacement ordinates corresponding to the equivalent elastic period. In the context of displacement-based design, h is the coef®cient equivalent to the behaviour factor, q, in forcebased design. Deformation beyond the elastic range principally has the effect of increasing both the vibration period and the energy dissipation. While both the stiffness degradation and the dissipation of energy are considered in the behaviour factor, the factor h is only a function of dissipation of energy. This is due to the fact that the increase of vibration period is already taken into account in the de®nition of an equivalent elastic period of vibration TE. The coef®cient h may be expressed as a function of damping using for example the relationship given in Eurocode 8: s 21j 5 h 7 where h is equal to 1 when j is equal to 5%. In this work the elastic displacement spectra are for j equal to 1%. Thus the Eq. (5) above must be modi®ed as: sr s 21j 7 21j 6 h 7 3 211 The damping coef®cient may now be expressed as:
j 3h 2 2 2
7
In this study, coef®cients for the de®nition of h as a function of magnitude, distance, soil condition and period are obtained. Mean values for all ductility levels and both response models (EPP and HHS) are also calculated in order to obtain damping values pertaining to displacement-based design of inelastically responding structures. 6.2. Attenuation relationships Attenuation coef®cients for the displacement modi®cation factor (h ) for EPP and HHS models were calculated. Fig. 12 demonstrates the in¯uence of magnitude and distance on h . The in¯uence of the above-listed parameters
56
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
Fig. 12. In¯uence of: (a) magnitude; and (b) distance on h -factor evaluated for the EPP model.
on h can be neglected. This is a consequence of having the same dependence from input motion parameters that was observed for elastic and inelastic spectra. Displacement modi®cation factor is obtained as the ratio between elastic and inelastic displacement spectra, the in¯uence of input motion parameters on h -factor is therefore eliminated. 6.3. Average values The mean values of h -factor for different periods, magnitudes, distance and soil conditions were evaluated. The input motion parameters employed in this analysis are: ² magnitude in the range [5.5,7] with magnitude step of 0.5; ² distance less than 150 km and distance step 2 km; ² soil characteristics considered in the attenuation model.
spectral ordinates obtained for the HHS model in the case of elastic-perfectly plastic behaviour and the elastic spectral ordinates for damping equal to j E are compared. In this representation the inelastic spectral ordinates are those at equivalent elastic periods TE. When elastic and inelastic response displacements correspond to the same equivalent elastic vibration period TE, different ductility requirements correspond to different maximum displacements. Thus, comparing the inelastic displacements mentioned above with the elastic ones, different equivalent damping values are obtained for various levels of ductility. The equivalent damping value, which accounts for energy dissipation, is obtained for post-yield cycles of a given amplitude through the equation:
jE
EH 4pEEL
8
Not all the combinations of the above parameters have been used, because for earthquakes of low magnitude there is no need to consider long distances. In order to select magnitude and distance pairs of engineering signi®cance, the following limits have been set:
where EH is the energy dissipated in a full cycle of load reversals and EEL the elastic strain energy. Eq. (8) can be written for the EPP model as: 1 jE a 1 2 9 m
² for magnitude less than 5.5 distances over 50 km are excluded; ² for magnitude less than 6 distances over 75 km are excluded; ² for magnitude less than 6.5 distances over 100 km are excluded.
where a is 0.64 for the EPP hysteretic model when all the cycles of load reversals have the same amplitude up to the target ductility. Using the results obtained from the above procedure, values of a can be re-evaluated for a more realistic de®nition of j E. Eq. (9) which relates the equivalent
The mean values and the standard deviation of h obtained above are calculated considering the de®nition of the reduction coef®cient given in Eq. (4). The standard deviations calculated herein represent the dispersion of the average values of h on the period, when magnitude, distance and soil condition have changed. These values are presented in Table 2. Using Eq. (7) damping coef®cients j E corresponding to the same parameters were evaluated. In Fig. 13 the inelastic
Table 2 Average (kxl) and standard deviation (s ) values of h -factor
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
kxl
s
kxl
s
kxl
s
kxl
s
1.83 2.07 1.97 2.27 2.41
0.07 0.13 0.11 0.18 0.21
2.23 2.39 2.25 2.73 3.12
0.13 0.19 0.17 0.31 0.42
2.45 2.55 2.39 3.15 ±
0.18 0.23 0.21 0.44 ±
2.65 2.70 2.51 ± ±
0.23 0.28 0.26 ± ±
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
57
Fig. 13. (a) Inelastic displacement spectra for different ductilities; and (b) elastic displacement spectra for equivalent viscous dampings (HHS model for K3 0; Ms 6; d 10 km; soft soil).
ing reach the limit curve, leading to a greater dissipation of energy.
damping value and the ductility factor is assumed for the HHS model too. Therefore, differences in hysteretic behaviour are represented by variations in a . The equivalent damping values j E and the corresponding a coef®cients are reported in Table 3. Assessment of the mean values of h lead to the following observations:
For long period systems the mean values of h -factor obtained above are unconservative, since they are in¯uenced by the high values characteristic of the response modi®cation factor of stiff systems. In order to improve the estimation of h -factor for long period systems, mean values were evaluated comparing the energy of inelastic and elastic displacement spectra, both considered for an equivalent period TE. The displacement modi®cation factor is thus given by the following expression:
1. Lower h is obtained with the EPP model than with the HHS when applied to a perfectly plastic case. This is due to the higher initial stiffness of the HHS model that uses a secant stiffness of 50% of the initial stiffness. Therefore, for the same maximum displacement the ductility demand for the EPP case is twice that of HHS case, provided the same initial stiffness is used. 2. In the hardening case lower h was obtained than in the elasto-plastic case, because the equivalent system corresponding to hardening is stiffer than that corresponding to the elasto-plastic case. 3. In the softening case the average h value is higher than in both the EPP and the hardening cases. This is due to the high values of the displacement modi®cation factor for stiff systems with softening behaviour. In terms of dissipation of energy, high h values correspond to high energy dissipation. Stiff systems tend to have a large number of load reversals. Therefore, for softening systems a large number of cycles of loading and reload-
ZT EMax
h
0 ZTEMax 0
SDEL SD
10
IN
Long period ordinates become of greater importance than short period ordinates, since the area under both the elastic and inelastic spectra in the short period range is only a small percentage of the total area. The results of this investigation for the range of ductilities taken into account are given in Tables 4 and 5. The mean values of h found from Eq. (10) are lower than those reported in Table 2. Therefore, the latter values are
Table 3 Equivalent damping j E and a values
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
j E (%)
a
j E (%)
a
j E (%)
a
j E (%)
a
8 11 10 13 15
0.16 0.22 0.19 0.27 0.31
13 15 13 20 27
0.19 0.23 0.20 0.31 0.41
16 18 15 28 ±
0.21 0.23 0.20 0.37 ±
19 20 17 ± ±
0.23 0.24 0.20 ± ±
58
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
The formulation of displacement modi®cation factor proposed is: ( h h if TE # T 11 h 1 1 h 2 1 T=T if TE . T
Table 4 Average (kxl) and standard deviation (s ) values of h -factor calculated with Eq. (10)
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
kxl
s
kxl
s
kxl
s
kxl
s
1.53 1.81 1.78 1.84 1.85
0.11 0.19 0.18 0.21 0.21
1.66 2.04 2.05 1.98 1.93
0.15 0.29 0.28 0.27 0.27
1.72 2.12 2.17 1.98 ±
0.17 0.33 0.33 0.29 ±
1.74 2.21 2.30 ± ±
0.18 0.39 0.40 ± ±
where T is the equivalent elastic period at the border of the increasing branch of the inelastic displacement spectra and h is a constant value which de®nes the displacement modi Considering the inelastic displace®cation factor up to T: ment spectra calculated in this work, an adequate value of T; for all the levels of ductilities and hysteretic behaviours used, seems to be 2 s. Therefore the constant branch of the functions de®ning h is operative for most practical applications. The assumed values of h are the mean values calculated via Eq. (10) and reported in Table 4. As a consequence of this choice the functions are conservative in the short period range. However, this is reasonable because in the short period range the displacements are small and therefore the accuracy of the estimates is not crucial. Nevertheless, h must be more conservative in the short period range because the displacement spectra present an increasing branch. Erroneous evaluation of the period of vibration can lead to an under-estimation of the required displacement capacity. This cannot occur in the long period range that is characterised by practically horizontal line of the displacement spectrum. Fig. 15 shows the approximate curves representing Eq. (11). It is evident that the approximate curves do not vary signi®cantly as a function of ductility and hysteretic behaviour.
conservative as they are not in¯uenced by the behaviour of systems in the short period range. 6.4. Period-dependent response modi®cation functions Period-dependent h functions were evaluated. The average and the standard deviation values of h -factor obtained changing input motion parameters, were calculated for all periods and ductilities considered in this study. The results of these analyses are represented in Fig. 14. The hysteretic behaviour and the ductility have a strong in¯uence on the value of h -factor, but this is evident only in the short period range. For these periods h is higher for softening systems than for perfectly elastic and hardening ones. This is due to the fact that short period systems tend to have an increased number of load reversals. Therefore, for softening system hysteresis cycles reaching the post-elastic branch of the force-displacement primary curve occur more after, thus increasing the energy dissipation. The displacements of both elastic and inelastic systems tend to the ground displacement in the long period range, independently of the hysteretic behaviour. In order to obtain usable values of h in the whole period range, they are expressed as a function of period as follows:
7. Residual inelastic displacements Perfectly feasible designs may undergo large irreversible displacements leading to a permanent off-set. This would impair the function of the structure. It is therefore of practical signi®cance to derive relationships between various inputs and response parameters and the permanent irrecoverable displacement of structures. This has been undertaken and the results are herein discussed. The response characteristics chosen are for softening behaviour with stiffness K3 of 220 and 230%. This is because in hardening and perfectly plastic cases the residual displacements are less signi®cant than in the case of softening behaviour (Kawashima et al. [32], MacRae and Kawashima [33]). The
1. a horizontal portion in the period range limited by the equivalent elastic period in which the inelastic displacement spectra reach the peak value; 2. a decaying branch in the long period range. This function must tend to 1 when the period increases. This is reasonable because both the elastic and inelastic displacements converge to the peak ground displacement if the SDOF system is in®nitely ¯exible.
Table 5 Equivalent damping j E and a values corresponding to h -factor reported in Table 4
m2
EPP K3 0 K3 10%Ky K3 220%Ky K3 230%Ky
m3
m4
m6
j E (%)
a
j E (%)
a
j E (%)
a
j E (%)
a
5 8 8 8 8
0.10 0.16 0.15 0.16 0.17
6 10 11 10 9%
0.09 0.16 0.16 0.15 0.14
7 11 12 10 ±
0.09 0.15 0.16 0.13 ±
7 13 14 ± ±
0.08 0.15 0.17 ± ±
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
59
Fig. 14. (a) In¯uence of ductility; and (b) hysteretic behaviour on h -factor.
Fig. 15. (a) In¯uence of ductility; and (b) hysteretic behaviour on period-dependent h -factor functions.
Fig. 16. Ratio between inelastic and residual displacement spectra for softening behaviour Ms 6; d 10 km; soft site).
regression coef®cients for the de®nition of residual displacement spectra were calculated. Fig. 16 shows the ratio between maximum inelastic displacement and residual displacement. The residual displacements are a high percentage of the maximum displacements reached by the systems. In the aforementioned ®gure they are strongly dependent on
the ductility requirements and the post-elastic stiffness, as observed in previous work (Kawashima et al. [32], MacRae and Kawashima [33]). If the designer considers that such permanent displacements are unacceptable, another iteration in the displacement design cycle, described in Section 1 above, is necessary.
60
B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61
8. Conclusions Seismic assessment and design based on displacements provides a conceptually appealing alternative to conventional force-based design. Since one of the commonly used deformational quantities employed to assess the satisfaction of limit states are displacements, the only question that arises from considering displacement-based design is why was it not adopted earlier. Seismic design displacement spectra are three-parameter demand representations, the three parameters being effective period, relative response displacement and equivalent damping ratio. In three recently published works (Bommer and co-workers [9,10], Tolis and Faccioli [11]) displacement spectra (as a function of magnitude, distance and site condition) and modi®cations to Eurocode 8 spectrum were proposed. However, the derived spectra would not be used unless robust and veri®able relationships between global ductility factor and corresponding equivalent damping are derived. Also, in the latter papers, elastic highly damped spectra were derived to represent inelastic response. The paper introduces a new form of inelastic displacement spectra and displacement reduction factor, as well as residual displacement spectra. The results of non-linear analyses undertaken have been employed for the de®nition of an equivalent elastic system (substitute structure). The equivalence between elastic and inelastic systems is interpreted in terms of period of vibration and dissipation capacity. There is a simple relation between the initial period and equivalent elastic period of vibration, which is a function of required ductility and primary curve shape, while for the energy dissipation a reduction factor h has been proposed. This may be related to equivalent elastic damping, which can conveniently describe the energy dissipation. The displacement modi®cation factor h for displacementbased approaches is the reduction coef®cient corresponding to the behaviour factor (R or q) in force-based methods. The advantage of h over the behaviour factor is that for the de®nition of the former a correct identi®cation of ductility capacity and hysteretic behaviour of the structure is not important, because their in¯uence is already taken into account via an equivalent elastic vibration period. On the other hand, for the behaviour factor that takes into account the effects of plasticity for a force-based design or assessment method, a strong in¯uence from the ductility has been demonstrated. This is because the behaviour factor considers both the increase in vibration periods and the dissipation of energy due to inelastic behaviour. The evaluation of h is therefore an indispensable ingredient in the displacementbased design approach that was hitherto unavailable. Acknowledgements The writers would like to express their gratitude to Mr G.O. Chlimintzas for his help with the strong-motion
dataset. The regression program was kindly provided by Dr S.K. Sarma, whilst support was given by Mr D. Lee during the implementation of the hysteretic model. All the above are from Imperial College. Funding for the stay of the primary author at Imperial College was provided by the EU network programs ICONS and NODISASTR.
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[28] Vidic T, Fajfar P, Fischinger M. Consistent inelastic design spectra: strength and displacement. Earthquake Engng Struct Dyn 1994;23:507±21. [29] Miranda E, Bertero VV. Evaluation of strength reduction factor for earthquake-resistance design. Earthquake Spectra 1994;10:357±79. [30] Krawinkler H, Nassar AA. Seismic design based on ductility and cumulative damage demand and capacities. In: Fajfar P, Krawinkler H, editors. Nonlinear seismic analysis and design of reinforced concrete buildings. New York: Elsevier, 1992. [31] Newmark NM, Hall WJ. Earthquake spectra and design. EERI monograph series, EERI, Oakland, 1982. [32] Kawashima K, MacRae G, Hoshikuma J, Nagaya K. Residual displacement response spectra. J Struct Engng, ASCE 1998;124:523±50. [33] MacRae G, Kawashima K. Post-earthquake residual displacement of bilinear oscillators. Earthquake Engng Struct Dyn 1997;26:701±16.