Evaluating a rigid-plastic method to estimate the earthquake ductility demand on structures

Transcription

1 Earthquake Resistant Engineering Structures VIII 261 Evaluating a rigid-plastic method to estimate the earthquake ductilit demand on structures M. C. orcu & G. Carta Dipartimento di Ingegneria Strutturale, Universit of Cagliari, Ital Abstract In order to evaluate the reliabilit of a rigid-plastic method in estimating the earthquake displacement ductilit demand, the present paper applies the method to hundreds of different elastic-plastic oscillators under more than thirt recorded earthquakes. The mean ratio of the predicted value over the exact value of the displacement ductilit demand is computed and plotted as a function of the vibration period of the oscillator for different values of the ield acceleration. The results show that, whatever the oscillator and the earthquake, the rigidplastic method leads to a generall conservative estimate of the inelastic displacement demand. Mean errors less than 15% are found both for comparativel short-period oscillators and for comparativel long-period oscillators. For medium-period oscillators, the relative mean error is generall less than 30%, even for ver high levels of ductilit demand. Some advantages of the rigid-plastic method with respect to other approximate methods are also discussed in the paper. Kewords: earthquake ductilit demand, seismic inelastic displacement prediction, rigid-plastic method. 1 Introduction The assessment of the earthquake ductilit demand on structures is often carried out b means of approximate methods, most of which are based on the theor of linear elastic oscillators, cf. e.g. Rosemblueth and Herrera [1]; Gulkan and Sozen [2]; Iwan [3]; Kowalsk et al. [4]; Newmark and Hall [5]; Miranda [6]. An alternative method was proposed b aglietti and orcu [7] and subsequentl improved b orcu and Carta [8, 9], which predicts the imum plastic displacement of an elastic-plastic oscillator from that of a rigid-plastic WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress doi: /eres110221

2 262 Earthquake Resistant Engineering Structures VIII oscillator possessing the same ratio between ield strength and mass (ield acceleration). The peak displacement of the latter can be obtained from the earthquake rigid-plastic pseudo-spectrum, which is a single-curve response diagram (aglietti and orcu [7]; Domingues Costa et al [10]; orcu and Mascia [11]). Section 2 recalls how the rigid-plastic method can be applied in practice. To evaluate the reliabilit of such a method, the present paper analses the results from hundreds of different elastic-plastic oscillators subjected to a significant variet of recorded ground motions. The mean ratio between estimated and calculated values of the displacement ductilit demand is evaluated as a function of the period T of the oscillator for different values of the ield acceleration. The results provided in Section 3 show that the rigid-plastic method is generall conservative. Moreover, it leads to small relative errors both in the short and in the long period range. Whereas in the medium period range the error ma reach 30%. In order to keep the errors of the rigid-plastic method below small percentage also in this range, an improvement of the empirical formula to be applied for medium period oscillator would be advisable. It should be noted, however, that these errors usuall refer to high levels of ductilit demand. Some favourable features of the rigid-plastic method are finall discussed in Section 4. The method ma be, in fact, faster to appl than other approximate methods available in current literature, which usuall require iteration procedures to estimate the ductilit demand on a structure. In addition, it singles out the range of periods in which the considered elastic-plastic oscillators ma plasticall ield under a given earthquake. This is a general result, which can be exploited b an method that aims to predict the earthquake inelastic displacement demand. Also when compared to other approximate methods, the rigid-plastic method is shown to provide good enough estimates, even when the plastic displacements are ver large. Other authors adopted a rigid-plastic approximation to model the response of ductile structures (e.g. Makris and Black [12]; Hibino et al. [13]). A rigid-plastic approach was also recentl proposed b Domingues Costa et al. [14], which predicts the imum plastic displacements of MDOF buildings b means of equivalent generalized SDOF sstems. Thanks to this, the rigid-plastic method applied in the present paper could also be exploited to assess the seismic ductilit demand of MDOF sstems. 2 The rigid-plastic method prediction The earthquake ductilit demand of an elastic-perfectl-plastic oscillator can be defined b means of the following ratio: u, (1) u also referred to as the ductilit factor or displacement ductilit ratio (Chopra [15]). Here u is the absolute value of the imum displacement of the WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

3 Earthquake Resistant Engineering Structures VIII 263 oscillator, while u is the absolute displacement at ield. When the elasticplastic oscillator deforms into the plastic range, u is greater than u and, consequentl, the ductilit factor becomes greater than unit. In this case, u can be decomposed as follows: u u u, (2) where u denotes the absolute value of the imum plastic displacement. The ductilit ratio then becomes: u 1, (3) u On the other hand, the absolute displacement at ield u is given b: u F 2 T a 2, (4) k 4π k, T, F and a being the stiffness, the natural period of vibration, the ield strength and the ield acceleration of the oscillator, respectivel. The ield acceleration is given b the ratio between the ield strength F and the mass M of the oscillator, and represents the imum acceleration that the oscillator ma reach during the motion (Chopra [15]; aglietti and orcu [7]). All displacements are here intended to be relative to the ground. In view of eq. (4), the ductilit factor can be also expressed as: 2 4π 1 u 2 (5) T a This equation shows that assessing the ductilit demand for an elastic-plastic oscillator requires that u be known. For a given earthquake, u depends on the oscillator vibration period T, damping ratio and ield acceleration a. It should be calculated b numerical integration of the non-linear equations of motion of the elastic-plastic oscillator. The rigid-plastic method proposed b orcu and Carta [9] provides a simpler, though approximate, wa to predict u. It leads to the following formulae: R u = u for T 0.13 T, (6a) R + p u u u for 0.13T T T, (6b) WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

4 264 Earthquake Resistant Engineering Structures VIII = R T u u T T T for T T T, (6c) u = 0 for T T, (6d) where a 2 2.5T T 1 T g 2 T T. 4 T T T u p = a In the above formula time should be expressed in seconds. R In equations (6a)-(6c) u represents the imum displacement that would be reached under the given earthquake b a rigid-plastic oscillator possessing the same ield acceleration as that of the elastic-plastic oscillator under R consideration. For a given earthquake u onl depends on a, and can be calculated b integrating the non-linear equations of motion of the rigid-plastic oscillator, which are simpler than those of the elastic-plastic one (aglietti and orcu [7]). Alternativel, and more quickl, it can be obtained from the rigidplastic pseudo-spectrum of the earthquake, (cf. aglietti and orcu [7]; orcu and R Mascia [11]; Domingues Costa et al. [10]). An instance of how obtaining u from a rigid-plastic pseudo-spectrum, for a given value of a, is provided in Figure 1. (7) R u 0.20 (m) arkfield, California, 90, 2004 Rigid-lastic seudo-spectrum a /g Figure 1: R Obtaining u from the earthquake rigid-plastic pseudo-spectrum. WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

5 Earthquake Resistant Engineering Structures VIII 265 Quantities T and T appearing in eqns (6) and (7) are two characteristic values of period that depend on the earthquake and are function of a and. The can easil be obtained b intercepting the displacement elastic response spectrum with the following curves (orcu and Carta [8]): Figure 2 shows how obtaining 2 at 2 R 8π u T a 2 2 4π at u (, ) 1, (8) 2 at u T, a (9) 2 4π T and T from an elastic response spectrum. EL u (m) arkfield, California, 90, 2004 Elastic Response Spectrum =10% Curve (8) Curve (9) T T T (s) Figure 2: Determining T and T from the elastic displacement response spectrum (for a 0.2g ). R Once that u is taken from the rigid-plastic pseudo-spectrum and the pair of characteristic periods T and T are found from the elastic response p spectrum, u can be predicted directl from eqns (6) and (7). An example on how to get such a prediction is given in Figure 3. p B introducing the estimated value of u into eqn (5), the ductilit factor is finall obtained. For the same instance considered in Figures 1-3, Figure 4 plots both the predicted and the exact values of the ductilit factor. Figure 4 shows that tends to unit as T tends to T. Indeed, for T T it is p u 0 and, therefore, 1, as follows from eqn (5). According to eqn (5), WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

6 266 Earthquake Resistant Engineering Structures VIII moreover, for an given value of a the ductilit factor tends to infinit as T tends to zero. This makes a direct comparison between predicted and exact values of inaccurate in the short-period range. For this reason, the ratio between predicted and exact values of will instead be considered in the next section, which allows for a comparison in the whole range of periods. u 0.20 (m) arkfield, California, 90, 2004 Rigid-lastic Method rediction 0.15 =10% a =0.2g exact values T T T T (s) Figure 3: redicting the peak plastic displacements through the rigid-plastic method Figure 4: exact values arkfield, California, 90, 2004 Rigid-lastic Method rediction =10% a =0.2g T (s) 0.13T T T redicting the ductilit factor b means of the rigid-plastic method. WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

7 Earthquake Resistant Engineering Structures VIII Mean relative errors in the rigid-plastic prediction In order to evaluate the relative errors that can be made when estimating the earthquake ductilit demand b means of the rigid-plastic method, the ratio between the earthquake displacement ductilit factor as estimated b means of the rigid-plastic method, sa, and the exact value of the same factor computed b a non-linear time-histor analsis will be considered below. In view of eq. (3), this ratio can be expressed as: u r u u u, (10) where u indicates the plastic displacement estimated through eqns (6) and (7) and u the calculated exact value. The ratio r gives the relative error we introduce when estimating the ductilit factor with the rigid-plastic method. A comparison between eqn (10) and eqn (3) shows that r also gives the ratio between the estimated and the calculated total displacements. As the estimated and the calculated values of the plastic displacement tend to coincide, the ratio r tends to unit, which means that no error is committed in estimating the ductilit demand, or, similarl, the total displacement. This obviousl happens if T equals zero (rigid-plastic behaviour), and also when T p is equal to T, since in both these cases u u. On the other hand, r is p equal to unit also when T T, since it is u u 0. In all these cases, the rigid-plastic method predicts the earthquake displacement demand exactl. Otherwise, some errors can be produced. Table 1: Earthquakes considered in the present investigation. 1 Ardal (Iran), LONG, Landers (California), LCN000, Cape Mendocino (Cal), ET090, Loma rieta (Cal), CLS000, Cartago (Costa Rica), LONG, Mammoth Lakes(Cal), LLUL000, Chamoli (India), N20E, Montenegro, N-S, Chi Chi (Taiwan), CHY041N, Morgan Hill (Cal), CYC195, Coalinga (California), D-TSM360, N. alm Springs (Cal), NS300, Duzce (Turke), DZC270, arkfield (California), C02065, Edgecumbe (New Zealand), N07W, arkfield (California), 90, El Salvador, LONG, San Fernando (California), S16E, Erzincan (Turke), N279, South Iceland, LONG, Friuli (Ital), E-W, Spitak (Armenia), GUK000, Gazli (Uzbekistan), E-W, Superstitn Hills (Cal), B-SU135, Imperial Valle (Cal), H-BCR230, Tabas (Iran), N74E, Irpinia (Ital), A-STU270, Tabas (Iran), TAB-LN, Kobe (Japan), N35W, Trinidad, B-RDE000, Kocaeli (Turke), ATS000, Victoria (Mexico), CE045, 1980 WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

8 268 Earthquake Resistant Engineering Structures VIII To evaluate the extent of these errors, a numerical investigation was carried out. B referring to the earthquakes listed in Table 1, the value of the ratio r as defined b eqn (10) was computed for elastic-plastic oscillators possessing different realistic values of a and a damping ratio. The latter is a tpical value for damping ratio when stress is at the ield point, cf. Chopra [15]. For each earthquake and for each value of a, different values of natural period, ranging from zero to T, were considered. For each value of T, the mean value of the ratio r, sa M r, was finall obtained. The resulting diagram is presented in Figure 5. It shows that, whatever the ield acceleration and whatever the natural period T of the elastic-plastic oscillator, the rigid-plastic method does on average provide a conservative estimate of the displacement ductilit factor (values of M r larger than one). In particular, the mean relative errors are generall lower than 30%. The are ver low for T 0.2s and alwas lower than 15% for T 0.75s. Mean ratio Mr a = 0.3g a = 0.2g a = 0.1g T(s) Figure 5: Mean ratio of predicted to calculated imum displacements for different values of a (=10%). 4 Evaluating the rigid-plastic method Most of the approximate methods proposed in current literature estimate the earthquake ductilit demand b means of parameters that are function of itself. As a consequence, the usuall require iteration procedures which ma also impl convergence problems, cf. e.g. Chopra and Goel [16]; Miranda and Akkar [17]. An in-depth evaluation of the accurac of some of these methods was done b Miranda and Ruiz Garcia [18] and b Akkar and Miranda [19]. WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

9 Earthquake Resistant Engineering Structures VIII 269 On the contrar, the rigid-plastic method does not involve an iteration procedure. Should the rigid-plastic pseudo-spectrum and the elastic response spectrum of the considered earthquake be available, the method predicts the inelastic displacement demand on an elastic-plastic oscillator b means of a direct procedure, see Section 2. What is more, the rigid-plastic method spots the range of periods in which the inelastic demand prediction actuall needs to be obtained under a given earthquake. This range is alwas given b 0 T T. The actual value of T, which is different for different earthquakes and for different values of a and ma be easil obtained from the elastic response spectrum of the considered earthquake, see Figure 2. General as it is, this result could be adopted b an approximate method that aims at estimating the earthquake ductilit demand on elastic-plastic oscillators. Within the above range, the rigid-plastic method gives a good estimate of the peak plastic displacement of comparativel short-period oscillators (sa T 0.25s) and comparativel long-period oscillators ( T 0.75s), see Figure 5. On the contrar, high errors are generall encountered in the short period range when other approximate methods are adopted, as can be inferred from the diagrams presented b Miranda and Ruiz Garcia [18] and b Akkar and Miranda [19]. For T 0.75s the rigid-plastic method prediction is on average comparable with that of other approximate methods (cf. e.g. the results presented b Miranda and Ruiz Garcia [18] and b Akkar and Miranda [19]). In the medium period range the rigid-plastic prediction ma be, however, less satisfactor. Here the empirical formula (7) plas a fundamental part for the rigid-plastic prediction. Most of the mean errors found in the range 0.25s T 0.75s (see Figure 5) can be actuall put down to the appliance of such a formula. It should be noted that in the same range of periods some other approximate methods might give a better prediction on average (cf. Miranda and Ruiz Garcia [18]; Akkar and Miranda [19]). This could denote the need for an improvement of formula (7). The following points should be noted, though. The mean ratio Figure 5 is relevant to assigned values of M r plotted in a, which ma entail ver high values of in the short-medium period range ( 10, as Figure 4 shows. This means that the rigid-plastic method is able to estimate ver high values of with reasonabl narrow mean errors. On the contrar, rather low values of are usuall assigned when evaluating the mean errors relevant to other approximate methods (for example ranging from 2 to 6 is considered b Miranda and Ruiz Garcia [18]). In addition, the errors relevant to these methods generall increase as increases, this being especiall so in the medium period range (cf. Miranda and Ruiz Garcia [18]; Akkar and Miranda [19]). This means that also in the medium period range the rigid-plastic prediction can in fact be good enough, even when compared to that obtained b means of other approximate methods. WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

10 270 Earthquake Resistant Engineering Structures VIII 5 Conclusions With the aim of evaluating the extent of the relative errors that can be made when the imum seismic displacement of an elastic-plastic oscillator is estimated b means of the rigid-plastic method, a numerical investigation was carried out in the present paper. The results show that this method leads -on average- to conservative and fairl good predictions, whatever the oscillator and the earthquake, and even when the plastic displacements are ver large. Mean relative errors lower than 15% are found both for short-period and long-period oscillators. For medium period oscillator the error is generall less than 30%. A profitable feature of the method is that of obtaining the ductilit demand prediction directl in the range of periods where inelastic displacements ma actuall occur under a given earthquake. Moreover, the rigid-plastic prediction does not involve an iterative procedure, as man other approximate methods do. For short-period oscillators the mean error is generall much lower than that relevant to most of the other approximate methods. For medium-period and longperiod oscillators, a comparable mean error ma be found with respect to other methods. Especiall as the ductilit demand increases. Acknowledgements The recorded accelerograms considered in this paper were obtained from: ESD The European Strong-Motion Database, EER Strong Motion Database, COSMOS Virtual Data Center, References [1] Rosemblueth, E. & Herrera, I., On a kind of hsteretic damping. Journal of Engineering Mechanics Division ASCE, 90, pp , [2] Gulkan,. & Sozen, M., Inelastic response of reinforced concrete structures to earthquakes motion. ACI Journal, 71, pp , [3] Iwan, W.D., Estimating inelastic response spectra from elastic spectra. Earthquake Engineering and Structural Dnamics, 8, pp , [4] Kowalsk, M., riestle, M.J.N. & McRae, G.A., Displacement-based design of RC bridge columns in seismic regions. Earthquake Engineering and Structural Dnamics, 24, pp , [5] Newmark, N.M., & Hall, W.J., Earthquake Spectra and Design, Earthquake Engineering Research Institute: Berkele, CA, [6] Miranda, E., Inelastic displacements ratios for structures on firm sites. Journal of Structural Engineering, 126, pp , [7] aglietti, A. & orcu, M.C., Rigid-plastic approximation to predict motion under strong earthquakes. Earthquake Engineering and Structural Dnamics, 30, pp , WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress

11 Earthquake Resistant Engineering Structures VIII 271 [8] orcu, M.C. & Carta, G., Rigid-plastic bound to the seismic inelastic response of flexible elastic-plastic oscillators. European Earthquake Engineering, 3, pp. 3 9, [9] orcu, M.C. & Carta, G., Rigid-plastic seismic analsis to predict the structural ductilit demand. International Journal of Applied Engineering Research, 4(3), pp , [10] Domingues Costa, J.L., Bento, R., Levtchitch, V. & Nielsen, M.., Simplified non-linear time-histor analsis based on the Theor of lasticit. roc. of the ERES 2005 Conference, WIT ress: Southampton, UK, pp , [11] orcu, M.C. & Mascia, M., Rigid-plastic pseudo-spectra: peak response charts for seismic design. European Earthquake Engineering, 3, pp , [12] Makris, N. & Black, C.J., Dimensional analsis of rigid-plastic structures under pulse-tpe excitations. Journal of Engineering Mechanics, 130(9), pp , [13] Hibino, Y., Toshikatsu, I., Domingues Costa, J.L. & Nielsen, M.., rocedure to predict the store where plastic drift dominates in two-store building under strong ground motion. Earthquake Engineering and Structural Dnamics, 38(7), pp , [14] Domingues Costa, J.L., Bento, R., Levtchitch, V. & Nielsen, M.., Rigidplastic seismic design of reinforced concrete structures. Earthquake Engineering and Structural Dnamics, 36, pp , [15] Chopra, A.K., Dnamics of Structures. Theor and Application to Earthquake Engineering, rentice Hall: New Jerse, [16] Chopra, A.K. & Goel, R.K., Evaluation of NS to estimate seismic deformation: SDF sstems. Journal of Structural Engineering, 126(4), pp , [17] Miranda, E. & Akkar, S.D., Evaluation of iterative schemes in equivalent linear methods. Earthquake Engineering Research Institute, [18] Miranda, E. & Ruiz Garcia, J., Evaluation of approximate methods to estimate imum inelastic displacement demands. Earthquake Engineering and Structural Dnamics, 31, pp , [19] Akkar, S.D. & Miranda, E., Statistical evaluation of approximate methods for estimating imum deformation demands on existing structures. Journal of Structural Engineering, 131, pp , WIT Transactions on The Built Environment, Vol 120, 2011 WIT ress