DANNY ARROYO MARIO ORDAZ

Transcription

1 Journal of Earthquake Engineering, :47 65, 7 Copyright A.S. Elnashai & N.N. Ambraseys ISSN: print / X online DOI:.8/ Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 Hysteretic Energy Demands for SDOF Systems Subjected to Narrow Band Earthquake Ground Motions. Applications to the Lake Bed Zone of Mexico City UEQE X Journal of Earthquake Engineering, Vol., No., February 7: pp. 9 Hysteretic D. Arroyo Energy and M. Ordaz Demands for SDOF Systems DANNY ARROYO Departamento de Materiales, Universidad Autónoma Metropolitana-Azcapotzalco, Reynosa Tamaulipas, México MARIO ORDAZ Instituto de Ingeniería, UNAM, Coyoacán, México This study proposes a way to estimate the hysteretic energy demands for oscillators subjected to narrow-band earthquake ground motions using an equivalent sinusoidal pulse. The parameters required to define the equivalent pulse are: the peak ground acceleration, the maximum of the Fourier amplitude spectrum, and the predominant period of the site. The equivalent pulse is utilized to compute the hysteretic energy demands associated to the design elastic pseudoacceleration spectra prescribed by the Mexico City Building Code and to construct damage index maps for systems designed according to this code and subjected to a seismic event with M = 8 and focal distance of 3 km. Keywords Hysteretic Energy; Pulses; Earthquake Ground Motion Duration; Narrow-band Earthquake Ground Motions; Damage Index Maps. Introduction Damage in structures under earthquake ground motions depends on the seismic demands that the structure experiences and on its structural capacities. Failure of the structure can be reached either when the peak displacement demand of the structure exceeds its capacity displacement under monotonic load or when its mechanical properties deteriorate due to several load cycles or brittle failure of some elements. Conventional earthquake resistant design methods asses the performance of structures during seismic events through their peak displacement. However, several researchers have shown the shortcomings associated to this design criterion, especially when long earthquake ground motions are considered [Park and Ang, 985; Fajfar, 99; Fajfar and Vidic, 994; Terán, 996; Terán and Jirsa, 5]. During long earthquake ground motions, several load cycles are expected and the mechanical properties of a structure can exhibit deterioration. This phenomenon can lead to structural failure even though the maximum displacement demand observed during the Received September 5; accepted 6 June 6. Address correspondence to Danny Arroyo, Departamento de Materiales, Universidad Autónoma Metropolitana- Azcapotzalco, Av. San Pablo 8, Col. Reynosa Tamaulipas, Mexico; aresda@correo.azc.uam.mx 47

2 48 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 earthquake ground motion never exceeds the displacement capacity of the structure under monotonic load. This phenomenon is known as low-cycle fatigue. The level of deterioration depends on the mechanical properties of the elements that compose the structure and on the amplitude, sequence, and number of load cycles. Hence, developing a rational design method to account for low-cycle fatigue is a complex issue. One important parameter in methods that account for low-cycle fatigue is the hysteretic energy demand that earthquake ground motions impose to structures [Park and Ang, 985; Fajfar, 99; Fajfar and Vidic, 994; Rodriguez, 994; Teran, 996; Manfredi, ; Riddell and Garcia, ]. This article proposes to estimate the hysteretic energy demands for 5% of damping on systems subjected to narrow band motions, through the use of an equivalent sinusoidal pulse that reflects the intensity and duration of actual earthquake ground motions. It is shown that the accuracy of the equivalent pulse is similar to existing expressions to compute hysteretic energy demands. The parameters required to define the equivalent pulse have been comprehensively studied by several researchers for the lake bed zone of Mexico City [Esteva and Rosenblueth, 964; Castro et al., 988; Ordaz et al., 994; Pérez et al., 999; Singh et al., 988; Ordaz et al., 997]. These studies allow defining equivalent pulses for different sites in the lake bed zone of Mexico City, where very few records have been recorded during intense seismic events. The equivalent pulses are applied to the following.. Define an equivalent pulse associated to the pseudoacceleration design spectra prescribed by the Mexico City Building Code for different sites in the lake-bed zone of Mexico City and to estimate the hysteretic energy demands associated to those design spectra.. Construct damage index maps for systems designed according to the Mexico City Building Code, for a seismic event with M = 8 and R = 3 km.. Previous Research In a companion article, it is shown that a sinusoidal pulse can be utilized to estimate the maximum inelastic strength and displacement demands for elastoplastic oscillators subjected to narrow-band motions [Arroyo and Ordaz, 5]. Corrective motions should be added at the beginning and at the end of the sinusoidal pulse so it gives a realistic representation of ground motion during an earthquake [Tarquis, 988; Arroyo and Ordaz, 5]. A discussion on the effect of the corrective motions on the dynamic response of elastoplastic oscillators subjected to sinusoidal pulses, and on the advantages of the corrected pulses over simple sinusoidal ones, when they are used as earthquake ground motion models, can be found in the companion article. The corrected sinusoidal pulse (x g ) is defined in Eq. (.); although not shown, velocity and acceleration corrected pulses can be obtained by differentiating Eq. (.) with respect to time: a+ at+ a3t + a4t 3 + a5t 4 + a6t 5 if t t Amax xg ( t) = sin( Ω( t t )) if t < t t + ntg Ω b btm b3tm b4tm + b5tm + b6tm if t + ntg < t < t + ntg + t (.)

3 Hysteretic Energy Demands for SDOF Systems 49 where Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 T g = π Ω (.) is the predominant period of the ground motion, and n is the number of sinusoidal cycles considered. According to Tarquis [988], the following values for the different parameters in Eq. (.) can be found: a = a = a3 = a a 5 a 4 6 4A = Ωt max 7A = max Ωt 3A = Ωt 3 max 4 t = t t nt m g (.3) (.4) (.5) (.6) (.7) b = x ( t = t + nt ) g b = v ( t = t + nt ) g g g (.8) (.9) b 4 b b b 3 ag( t = t + ntg) = 3bt + 6bt + b = t bt 3 + 8bt + 5b = 4 t bt 3 + 3bt + 6b = 5 t t = t =. 76T g. (.) (.) (.) (.3) (.4) According to Eq. (.), the whole duration of the pulse is given by: ttot = t + ntg + t. (.5) In this article, we have defined the pulse duration as the number of cycles (n) of T g seconds considered in Eq. (.5)

4 5 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 3. Relationship Between Duration and Fourier Amplitude Spectrum for Sinusoidal Pulses Consider the following sinusoidal pulse: at () = A sin( Ω t). (3.) In order to give a finite duration (t d ) to the pulse, Eq. (3.) could be multiplied by the Heaviside function, H(t), defined as: Then, the Fourier transform of a sinusoidal pulse with duration t d is given by: The function a(t) can be rewritten in its exponential form as follows: Substituting Eq. (3.4) in Eq. (3.3) and computing the Fourier transform we find: max td if t < Ht () = t. d if t > (3.) iω t A( ω ) = H() t a() t e dt. (3.3) Amax Ωit it at () = i( e e Ω ). (3.4) A A A( ) max max ω = ih( ω Ω) + ih( ω + Ω ) (3.5) where H(w) is the Fourier transform of the Heaviside function which is given by [Hsu, 97]: ω t H( ω ) = sin d. ω (3.6) Substituting Eq. (3.6) in Eq. (3.5) we obtain: Amax td A td A( ) i sin ( ω Ω ) max ω i sin ( ω = ) ( ω ) + ( ω + ) Ω Ω Ω. (3.7) Finally, the Fourier amplitude spectrum is given by: A A( ω ) = max td td sin ( ω Ω ) A sin ( ω + Ω ) max. ω Ω ω + Ω (3.8) It can be shown that the maximum of this spectrum (max A(w) ) occurs when w = W and is given by: td sin( Ωtd) max A( ω ) = A max. Ω (3.9)

5 Hysteretic Energy Demands for SDOF Systems 5 We note that the second term in Eq. (3.9) tends to decrease as the value of W increases so for values of T g lower than sec max A(w) is approximately: Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 Amax t max A( ω ) d. (3.) According to Eq. (3.), if max A(w) and A max are known, the pulse duration can be estimated as follows: A td = max ( ω ). A From the results shown in this section, it can be noted that the parameters required to define an equivalent sinusoidal pulse associated to a narrow-band earthquake ground motion are A max, T g, and max A(w). Furthermore, the number of equivalent sinusoidal pulses associated to a narrow-band earthquake ground motion can be estimated as: n eq max (3.) td =. (3.) T g 4. Use of Equivalent Sinusoidal Pulses to Estimate Hysteretic Energy Demands of Oscillators Subjected to Narrow-band Earthquake Ground Motions The equivalent sinusoidal pulse was utilized to estimate the hysteretic energy per unit mass (E Hm ) dissipated by oscillators subjected to narrow-band accelerograms. Table shows the characteristics of each record utilized. The classification of accelerograms was carried out by inspection of their elastic energy input spectra for 5% damping. When the peak spectral ordinates were concentrated in a short range of periods, the record was considered as narrow band; in the companion articles the elastic energy spectra for each record utilized can be found. The values of t d and n eq shown in Table were computed with Eqs. (3.) and (3.); max A(w) was obtained from the smoothed Fourier amplitude spectrum of each record and T g was set as the period associated to the maximum of the elastic velocity spectrum for 5% of damping. An equivalent pulse related to each record was generated according to Eq. (.). E Hm spectra for each record were calculated and compared with those obtained from the corresponding equivalent pulse; elastoplastic and degrading hysteretic models were considered. Two different levels of hysteretic degradation were considered, according to the hysteretic model proposed by Kunnath et al., [99]. This model requires the definition of three parameters; the parameters for the models considered and its implications are shown in Table, while Fig. shows a sketch of the degrading hysteretic models utilized. Values of structural period, T, in the range of. to 5 s and maximum ductility demands (m) of.5,, 3, 4, and 6 were considered. Estimations obtained from two well-known rules to compute E Hm, obtained from statistical analyses, were also computed. The purpose of the comparisons shown in this section is to demonstrate that the accuracy of the estimations obtained from equivalent pulses are similar to the accuracy of available expressions utilized to compute hysteretic energy demands. The comparisons should be carefully interpreted, since the rules considered were not specially developed for narrow band motions.

6 5 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 TABLE List of real earthquake records utilized Station Date Comp A max (cm/s ) T g (s) max A(w) (cm/s) t d (s) n eq SCT 9/9/85 EW SCT 9/9/85 NS Roma A //94 EW Roma A //94 NS Córdoba 4/9/95 EW Córdoba 4/9/95 NS Redwood City /8/ Redwood City /8/ Banja Luka-Borik 9 4/5/79 NS Banja Luka-Borik 9 4/5/79 EW Roma A 6/5/99 NS Roma A 6/5/99 EW Sector Popular 9/3/99 NS Sector Popular 9/3/99 EW Córdoba 9/3/99 NS Córdoba 9/3/99 EW Buenos Aires 9/3/99 NS Buenos Aires 9/3/99 EW Cibeles 9/3/99 NS Cibeles 9/3/99 EW SCT 9/3/99 NS SCT 9/3/99 EW TABLE Degrading hysteretic models considered Model Parameter a Parameter b Parameter g Value Implication Value Implication Value Implication Important level of stiffness degradation Important level of stiffness degradation.5 Concrete ductile frames.3 Non-ductile concrete frames Pinching is not considered Pinching is not considered a) Model F b) Model F x x FIGURE Degrading hysteretic models considered.

7 Hysteretic Energy Demands for SDOF Systems 53 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 The rules considered were: (a) Fajfar and Vidic [99] proposed the following expression to compute E Hm for bilinear non degrading oscillators with % of post-yield stiffness. where SA is the pseudoacceleration spectrum and g is defined as: with z T = E Hμ = γμsa R ω 9. T T T T 65. z g μ For bilinear non degrading oscillators c g =.4, c m =.7, c a = c v =, and c d =.8, PGV, and PGD are the peak ground velocity and displacement, respectively. (b) Manfredi [] proposed the following expression to compute E Hm for elastoplastic oscillators: (4.) γ = zt zμ zg (4.) z μ ( μ ) = μ cμ at dt = () Amax PGV T T cpgv v = π ca a max if T T if T T T if T T c g cpgd d = π. cpgv v (4.3) (4.4) (4.5) (4.6) (4.7) E = ( μ ) n Hμ c e SA ω R ne = +. 8( Rμ ) 3/ 5 ID δ / 6 τ / μ (4.8) (4.9) I D T τ = T = dur at () dt A max PGV if T T if T > T. (4.) (4.)

8 54 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 where m c is the maximum cyclic ductility demand, and for x =.5 the parameter d is equal to unity. For simplicity, in this article is assumed that m c is equal to m +. This assumption becomes a good approximation when the response curve loaddisplacement tends to be symmetric. The accuracy of the different estimations of E Hm was compared through the mean logarithmic error defined in Eq. (4.). ε ln = qmp m q j= i= k= E ln E where q, m, p are the number of records, the number of ductilities, and the number of periods considered, respectively, and E Hμ is the estimated value of E Hm. The results obtained are summarized in Figs. and 3. As it can be observed, for elastoplastic oscillators the accuracy of the equivalent pulses (EQP) is similar to that of the different rules considered. Since the Fajfar and Vidic and Manfredi expressions were developed for non degrading systems, they are only compared with the results obtained from elastoplastic oscillators. Similar trends for e ln were observed for the equivalent pulses and Manfredi s rule. In the long-period range some differences were observed between the equivalent pulse and the rule of Fajfar and Vidic. However, it must be recognized that the proposal of Fajfar and Vidic was intended for bilinear oscillators. Figure 3 shows that the accuracy of the equivalent pulse to estimate E Hm for oscillators with degrading behavior is comparable to that of the elastoplastic case. Nevertheless, some differences can be observed: in the long-period range the values of e ln tend to be larger for the elastoplastic case respect to the degrading hysteretic models, while for periods lower than one second e ln tends to be greater for the degrading hysteretic models. p Hμ Hμ (4.) ε ln EQP Fajfar Manfredi µ = 3 4 T(s) 5 ε ln µ =3 EQP Fajfar Manfredi 3 4 T(s) 5 ε l EQP Fajfar Manfredi µ =4 3 4 T(s) T(s) 5 FIGURE Comparison between logarithmic errors for different estimations of E Hm for elastoplasctic model, x = ε ln EQP Fajfar Manfredi µ =6

9 Hysteretic Energy Demands for SDOF Systems 55 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 ε ln μ = 4 μ=6 μ = μ= ε ln μ = 4 μ = 6 μ = μ = 3 Model Model T(s) T(s) 5 FIGURE 3 Logarithmic errors associated to estimations of E Hm obtained from equivalent pulses for degrading hysteretic models, x =.5. It has been shown, so far, that E Hm estimations obtained from the equivalent pulse proposed have similar accuracy than estimations related to rules proposed by other researchers. It is important to note that the results presented in this section are valid only for oscillators with hysteretic behaviour without pinching. In this article, we have applied the equivalent pulse to estimate hysteretic energy demands only for 5% damped systems; more research is needed to validate the use of the equivalent pulse for other damping values. 5. Hysteretic Energy Demands Associated to Design Elastic Pseudoacceleration Spectra Prescribed by the Mexico City Building Code The parameters required to define the equivalent pulse proposed are T g, A max, and max A(w). The Mexico City Building Code proposes a map to estimate T g for different sites of the lake-bed zone of Mexico City. This map is not reproduced in this article but it can be found in the current version of the Mexico City Building Code [4]. T g values in the lake-bed zone are in the range of. 4.5 s. Also, the code proposes the following equation to estimate the value of A max normalized by the acceleration of gravity (g) for different sites in the lake bed zone: a ( Tg 5. ) if 5. Tg 5. =. (5.) 5. if Tg > 5. According to Eq. (5.), A max depends on T g and is in the range of..5 g. Then, the Mexico City Building Code explicitly defines two of the three parameters required to obtain the equivalent pulse, that is, T g and A max. Der Kiureghian and Neuenhofer [99] developed a method to obtain the power spectral density function (G(w)) associated to a response spectrum. Recently, Pérez [] applied this method to obtain the power density functions associated to the elastic pseudoacceleration design spectra of the Mexico City Building Code. Figure 4 shows the pseudoacceleration design spectra for different values of T g, according to the Mexico City Building Code, and its corresponding G(w); details on the computation of this spectra can be found in Pérez []. From the G(w) spectra the value of max A(w) can be obtained as follows: max A( ω ) τ max( G( ω )). = (5.)

10 56 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 SA (cm/s ) T(s) SA (cm/s ) SA (cm/s ) T s =.5 T s = T(s) T s = T(s) G(ω) (cm /s 3 ) G(ω) (cm /s 3 ) G(ω) (cm /s 3 ) f (Hz) T s =.5 f (Hz) T s = T s = 3 f (Hz) SA (cm/s ) T s = T(s) G(ω) (cm /s 3 ) T s = 4 f (Hz) FIGURE 4 SA design spectra for different values of T g according to the Mexico City Building Code and their related G(w), x =.5. In Eq. (5.), t is the duration between the times where the 5 and 95% of the Arias intensity is developed. Although the design spectra of the Mexico City Building Code are related to seismic events with different durations, it was decided to use a value of t associated to an earthquake with M = 8 and R = 3 km, since the amplitudes of the design spectra in the lake-bed zone highly depend on such type of earthquakes [Ordaz and Reyes, 999]. Note that t is related to t d. The value of t d is the duration of an equivalent pulse which resembles the energy content of an earthquake ground motion; this ground motion has a duration between the times where the 5 and 95% of the Arias intensity is developed equal to t. Furthermore, the value of t is required only if the Fourier amplitude spectrum

11 Hysteretic Energy Demands for SDOF Systems 57 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 is unknown but the elastic SA spectrum is available; if the Fourier amplitude spectrum is known t d can be directly computed from Eq. (3.). Table 3 shows the number of equivalent pulses associated to different design spectra of the Mexico City Building Code and the values of the different parameters involved in the computations. According to Pérez [], the t values shown in Table 3 were defined as a fraction of the duration obtained from the expression proposed by Reinoso and Ordaz [] for the lake bed zone of Mexico City. With the information shown in Table 3, an equivalent pulse associated to a design spectrum can be constructed. This pulse can be used to: (a) estimate the E Hm demands related to the design spectra specified in the code for the lake-bed zone of Mexico City, and, (b) to estimate strength and peak displacement demands for a given ductility, as it is shown in the companion article. Figure 5a compares strength per unit of mass spectra for systems with m = 4. This figure includes spectra obtained from: (a) the SCT-EW accelerogram (SCT-EW), which was recorded during the 985 Michoacan Earthquake; (b) the equivalent pulse associated to the elastic design spectrum for a site with T g = s (EQP); and (c) the design spectrum for TABLE 3 Equivalent pulses associated to different SA design spectra of the Mexico City Building Code T g A max (cm/s ) max G(w) (cm /s 3 ) t (s) max A(w) (cm/s) t d (s) n eq F/m (cm/s ) SCT EW NTC-S EQP μ = T(s) 5 E Hμ (cm /s ) SCT-EW EQP NTC-S μ = 3 4 T(s) 5 xmax (cm) SCT EW EQP NTC-S μ = T(s) 5 E Hμ (cm /s ) SCT-EW EQP NTC-S μ = T(s) 5 FIGURE 5 Comparison between spectra for the SCT-EW record, the spectra for Mexico City Building Code, and the spectra obtained from the equivalent pulse associated to SA design spectrum for a site with T g of two seconds according to the Mexico City Building Code, x =.5.

12 58 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 a site with T g = s (NTC-S). As shown in Table, the SCT-EW record has a peak ground acceleration of 69 cm/s and T g = s. For systems with T lower than.5 s, the equivalent pulse yields strength demands comparable with those specified by the design spectrum, while for systems with T greater than.5 s the strength demands obtained from the equivalent pulse are larger than the ordinates of the design spectrum. Opposite trends are observed when comparing the SCT-EW strength spectrum and the strength spectrum obtained from the equivalent pulse. For systems with T lower than s, the ordinates of the pulse spectrum are greater than those of the SCT-EW spectrum, while for systems with T greater than s the two spectra become comparable. Significant differences are observed in the short period range, since the peak ground acceleration for the SCT-EW record is lower than the peak ground acceleration of the equivalent pulse and the design spectrum, as it can be observed in Tables and 3. Figure 5b compares peak displacement spectra for system with m = 4. This figure includes the same spectra than Fig. 5a. For systems with T lower than.5 s, the equivalent pulse yields peak displacements comparable with the displacements specified by the design spectrum, while for systems with T greater than.5 s, the peak displacements obtained from the equivalent pulse are lower than the ordinates of the design spectrum. As it can be observed, the SCT-EW displacement spectrum and the displacement spectrum obtained from the equivalent pulse are very similar. Figures 5c and 5d compare E Hm elastoplastic spectra for the SCT-EW record and the spectra obtained from the equivalent pulse associated to the elastic design spectrum for a site with T g = s (EQPNTC-S). Although both spectra follow similar trends, significant differences are observed in the range between.5 and s, where the equivalent pulse yields larger values of E Hm than those observed for the SCT-EW record. This difference is due to the fact that the equivalent pulse has greater peak ground acceleration than the SCT-EW record, as it has been indicated. In the comparisons shown in Fig. 5, the differences arise due to: (a) the spectra for the SCT-EW record are related to a single seismic event, while the equivalent pulse was obtained from a design spectrum that was defined considering many seismic events; and (b) the SA elastic design spectra of the Mexico City Building code were only developed to compute strength demands, in spite of which, the trends observed for the equivalent pulse E Hm spectra and the spectra obtained with the SCT-EW record are similar. 6. Damage Index Maps for Systems Designed According to the Mexico City Building Code During the last century, several earthquakes have caused severe damage to structures localized in the lake-bed zone of Mexico City. Unfortunately, very few accelerograms have been recorded during these intense seismic events. During the great Michoacan earthquake in 985, only six stations in the lake bed-zone of Mexico City recorded the event. After the Michoacan earthquake many sites of the Valley of Mexico were instrumented. Nowadays, more than stations are localized in its lake-bed zone. However, this instrumentation has only recorded low and moderate seismic events. Hence, the conditions that could be observed during intense seismic events can only be estimated, and many researchers have conducted studies to this aim. Esteva and Rosenblueth [964] proposed an attenuation relation to compute A max for firm soil sites of Mexico City. Other researchers have proposed attenuation rules for the

13 Hysteretic Energy Demands for SDOF Systems 59 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 Fourier spectrum at a reference site [Castro et al., 988; Ordaz et al., 994; Pérez et al., 999] and empirical transfer functions [Singh et al., 988] to compute the Fourier amplitude spectrum for the lake bed of Mexico City. Using this information, and an interpolation procedure, it is possible to estimate several characteristics of future ground motions at almost any place in Mexico City [Ordaz et al., 997]. Based on these studies, the parameters required to define the equivalent pulses can also be estimated for a given seismic event, characterized by its magnitude (M) and its focal distance (D). In order to estimate the hysteretic energy demands expected during an intense seismic event, the parameters required to obtain equivalent pulses for different sites in the lake bed zone (A max, T g, and max A(w) ) were computed assuming M = 8 and D = 3 km. The computations were carried out with a computer program developed by Ordaz et al., [997]. Once these parameters have been computed, the equivalent pulses can be determined, as explained above. The results obtained are summarized in Fig. 6 which shows the following contour maps, associated to the seismic event considered: (a) t d in the North-South (N-S) direction, (b) t d in the East-West (E-W) direction; (c) E Hm for systems with T = T g and m = 4 in the N-S direction; and (d) E Hm for systems with T = T g and m = 4 in the E-W direction. Elastoplastic behavior is considered, although, as it has been shown, equivalent pulses can be used for degrading hysteretic systems. In Figs. 6(a) and 6(b), contours for t d of 5,, and 5 s are plotted. Also, Figs. 6(c) and 6(d) show contours for E Hm of,,,, 3,, and 4, cm /s, for systems with T = T g. Systems in resonance were chosen since the highest E Hm demands are expected for these systems. Greater values of t d are observed near the centre of the lake-bed zone. Also, higher E Hm demands are observed in the E-W direction. As it can be observed, high E Hm values are observed in the following zones of Mexico City (numbers in parenthesis are the approximated geographical coordinates of each zone): Tlahuac ( 99., 9.8), Xochimilco ( 99., 9.8), Tasqueña ( 99., 9.37), Aeropuerto ( 99.5, 9.45), and Center ( 99., 9.45). In order to study the effect of the E Hm demands on RC frame structures localized in the lake-bed of Mexico City, designed according the Mexico City Building Code, contour maps of damage indexes were constructed. This structural type was chosen because during the Michoacan earthquake a great number of mid-rise structures of this kind exhibited severe damage. Numerous damage indexes have been proposed in the past; a detailed description of the available indexes can be found in Williams and Sexsmith [995] and in Mehanny and Deirlein []. We used the damage index proposed by Park and Ang [985] (ID PA ), which is defined in Eq. (6.): ID PA xmax EH = + β (6.) x F x u where x max is the maximum displacement demand, E H is the hysteretic energy demand, F y is the yield strength of the system, x u is the maximum displacement capacity under monotonic load, and b is a parameter which controls the degradation of the structural properties. Although several researchers have identified some inconsistencies in the ID PA [Williams and Sexsmith, 995; Terán, 996; Mehanny and Deirlein, ; Bozorgnia and Bertero, 3], we believe it serves our purposes, since it is widely used and it has been calibrated against numerous experimental tests of concrete elements. y u

14 6 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March a) t 4 (s) N S b) t 4 (s) E W c) E mp (cm /s ) N S d) E mp (cm /s ) E W FIGURE 6 (a),(b) Equivalent duration contours for the lake bed zone of Mexico City, (c), (d) Hysteretic energy contours for the lake zone of Mexico City (T = T g, m = 4, and x =.5). Symbols adopted: (i) Geographical limits of Mexico City are plotted in hidden thick line, (ii) The inner continuous line is the boundary of the lake bed zone, (iii) Grid zones are firm soil zones inside the lake bed zone For a given T, the different parameters required to compute ID PA were obtained as follows: F y was obtained in accordance with the Mexico City Building Code. In this code the design value of F y is obtained reducing the elastic design spectrum by two factors. The first factor accounts for inelastic behavior and the second factor accounts for overstrength. The reduction by inelastic behavior depends on T and on the

15 Hysteretic Energy Demands for SDOF Systems 6 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 deformation capacity of the system, characterized by parameter Q. For ductile concrete framed buildings, the code specifies Q = 4 while for nonductile concrete framed buildings the code specifies Q =. In this article, it is assumed that the expressions proposed by the code to compute the overstrength factor are exact; thus F y was computed reducing the elastic spectrum only for inelastic behavior. Defining the value of x u for actual structures is a complex issue and more research is certainly needed. In this study, x u was characterized through the ductile capacity (m u ) defined as the ratio between x u and the initial yield displacement of the system. For ductile frames a value of m u = 6 was considered. For non ductile frames, a values of m u = 3 was used. These values are regarded as the upper bound of the ductile capacities of these types of structures. For ductile concrete framed buildings a value of b =.5 was considered, while for non ductile concrete framed buildings a value of b =. was used. There are differences between the dynamic response of SDOF systems and actual buildings, depending on several factors that are difficult to take into account, such as irregularities, higher mode effects, and failure mechanism. However, some researchers have demonstrated that results obtained from SDOF systems can be used to evaluate seismic demands for regular MDOF systems if higher mode effect can be neglected [Tso et al., 99; Chou and Uang, 4; Teran, 5]. Hence, the results presented in this section can be used as indicator of the seismic behavior of regular mid-rise concrete framed buildings responding mainly in their first mode shape. The results obtained are summarized in Figs. 7 and 8. Figure 7 shows contour maps of ID PA for ductile concrete framed buildings while Fig. 8 shows contour maps for non ductile concrete framed buildings. Values of T of,.5,, and 3 s were considered in each case. In Figs. 7 and 8, contours for ID PA = are plotted. As it can be observed, at some sites of the lake-bed zone, systems designed according to the Mexico City Building Code can exhibit ID PA greater than unity, especially for ductile concrete framed buildings. The zones with ID PA > are Xochimilco ( 99., 9.8), Tasqueña ( 99., 9.37), Tlahuac ( 99., 9.8), Aeropuerto ( 99.5, 9.45), and Centre ( 99., 9.45). As it can be observed, the Center zone with ID PA > matches with the zone of severe damage concentration observed during the Michoacan earthquake. It is important to note that in 985 most of the mid-rise concrete framed buildings were localized in or near to the Center of the City. The results shown in this section suggest that in specific sites of Mexico City regular mid-rise concrete framed structures could exhibit severe damage during intense seismic events and that the Building Code requirements might be unconservative. However, more analytical and experimental research is needed to validate the findings presented in this paper. Only then, a rational modification to the current version of the Mexico City Building Code could be proposed. 7. Conclusions An equivalent sinusoidal pulse useful to compute hysteretic energy demands on five percent damped oscillators subjected to narrow band earthquake ground motions has been presented. The parameters needed to define the equivalent pulse are: the peak ground acceleration, the predominant period of the site and the maximum of the Fourier amplitude spectrum. The equivalent pulse has been used to compute the hysteretic energy demands that a magnitude 8 event with focal distance of 3 km would impose on SDOF elastoplastic

16 6 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March a) T =. s b) T =.5 s c) T =. s d) T = 3. s FIGURE 7 Damage index contours for systems designed according the Mexico City Building Code (Q = 4, m u = 6, b =.5, and x =.5). Symbols adopted: (i) Geographical limits of Mexico City are plotted in hidden thick line, (ii) The inner continuous line is the boundary of the lake bed zone, (iii) Grid zones are firm soil zones inside the lake bed zone, (iv) Zone of severe damage concentration observed during the 985 Michoacan Earthquake is plotted in continuous thick line. structures designed with the Mexico City Building Code. This allowed us to construct contour maps of damage index for systems designed according this building code. The contour maps presented try to estimate the behavior of regular concrete framed buildings localized in the lake-bed zone of Mexico City. Several assumptions were made

17 Hysteretic Energy Demands for SDOF Systems Downloaded By: [Arroyo, Danny] At: 3:4 8 March a) T =. s b) T=.5 s a) T=. s b) T=3. s FIGURE 8 Damage index maps for systems designed according the Mexico City Building Code (Q =, m u = 3, b =., and x =.5). Symbols adopted: (i) Geographical limits of Mexico City are plotted in hidden thick line, (ii) The inner continuous line is the boundary of the lake bed zone, (iii) Grid zones are firm soil zones inside the lake bed zone, (iv) Zone of severe damage concentration observed during the 985 Michoacan Earthquake is plotted in continuous thick line. in order to obtain this damage index maps, the most important regarding overstrength and ductile capacity of actual concrete buildings. However, the results suggest that in specific sites of Mexico City concrete framed structures could exhibit severe damage during intense seismic events.

18 64 D. Arroyo and M. Ordaz Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 However, more analytical and experimental research is needed to validate the findings presented in this article. Only then, a rational modification to the current version of the Mexico City Building Code could be proposed. Although the procedure described in this article was applied to the lake-bed of Mexico City, it could be applied to other zones where narrow band motions are expected. Acknowledgments The authors gratefully acknowledge the assistance of Arturo Pérez in some numerical computations, and the encouragement and suggestions of Prof. Amador Terán. The observations and suggestions of two anonymous reviewers are also acknowledged. References Arroyo, D. and Ordaz, M. [5] Use of corrected sinusoidal pulses to estimate inelastic demands of elasto-perfectly plastic oscillators subjected to narrow-band motions, submitted for publication to Journal of Earthquake Engineering. Bozorgnia, Y. and Bertero, V. V. [3] Damage spectra: characteristics and applications to seismic risk reduction, Journal of Structural Engineering, 9(), Castro, R., Singh, S. K., and Mena, E. [988] The México earthquake of September 9, 985- An empirical model to predict Fourier amplitude spectra of horizontal ground motion, Earthquake Spectra, 4(4), Chou, C. C. and Uang, C. M. [4] Evaluating distribution of seismic energy in multistory frames, 3 th World Conference on Earthquake Engineering, Vancouver Canada, Paper N 4. Der Kiureghian, A. and Neuenhofer, A. [99] A response spectrum method for multiple-support seismic excitations, Report No. UCB/EERC-9/8, University of California at Berkeley. Esteva, L. and Rosenblueth, E. [964] Espectros de temblores a distancias moderadas y grandes, Bulletin of the Mexican Society of Earthquake Engineering, (), -8 (in Spanish). Fajfar, P. [99] Equivalent ductility factors taking into account low-cycle fatigue, Earthquake Engineering and Structural Dynamics,, Fajfar, P. and Vidic, T. [994] Consistent inelastic design spectra: hysteretic and input energy, Earthquake Engineering and Structural Dynamics, 3, Hsu, H. P. [97], Fourier Analysis, Simon and Schuster, Inc. New York. Kunnath, S. K., Reinhorn, A. M. and Park, Y. J. [99], Analytical modeling of inelastic seismic response of R/C structures, Journal of Structural Engineering, 6(4), Manfredi G. [] Evaluation of seismic energy demand, Earthquake Engineering and Structural Dynamics, 3, Mehanny, S. S. and Deierlein, G. G. [], Modeling of assessment of seismic performance of seismic of composite frames with reinforced concrete columns and steel beams, Report No. 35, The John A. Blume Earthquake Engineering Center, University of Stanford. Mexico City Building Code [4], Normas técnicas complementarias para el diseño por sismo, Gaceta Oficial del Distrito Federal, Tomo II, No. 3-Bis, 4 (in Spanish). Ordaz, M., Pérez-Rocha, L. E., Reinoso, E., and Montoya, C. [997], Sistema de cómputo para el cálculo de espectros esperados en la ciudad de México, XI Mexican Conference on Earthquake Engineering, Veracruz, México, , (in Spanish). Ordaz, M., and Reyes, C. [999], Earthquake hazard in Mexico City: observations versus computations, Bulletin of the Seismological Society of America, 89(5), pp Park, Y. J. and Ang A. H. [985] Mechanistic seismic damage model for reinforced concrete, Journal of Structural Engineering, (4), Pérez, L. E., Ordaz, M., and Singh, S. K. [999] Modelo de atenuación para sismos de distinto origen, XII Mexican Conference on Earthquake Engineering, Morelia, México, 6 7 (In Spanish). Pérez, I. A. [], Análisis de la respuesta sísmica de apéndices y aplicaciones al Valle de México, Master Thesis, Universidad Nacional Autónoma de México (in Spanish).

19 Hysteretic Energy Demands for SDOF Systems 65 Downloaded By: [Arroyo, Danny] At: 3:4 8 March 7 Reinoso, E. and Ordaz, M. [] Duration of strong ground motion during Mexican earthquakes in terms of magnitude, distance to the ruptura area and dominant site period, Earthquake Engineering and Structural Dynamics, 3, Riddell, R. and Garcia, J. E. [] Hysteretic energy spectrum and earthquake damage, Seventh US Conference on Earthquake Engineering, Boston. Rodriguez, M. [994] A measure of the capacity of earthquake ground motions to damage structures, Earthquake Engineering and Structural Dynamics, 3, Singh, S. K., Lermo, J., Domínguez, T., Ordaz, M., Espinosa, J. M., Mena, E. and Quass, R. [988] The México earthquake of september 9, 985- A study of amplification of seismic waves in the valley of México with respect to a hill zone site, Earthquake Spectra, 4(4), Tarquis, F. [988] Structural response and design spectra for the 985 Mexico City Earthquake, Ph. D. Dissertation, University of Texas, Austin. Terán, A. [996], Performance-based earthquake-resistant design of framed buildings using energy concepts, Ph. D. Dissertation, University of California, Berkeley. Terán, A. [4] On the use of spectra to establish damage control in regular frames during global predisign, Earthquake Spectra, (3), 995. Terán, A. and Jirsa, J. O. [5], Considerations for the formulation of a seismic design methodology that accounts for the effect of low cycle fatigue, Earthquake Spectra, (3), Tso, W. K., Zhu, T. J., and Heidebrecht [993] Seismic energy demands on reinforced concrete moment-resisting frames, Earthquake Engineering and Structural Dynamics,, Williams, M. S. and Sexsmith, R. G. [995] Seismic damage indices for concrete structures: a state of the art review, Earthquake Spectra, (),