Effect of seismic sequences in reinforced concrete

Soil Dynamics and Earthquake Engineering 63 (2014) 56–68

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Effect of seismic sequences in reinforced concrete frame buildings located in soft-soil sites Jorge Ruiz-García a,n, Marco V. Marín a, Amador Terán-Gilmore b a b

Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C, Planta Baja, Ciudad Universitaria, 58040 Morelia, Mexico Departamento de Materiales, Universidad Autónoma Metropolitana-Azcapotzalco, 02200 México D.F., Mexico

art ic l e i nf o

a b s t r a c t

Article history: Received 23 February 2013 Received in revised form 18 February 2014 Accepted 15 March 2014

On September 19 and 20, 1985, two strong subduction interface earthquakes struck Mexico City leading to a large stock of damaged, or even collapsed, reinforced concrete (RC) building structures located in soft soil sites of the former lake-bed zone. The aim of this study is to gain further understanding on the effects of soft-soil seismic sequences on the seismic response of RC framed-buildings. This investigation employed artificial sequences since only two real sequences were gathered during the 1985 earthquakes. The nonlinear response, in terms of peak and residual lateral inter-story drift demands of four RC buildings having 4, 8, 12, and 16 stories, was evaluated. Results show that the relationship of the damaged period of the building (i.e. period of the building after the mainshock) to the predominant period of the aftershock, which is a measure of its frequency content, has a significant impact in the building response. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Aftershocks Seismic sequences Frequency content Peak drift demands Residual drift demands

1. Motivation Man-made structures located in seismic regions are not exposed to a single seismic event (i.e. mainshock), but to a seismic sequence consisting of foreshocks, the mainshock, and aftershocks. Under some circumstances, aftershocks could trigger larger peak lateral displacement demands and/or larger permanent displacements than those experienced during the mainshock. As a consequence, aftershocks could increase structural damage or, even, drive the structure without major damage to demolition due to excessive permanent displacements. A clear example of this scenario was observed after the September 19, 1985 Michoacan earthquake (Mw ¼8.0) and the following aftershock on September 20 (Mw¼ 7.6) that struck Mexico City [1]. In addition, it is well documented that medium-rise buildings located in the old bedlake zone of Mexico City, mainly reinforced concrete (RC) buildings with frame-based structural systems, having between 8 and 16 stories suffered moderate-to-severe structural damage as a consequence of the mainshock [1,2]. Since a strong aftershock shook the city the following day, many buildings increased their state of damage, or suffered excessive permanent displacements. Thus, several dozen damaged RC buildings had to be demolished after the earthquakes because of the technical difficulties involved in

n

Corresponding author. Tel.: þ 52 443 2278545; fax: þ52 443 3041002. E-mail address: [email protected] (J. Ruiz-García).

http://dx.doi.org/10.1016/j.soildyn.2014.03.008 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

straightening and repairing buildings with large permanent drifts and of the threat of future aftershocks [1]. In spite of the 1985 experience, no study known to the authors has been conducted to investigate the effect of aftershocks in the response of the buildings located in soft soil sites, in such a manner as to caution practicing engineers about the importance of considering full seismic sequences during earthquake-resistant design. This paper presents the results of an analytical investigation whose main goal is to provide a general understanding of how RC frames respond to seismic sequences generated in very soft soil conditions; and particularly, to identify the main parameters that help explain under what circumstances may an aftershock has a detrimental effect that should be taken explicitly into account during earthquake-resistant design. Although it has been shown that artificial sequences could lead to a very different response than that from the real sequence [3–5], it was found that only two real mainshock–aftershock acceleration time-histories were available from the Mexican Strong Motion Database [6]. Because of this, the investigation employed two sets of artificial seismic sequences that on one hand, tried to represent the seismic environment of the lake-bed zone of Mexico City; and on the other hand, provided an opportunity to study the effect that the characteristics of the aftershock relative to those of the mainshock have on the dynamic response of four RC framed buildings. The buildings were designed according to the 1997 Mexican Building Code to withstand, through the development of ductile behavior, the design ground motion corresponding to the

J. Ruiz-García et al. / Soil Dynamics and Earthquake Engineering 63 (2014) 56–68

Lake Zone of Mexico City. While the dimensions of the frames are considered to be representative of those used in Mexico after the 1985 Michoacan earthquake, each building was designed by a different undergraduate student in an attempt to introduce in them the variation that would be observed in the structural properties of buildings designed by different engineers. Because of the overall view adopted for the study, the results presented herein are constrained to case-study buildings that are fixed at their base, which implies that soil–structure interaction effects were considered negligible. Within this context, it can be said that this interaction may increase or decrease the lateral deformation demands on structural systems built on soft soils, and that this depends on the structural systems adopted for the super-structure and foundation, the dynamic interaction between these systems and the ground motion, and on the depth at which the firm soil deposits can be found. Although the influence of the soft soil site conditions in the seismic response of the buildings is taken into account herein through the frequency content of the earthquake ground motions, it is important for future studies to address the effect of soil–structure interaction.

2. Reinforced concrete buildings and seismic sequences 2.1. Design and modeling of case-study buildings Four regular three-bay RC buildings having standard occupancy and different number of stories (N ¼4, 8, 12, and 16) were considered [7]. In all buildings, the first-story has a height of 4.5 m while the remaining stories have 3.0 m. Nominal strengths of 4200 kg/cm2 and 250 kg/cm2 were used for the yield stress of reinforcement steel and the compressive stress of concrete, respectively. Fig. 1 shows the typical plan view of the case-study buildings. It was assumed that the buildings were located in soft soil sites of Mexico City. Elastic design spectrum was reduced by a response modification factor equal to 4, which requires designers to provide ductile detailing to the elements. A force-based design, which is customarily in Mexican design practice, was employed for preliminary sizing of structural elements. Final sizing and detailing was determined to satisfy a lateral drift requirement of 0.6%. Detailed description of the design process of the buildings can be found in Teran-Gilmore [7]. The buildings were analyzed using the nonlinear dynamic analysis computer program RUAUMOKO [8]. Only half of each building was modeled due to their symmetry in plan, which means that one exterior and one interior frame were modeled as two-dimensional centerline models. The columns at the ground were assumed fixed at their bases which imply that soil–structure interaction was neglected. Both frames were attached through rigid frame elements to experience the same lateral deformation Analyzed frames

57

at each floor. Beams and columns were modeled as frame elements which concentrate their inelastic response in plastic hinges located at their ends. Flexural moment capacity for beams and columns was determined using nominal material properties. Additional strength and stiffness due to floor slab contribution in beams was taken into account according to the recommendations of Pantazopolou and French [9]. Moment–curvature curves considering cracked sections were obtained for each beam, while axial load–flexural bending interaction diagrams were considered in the columns. Since beams and columns are expected to behave predominantly in flexure, a stiffness-degrading Takeda-type moment–curvature relationship, as included in RUAUMOKO [8], with strain-hardening ratio equal to 0.1% was considered to model their hysteretic behavior. This hysteretic behavior was assumed since the RC elements are expected to be provided with adequate steel reinforcement detailing that preclude strength degradation and pinching due to high shear stresses, slippage of steel bars or other phenomena; and because it is considered to provide a reasonable estimate of the response of the RC frames for the deformation demands that are expected in them [10]. Two levels of unloading stiffness degradation, controlled by the parameter α in the RUAUMOKO library [8], were considered: low (α ¼0.1) and high (α ¼0.5). The selected values imply that member hysteretic behavior with α ¼0.5 has smaller energy dissipation capacity than that defined with α ¼ 0.1, as later illustrated in Fig. 9. As noted in [10], the level of unloading stiffness mainly depends on the level of axial load ratio, which means that beam elements would have different unloading stiffness parameters than column elements. Nevertheless, it was decided to use the same parameter for beams and columns in this study. Parameter β in the RUAUMOKO library [8], which controls the reloading stiffness in the Takeda-type moment–curvature relationship, was set as 0.0 (i.e. hysteresis loop during reloading moves toward the peak displacement point of unloading reached in the previous cycle) in all analysis. It should be mentioned that previous studies carried out in single-degreeof-freedom systems have noted that the selection of parameters α and β has minor influence on the peak displacement demands, but selection of both parameters has significant influence on the amplitude of residual displacement demands [11,12]. For instance, low α and low β values (e.g., in the range of 0.0–0.2), lead to larger residual displacements than those when β values are high (e.g., in the range of 0.6–0.8) as noted in Refs [11,12]. Values of parameter β in the range of 0.0–0.2 are suggested for real RC elements [11]. Rayleigh damping equal to 5% of critical was assigned to the first and second modes for the 4-story building, while this damping was applied to the first and fourth modes for the 8-, 12-, and 16-story buildings. During the analysis, local P-delta effects were included (i.e. large displacement analysis). Main dynamic and mechanical properties of the frames, obtained from modal and nonlinear static analyses, are summarized in Table 1. 2.2. Set of mainshock–aftershock seismic sequences Attention is focused herein on the effects of aftershocks on the inter-story drift demands induced to RC buildings located in soft Table 1 Fundamental period of vibration, T1, yield strength coefficient, Cy, roof yield drift, θy, normalized modal participation factor, Γ1ϕ1,roof, and normalized first mode effective mass, M1/MT, obtained for each frame model.

Fig. 1. Plan view of existing RC buildings considered in this study (units in meters).

MODEL

T1 (s)

Cy

θy (%)

Γ1ϕ1,roof

M1/MT

C-4N C-8N C-12N C-16N

0.81 1.32 1.40 1.74

0.32 0.19 0.21 0.20

0.54 0.56 0.48 0.57

1.25 1.32 1.36 1.43

0.91 0.81 0.79 0.74

58


soil sites. For this purpose, real (i.e. as-recorded) mainshock– aftershock acceleration time histories are needed for performing nonlinear dynamic analyses. However, it was found that only two real mainshock–aftershock acceleration time-histories recorded at Central de Abastos (CDAF) station during the September 19 and 20, 1985, earthquakes were available from the Mexican Database of Strong Motions [6]. It should be noted that CDAF station is located on a soft soil site of Mexico City. While Fig. 2 illustrates the recorded acceleration time-histories, Table 2 reports relevant ground motion features of the sequences (e.g. the peak ground acceleration, PGA, peak ground velocity, PGV, and predominant period of motion, Tg). Tg was defined as the period at which the maximum ordinate of a five percent damped relative velocity spectrum occurs [13]. For the soft soil deposits of Mexico City, Tg has been found to be closely related to the dominant period of the soil deposit computed from one-dimensional elastic models assuming that the response of the soil deposit is dominated

Acceleration (cm/s2)

100

9/20/1985

9/19/1985

50 Time [s]

0 0

50

100

150

200

250

300

-50 -100


100

50 Time [s] 0 0

50

100

150

200

250

300

-50 -100

Fig. 2. Acceleration time histories of the mainshock–aftershock sequences recorded during the 1985 Michoacan earthquakes at CDAF station: (a) N00E component and (b) N90E component.

by vertically propagating shear waves in a layered deposit, and the second mode of vibration is approximately one-third of the fundamental period of vibration [14]. It should be noted that the peak ground velocity (PGV) of the aftershock ground motion is around 35% of that of the mainshock ground motion, and that Tg of the mainshock is longer than that of the corresponding aftershock in the NS component. In addition, Fig. 3 shows that the elastic spectra for both the mainshock and aftershock ground motions have narrow band features. Since two seismic sequences are insufficient for developing conclusions about the effects of aftershocks, artificial seismic sequences that represent the ground motion features of real sequences should be developed. Two approaches have been commonly employed in the absence of real sequences: (1) backto back approach and (2) randomized approach. The first approach consists of repeating the real mainshock, at scaled or identical amplitude, as an artificial aftershock, which assumes that the ground motion features such as frequency content and strong motion duration of the mainshock and aftershock(s) are the same. The second approach consists of assembling a set of real mainshocks, and generating artificial sequences by selecting a mainshock and simulating the remaining aftershocks by repeating the mainshock waveformat repeatedly, at reduced or identical amplitude, with no change in spectral content as an artificial aftershock. Recent studies have demonstrated that the first approach is unrealistic and leads to a totally different response as compared with real sequences [3–5]. Therefore, the randomized approach was employed in this investigation. For generating artificial seismic sequences, a first set of 8 acceleration time histories gathered at recording stations located on soft soil sites of Mexico City were selected. This set of ground motions, denoted Set A in Table 2, includes 4 records having Tg values around 3.0 s and other 4 having Tg values close to 2.3 s. Only the records having Tg around 3.0 s were employed as mainshock ground motions (as outlined in Table 2 with the letter M), which mean that the dominant period of the mainshock is close or longer than that of the aftershock; fact that is consistent with what was observed from the real sequences. The PGV was selected as ground motion intensity measure since it is highly correlated with the

Table 2 List of earthquake ground motions employed to derive the artificial seismic sequences considered in this investigation.

Set A M1 M2 M3 M4

Set B M1 M2 M3 M4 M5 M6 M7 M8

Date

Ms

Station name

Station ID [6]

Comp.

19/09/1985 19/09/1985 20/09/1985 20/09/1985

8.1 8.1 7.6 7.6

Central Central Central Central

CDAF CDAF CDAF CDAF

N00E N90E N00E N90E

25/04/1989 25/04/1989 25/04/1989 25/04/1989 25/04/1989 14/04/1989 25/04/1989 14/04/1989

6.9 6.9 6.9 6.9 6.9 7.1 6.9 7.1

Villa del mar Villa del mar Jamaica Rodolfo Menéndez P.C.C. Superficie Córdova Liverpool Roma-B

29 29 43 48 25 56 58 RB

24/10/1993 25/04/1989 25/04/1989 25/04/1989 10/12/1994 14/09/1995 10/12/1994 19/09/1985

6.6 6.9 6.9 6.9 6.3 7.1 6.3 8.1

U. Colonia IMSS U. Colonia IMSS San Simón Roma Roma Roma SCT SCT

44 44 53 RO RO RO SC SC

de de de de

Abastos Abastos Abastos Abastos

PGA (cm/s²)

PGV (cm/s)

Tg (s)

66.3 95.9 40.1 30.2

26.4 36.7 12.1 9.3

2.88 2.96 2.28 3.01

EW NS NS EW EW EW EW EW

46.5 49.4 35.2 47.7 42.5 45.2 40.0 23.6

15.3 22.0 15.6 18.8 15.4 11.2 12.4 4.8

2.96 2.96 3.04 2.89 2.32 2.33 2.29 2.30

EW EW EW EW EW EW EW EW

15.0 43.5 33.0 56.0 12.3 34.6 15.0 167.9

2.5 7.5 7.8 12.0 2.2 7.0 4.0 61.1

1.34 1.28 1.56 1.27 1.39 1.31 1.89 2.03


energy demands imposed to the structures located in very soft soil sites [7,15]. Thus, each earthquake record employed as a mainshock was linearly scaled to reach the PGV registered at the wellknown East–West component of the Secretaría de Comunicaciones y Transportes (SCT) station during the September 19, 1985, earthquake. In addition, the 8 records were scaled to reach 35%, 70% and 100% of the PGV of the SCT ground motion when employed as aftershock earthquake ground motions. Therefore, 28 artificial sequences were generated randomly for each VA/VM ratio (i.e. 0.35, 0.70, and 1.0; where VA and VM are the PGV of the aftershock and the mainshock, respectively) since no repetition of the mainshock waveformat was allowed in the set. The set of sequences established according to what has been discussed in this paragraph was denoted Set A. It is worth mentioning that large

Sv (cm/s)

scaling factors (i.e. greater than 10) might be necessary for reaching VA/VM ratios equal to 0.7 and 1.0. A recent study by Quiroz et al. [22] showed that the use of large scaling factors to linearly scale soft soil records results in biased estimates of the maximum deformation demands (i.e. underestimation of the demand) for systems exhibiting T/Tg ratios smaller than 0.5. This would be the case of the 4-story building, and the 8-story building when it is subjected to motions having Tg equal to 3.0 s. However, in general, the results can be considered reasonable in spite of the large scaling factors that are used in this investigation. Fig. 4 shows six artificial sequences (note that the figure contemplates the three VA/VM ratios for two combinations of motions). It should be noted, as illustrated in the right-bottom plot, that for sequences with a VA/VM ratio equal to one, it is

Sv (cm/s)

200

200 Mainshock

150

59

Mainshock

150

Aftershock

100

100

50

50

0 0.0

1.0

2.0

3.0

4.0

Aftershock

0 0.0

1.0

2.0 T (s)

T (s)

3.0

4.0

Fig. 3. Velocity response spectra computed from the mainshock and aftershock earthquake ground motions recorded during the 1985 Michoacan earthquakes at CDAF station: (a) N00E component and (b) N90E component.

Tg =2.96s

300

Tg =2.96s

0

0

100

200

300

400

500

VA/VM=0.35

-300



300

0

100

200

300

400

500

VA/VM=0.35

-300 300

0 0

100

200

300

400

500

VA/VM=0.70

-300



Tg =2.3s

0

300

0 0

100

200

300

400

500

VA/VM=0.70

-300

300

300

0

0

100

200

300

400

500

VA/VM=1.00

-300



Tg =2.96s

0

0

100

200

300

400

500

VA/VM=1.00

-300

Fig. 4. Examples of artificial sequences derived from mainshocks included in Set A.

Sv (cm/s)

350 300 250 200 150 100 50 0 0.0

Sv (cm)

N00E N90E mean set A

1.0

2.0 T(s)

3.0

4.0

350 300 250 200 150 100 50 0 0.0

mean set B

1.0

2.0

3.0

4.0

T(s)

Fig. 5. Comparison of velocity response spectra computed from the sequences recorded during the 1985 Michoacan earthquakes at CDAF station and the sets of artificial sequences derived from (a) Set A and (b) Set B.

60


possible for the aftershock to exhibit a larger PGA than that of the mainshock (although both motions were scaled to reach the same PGV). It should be mentioned that for performing nonlinear dynamic analyses, there is a time-gap of 40 s. having zero acceleration ordinates between the as-recorded mainshock and the artificial aftershock acceleration time-history to ensure that the systems reach their rest position after free-vibration. Similarly, aftershock acceleration time-history included zero acceleration ordinates at the end of the excitation. Additionally and according to the scope formulated for the paper, it is of interest to examine the seismic response of the casestudy buildings under seismic sequences with different frequency contents. Therefore, a second set of artificial sequences was generated from 8 acceleration time-histories recorded during 5 historical earthquakes in accelerographic stations placed on soft soil sites of Mexico City, as listed in Table 2. All earthquake ground motions were scaled to reach the PGV of the record captured at SCT station. For each one of the 8 scaled records, 7 sequences were generated by using it as a mainshock and the remainder 7 motions as aftershocks. This process was repeated to generate a total of 56 artificial seismic sequences, which are comprised in Set B. In should be noted that under this randomized approach, the PGA and/or the Tg of the aftershock ground motion can be larger than those/that of the mainshock ground motions. Particularly, while a subset of 28 artificial seismic sequences has PGA of the mainshock larger than that of the corresponding aftershocks, 6 out of these 28 sequences have Tg of the mainshock longer than that of the aftershock. A comparison between the velocity spectra obtained from each sequence recorded at CDAF station (scaled to reach the PGV recorded at SCT station) and the corresponding mean spectrum from Sets A and B is shown in Fig. 5a and b, respectively. It can be observed that the artificial sequences of

z/H

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-4N ( =0.1) M M+A_35% M+A_70% M+A_100%

1.0

2.0

3.0

4.0

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-8N ( =0.1)

1.0

2.0

3.0

4.0

Set A resemble, on average, the frequency content of the real sequences, while those included in Set B have shorter Tg than that of both real sequences.

3. Response under seismic sequences 3.1. Response to sequences corresponding to set A Initially, the influence of stiffness degradation in the response of the case-study buildings to the action of the mainshocks included in Set A and the sequences derived from them was examined. Figs. 6 and 7 show the height-wise distribution of median interstory drift ratio (IDR) for all buildings and both values of α, respectively. For comparison purposes, IDRs are shown as a function of normalized height, z/H, where z is the relative height from ground and H is the total height. It can be seen that for a VA/VM ratio equal to 0.35, the influence of αis negligible. However, the influence of stiffness degradation becomes more important as the relative intensity of the aftershock with respect to the mainshock increases, particularly for the 8- and 12-story buildings. Note that for VA/VM ¼0.7, the effect of high-level stiffness degradation is only significant for the 8-story frame. It is interesting to note that in general, the effect of the aftershocks is negligible for frames with low-stiffness degradation until the aftershocks are scaled to reach the same peak ground velocity as the mainshocks. The record-to-record variability of the IDR demands presented in the previous figures was examined through the coefficient of variation (COV). For instance, Fig. 8 shows the height-wise distribution of COV corresponding to all buildings and α ¼0.1. In general, it can be seen that the record-to-record variability increases as the VA/VM ratio increases. Although the COV values

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-12N ( =0.1)

1.0

IDR [%]

IDR [%]

2.0

3.0

4.0

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-16N ( =0.1)

1.0

IDR [%]

2.0

3.0

4.0

IDR [%]

Fig. 6. Heightwise distribution of median IDR demands for all buildings having low member's stiffness degradation; sequences derived from Set A.

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-4N ( =0.5) M M+A_35% M+A_70% M+A_100%

1.0

2.0 IDR [%]

3.0

4.0

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-8 N ( =0.5)

1.0

2.0 IDR [%]

3.0

4.0

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-12N ( =0.5)

1.0

2.0

IDR [%]

3.0

4.0

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-16N ( =0.5)

1.0

2.0

3.0

IDR [%]

Fig. 7. Heightwise distribution of median IDR demands for all buildings having high member's stiffness degradation; sequences derived from Set A.

4.0


z/ H

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

z/ H

C-4N ( =0.1) M+A_35% M+A_70% M+A_100%

0.2

0.4

C-8N ( =0.1)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.6

0.2

COV

0.4

0.6

z/ H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-12N ( =0.1)

0.2

COV

0.4

0.6

61

z/ H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

C-16N ( =0.1)

0.2

COV

0.4

0.6

COV

Fig. 8. Heightwise distribution of COV for all buildings having low member's stiffness degradation; sequences derived from Set A.

250

250 200

Strength

200

150

150

100

100

50 -15

1.4 1.2

-10

50

Disp

0 -5 -50 0

5

10

15

Strength

20

-15

-10

0 -5 -50 0

-100

-100

-150

-150

-200

-200

-250

-250

Sdn

1.4 1.2

Disp 5

10

15

20

Sdn

1 0.8

1 0.8

= = = =

0.6 0.4 0.2 0 0

2 3 4 6 1

0.6 0.4 0.2 0

T (sec) 2

3

4

5

T (sec) 0

1

2

3

4

5

Fig. 9. Effects of stiffness degradation on the displacement demands of systems subjected to mainshock ground motions recorded at the Lake Zone. (a) Low stiffness degradation, Model 1, (b) High stiffness degradation, Model 2, (c) Low stiffness degradation, spectra and (d) High stiffness degradation, spectra.

for the 8- and 12-story buildings may be relatively large for VA/VM ratios of 0.7 and 1.0, it should be noted that the use of artificial sequences lead to larger record-to-record variability than that when real sequences are employed [3]. In terms of achieving an overall understanding of the response of the individual frames, the effect of stiffness degradation is discussed next with the aid of Fig. 9. The mean normalized displacement spectra shown in the figure were obtained for 5% of critical damping and the seven mainshock ground motions under consideration in Table 3, which were recorded at sites in the Lake Zone with Tg around 2 s [15]. For this purpose two hysteretic behaviors corresponding to the cyclic response of RC structures (note there is no pinching of the hysteresis loops) were modeled using the three parameter model [16]. While the behavior corresponding to Fig. 9a (Model 1) corresponds to a Takeda-type hysteretic behavior, the one in Fig. 9b (Model 2) includes noticeable reloading and unloading stiffness degradation. The spectra shown in Fig. 9c and d correspond to normalized pseudodisplacement for constant ductility (Sdn). For a given value of ductility, the spectra shown in Fig. 9c consider the ratio between the mean displacement demand on SDOF systems that have a cyclic behavior characterized by Model 1, and that of corresponding systems (i.e.

having the same period) having elasto-perfectly-plastic hysteretic behavior. In the case of Fig. 9d, similar results are shown for Model 2. As shown in Fig. 9c and d, and beyond the natural variability expected in seismic response, the effect of stiffness degradation in terms of increasing the displacement demands tends to be negligible for a period close to zero, and tends to increase until it is maximized for a period close to Tg/2 (note that the abscissae in the figures are expressed in terms of T, and that the Tg of the ground motions summarized in Table 3 can be considered equal to 2.0 s). A further increase in period results in the effects of stiffness degradation decreasing until it becomes close to negligible for a period close to Tg. Once the period becomes larger than Tg, the displacement demands associated to degrading behavior are smaller than those corresponding to elasto-perfectly-plastic behavior. Also, it is noticeable that larger displacement demands are expected for Model 2. Considering the periods summarized in Table 1 for the buildings and a Tg close to 3.0 s for the seismic sequences (see Fig. 5a), the 4, 8, 12 and 16 story frames exhibit T/Tg ratios of 0.27, 0.44, 0.47 and 0.58. While the values of T/Tg of the frames help understand why stiffness degradation has a larger impact on the 8 and 12 story frames, the comparison of the spectra included in Fig. 9c and d help explain why

62


a larger level of unloading stiffness degradation leads to a larger effect during an aftershock. Within this context, note that in general the stiffness degradation effect studied in SDOF systems subjected to mainshocks has similar consequences for systems subjected to seismic sequences. The estimation of residual displacement demands is important for seismic assessment since the decision of repairing or demolishing a structure should be based on the technical difficulties involved in straightening leaned structures after earthquake excitation, which was the case after the 1985 Michoacan earthquakes [1]. In order to provide a context to the following results, it should be mentioned that FEMA 356 recommended seismic provisions for the assessment and rehabilitation of existing buildings in the U.S. specify limiting values on residual drift demands linked to system performance levels [17]. For instance, maximum residual interstory drift demand should not exceed 1% for Life Safety and 4% for Collapse Prevention performance levels. In addition, a recent field investigation in Japan highlighted that a residual inter-story drift of about 0.5% is perceptible for building occupants and a residual Table 3 Ground motions employed to derive Fig. 9. Date

MS

Station name

Comp.

PGA (cm/s²)

PGV (cm/s)

Tg (s)

19/09/1985 19/09/1985 20/09/1985 20/09/1985 25/04/1989 25/04/1989 25/04/1989

8.1 8.1 7.6 7.6 6.9 6.9 6.9

SCT Tlahuac Tlahuac Tlahuac Alameda Alameda Garibaldi

EW EW NS EW EW NS EO

167 118 49 51 46 37 52

61 35 13 15 15 10 17

2.0 2.1 2.0 1.9 2.1 2.1 2.1

C-4N ( = 0.1)

z/H

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

M M+A_35% M+A_70% M+A_100%

0.05

0.10

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

RIDR [%]

C-8N ( = 0.1)

0.05 0.10 RIDR [%]

inter-story drift of about 1.0% could cause human discomfort [18]. The distributions along height of median residual interstory drift demands, RIDR, for the frames including low-and-high member's unloading stiffness degradations, respectively, are illustrated in Figs. 10 and 11. Although the residual displacement demands computed in this study by using a Takeda type model tends to overestimate the amplitude of actual residual displacements as noted in [19,20], it is clear from the figures that the frames with high-unloading stiffness deterioration promote a self-centering behavior that constrains the permanent displacements, even under the strong aftershocks (i.e. sequences with VA/VM ratios of 0.7 and 1.0). Note that buildings with low unloading stiffness deterioration lead to larger RIDR demands. Unlike peak inter-story drift demands, the RIDR demands do not follow a clear trend as the intensity of the aftershocks increases. For example, median RIDR demands triggered by low intensity aftershocks are larger than those induced by high-intensity aftershocks in the lower stories of the 12-story building. As discussed in [21], this is a result of the large record-to-record variability inherent in the estimation of residual drift demands. While some conclusions can be offered in terms of the response of stiffness-degrading systems to mainshocks and seismic sequences; a primary goal of this paper is to identify relevant features of the ground motions that comprise the seismic sequences that are influential to the seismic response of RC buildings located in soft soil sites. Within this context, it can be said that the artificial seismic sequences have two main ground motion relationships: (1) the ratio of the intensity, measured in terms of PGV, of the aftershock to that of the mainshock (VA/VM), and (2) the ratio of the dominant period of the aftershock to that of the mainshock (TA/TM). The sequences corresponding to Set A exhibit VA/VM ratios of 0.35, 0.70, and 1.0; and TA/TM ratios of about 0.76 and 1.0. The distribution along height

z/H

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

C-12N ( = 0.1)

0.05

0.10

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

RIDR [%]

C-16N ( = 0.1)

0.05 0.10 RIDR [%]

0.15

Fig. 10. Heightwise distribution of median RIDR demands for all buildings having low member's unloading stiffness; sequences derived from Set A.

z/H

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

C-4N ( = 0.5) M M+A_35% M+A_70% M+A_100%

0.05

0.10

RIDR [%]

0.15

z/H

C-8N ( = 0.5)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

0.05

0.10

RIDR [%]

0.15

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

C-16N ( = 0.5)

0.05

0.10

RIDR [%]

0.15

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

C-16N ( = 0.5)

0.05

0.10

0.15

RIDR [%]

Fig. 11. Heightwise distribution of median RIDR demands for all frames having high member's unloading stiffness degradation; sequences derived from Set A.


z/H

1.0

0.9 0.8

z/H

VA/VM=0.35 Individual Mean Median

1.0

z/H


0.9 0.8

1.0

0.9 0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0 0.0

1.0 2.0 IDR [%]

3.0

0.0 0.0

1.0 2.0 IDR [%]

3.0

0.0 0.0

63


1.0 2.0 IDR [%]

3.0

Fig. 12. Heightwise distribution of median IDR demands for the 4-story building under seismic sequences with TA/TM around 1.0 and with three different values of VA/VM.

z/H 1.0 0.9 0.8

z/H 1.0

VA/VM=0.35

individual mean

individual

0.9

mean

0.8

Median

z/H 1.0

VA/VM=0.70

0.9 0.8

Median

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0 0.0

0.0 0.0

1.0 2.0 IDR [%]

3.0

1.0 2.0 IDR [%]

3.0

0.0 0.0

VA/VM=1.00

individual mean Median

1.0 2.0 IDR [%]

3.0


of IDR computed for the C-4N model is shown in Figs. 12 and 13 for the two TA/TM ratios. An increase in the intensity of the aftershocks results in an increase in the IDR demand, particularly in the bottom stories. Note that the TA/TM ratio has a strong influence becomes more important with an increase in the value of VA/VM. Particularly, sequences having a TA/TM of 0.76 (i.e. dominant periods of the aftershock are shorter than those of the mainshocks) trigger larger inter-story drifts than those derived from sequences with TA/TM ratios of about one. This observation can be explained since the damaged period of vibration of the building (i.e. its period of vibration at the end of the mainshock), estimated from a simple Fast Fourier Transform analysis of the inter-story acceleration response, becomes closer to the dominant period of the aftershock when TA of the aftershocks is close to 2.3 s. It is worth noting that the influence of TA/TM is more important to the response of the buildings than that of VA/VM; since the increment of IDR that results from an increase in VA/VM from 0.7 to 1.0 is small for TA/TM of 1.0 as opposed to what occurs for the case in which TA/TM is close to 0.76 (while there is an increment of median IDR of 44% in the ground story when TA/TM is around 0.76; this increment is about 23% for TA/TM of 1.0). The latter observation was confirmed after examining the lateral response of the other buildings. Particularly, Figs. 14 and 15 show the IDR demands for the 12-story building. Again as VA/VM increases, the IDR demands are larger for TA/TM of 0.76. In spite that the record-to-record variability of the IDR demands in Fig. 14 do not follow a wellestablished pattern (it is noticeable that this variability is larger for

VA/VM of 0.7 than for VA/VM of 0.35 and 1.0); it is possible to say that the frequency content of the aftershock is a relevant parameter when evaluating the effect of a seismic sequence on the response of a structure located on soft soil. 3.2. Response to sequences corresponding to set B In a second stage, all buildings were subjected to the action of the motions of Set B and the sequences derived from them. As discussed before, this set includes one motion recorded at the SCT station during the 1985 Michoacan earthquake. Figs. 16 and 17 show the median IDR demands for the buildings and both values of α. Although the artificial sequences lead to larger IDR demands than the mainshocks, these demands are larger for the buildings that exhibit high unloading stiffness degradation. Once more, Fig. 9 can be used to explain the response of the buildings to the sequences corresponding to Set B. By considering the periods summarized in Table 1 for the buildings and a Tg close to 1.3 s for the seismic sequences (see Fig. 5b), the 4-, 8-, 12- and 16-story buildings exhibit T/Tg ratios of 0.62, 1.01, 1.08 and 1.34, respectively. While the values of T/Tg of the buildings help understand why the seismic sequences have now a larger impact on the 4-story building, it also helps explain why the effect of the sequences is smaller than that of the sequences corresponding to Set A, and why a larger level of unloading stiffness degradation leads to a larger effect of the aftershock. Note that the IDR demands shown in Figs. 16 and 17 follow different trends than

64


z/H

z/H 1.0

VA/VM=0.35

1.0

Individual Mean Median

z/H

VA/VM=0.70

0.9

0.9

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.9 0.8

0.0

0.0

0.0 1.0

0.0

2.0

3.0

0.0

4.0

VA/VM=1.00

1.0

1.0

IDR [%]

2.0

3.0

0.0

4.0

1.0

IDR [%]

2.0

3.0

4.0

IDR [%]


z/H

z/H

VA/VM=0.35

1.0

Individual

0.9

Mean

z/H

VA/VM=0.70

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.8

Median

0.0

0.0

0.0

0.0

1.0

2.0 3.0 IDR [%]

0.0

4.0

VA/VM=1.00

1.0

1.0

2.0 3.0 IDR [%]

0.0

4.0

1.0

2.0 3.0 IDR [%]

4.0


z/H 1.0

C-4N ( =0.1) M M+A_100%

0.8

z/H 1.0

z/H 1.0

C-8N ( =0.1)

z/H 1.0

C-12N ( =0.1)

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.0 0.0

1.0

2.0 IDR [%]

3.0

4.0

0.0 0.0

1.0

2.0

3.0

4.0

IDR [%]

0.0 0.0

1.0

2.0 IDR [%]

3.0

4.0

0.0 0.0

C-16 N ( =0.1)

1.0

2.0

3.0

4.0

IDR [%]

Fig. 16. Heightwise distribution of median IDR demands for all buildings having low member's stiffness degradation and subjected to seismic sequences derived from Set B.

those included in Figs. 6 and 7. Particularly, while the former figures show a maximum IDR demand that tends to decrease with the height of the buildings, the latter figures indicate a tendency for the IDR demand to increase with height. An explanation for these tendencies can be provided by the spectra shown in Fig. 5 for the sequences corresponding to Sets A and B. While the spectral ordinates for the sequences corresponding to Set A tend to increase up to a period of 3.0 s, they tend to decrease for the sequences corresponding to Set B once the period of the buildings reach a value of 1.3 s.

A closer look at the response of the buildings to the sequences derived from Set B reveals, once more, that the T A =T M ratio plays an important role in the response, which is even more important than the intensity relationship between the mainshock and the aftershock. For instance, Fig. 18 shows the height-wise distribution of IDR demands triggered in the 8-story building (T1 ¼1.32 s) when subjected to 3 sub-sets of sequences derived from the maishocks M2, M7 and M8 included in Set B. The response under the corresponding mainshock is also shown in red dashed line. To help in providing an explanation of the effect of the aftershocks,


z/H 1.0

z/H 1.0

C-4N ( =0.5) M M+A_100%

z/H 1.0

C-8N ( =0.5)

C-12N ( =0.5)

65

z/H 1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.8

0.0 0.0

1.0

2.0

3.0

0.0 0.0

4.0

1.0

2.0

3.0

4.0

0.0 0.0

1.0

3.0

4.0

0.0 0.0

1.0

IDR [%]

IDR [%]

IDR [%]

2.0

C-16N ( =0.5)

2.0

3.0

4.0

IDR [%]

Fig. 17. Heightwise distribution of median IDR demands for all buildings having high member's stiffness degradation and subjected to seismic sequences derived from Set B.

z/H

z/H

1.0

1.0

M2+A M2 Mean

0.9

0.8 0.7

0.8 0.7

0.7 0.6

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.0 0.0

0.0 0.0

1.0

2.0

3.0

4.0

5.0

1.0

2.0

IDR [%]

3.0

IDR [%]

M8

0.8

0.5

1.48

M8+A

0.9

1.09

0.6

0.5

1.0

M7+A M7 Mean

0.9

1.61

0.6

z/H

4.0

5.0

0.0 0.0

1.0

2.0

3.0

4.0

5.0

IDR [%]

Fig. 18. Heightwise distribution of IDR demands for the 8-story building under three subsets of artificial seismic sequences generated from the mainshocks M2, M7 and M8 of Set B listed in Table 2.

a simple Fast Fourier Transform analysis (FFT) of the inter-story acceleration response due to the mainshock was employed to obtain an estimate of the damaged period of vibration of frame (i.e. period of vibration at the end of the mainshock excitation). In addition, velocity response spectra corresponding to the ground motions included in each subset are shown in Fig. 19. Under the first subset with mainshock M2 (note that most of the TA/TM ratios in this subset are larger than one), six out of seven sequences triggered larger inters-story drifts. For example, one of the largest responses is recorded when the M8 acts as aftershock, having TA/TM ¼1.61, although the peak ground acceleration (PGA) of the artificial aftershock is smaller than that of the mainshock. This could be explained since the FFT analysis revealed that the mainshock slightly elongated the frame's first mode period of vibration and, thus, the frame's period of vibration at the end of the mainshock will become closer to the aftershock's dominant period as shown in Fig. 19a. It is interesting to note that when M8 is employed as mainshock, the artificial sequences did not increase the interstory drifts even though most of the artificial aftershocks have PGA larger than that of the mainshock (e.g. simulating M7 as aftershock as shown in Fig. 19c). This could be explained since mainshock M8 induced highly nonlinear response in the structure which lead to increase the frame's period of vibration significantly (until about 2.0 s using the FFT analyses) and, thus, the relationship between the damaged period of vibration of the frames and the aftershock's dominant period is larger than one (i.e. the period of vibration of the building tends to move beyond the dominant period of the aftershocks) since all artificial sequences in the subset have TA/TM ratios smaller than one. Finally, the response under subset with mainshock M7 showed that interstory drift only increased when M8 is employed

as artificial aftershock. Again, this could be explained since the estimated period of vibration of the frame at the end of mainshock M7 was about 1.9 s and, thus, since the only aftershock with longer dominant period was M8 (with TA/TM ¼ 1.09 as illustrated in Fig. 19b), the drift response increased as shown in Fig. 18. It should be noted that other aftershocks in this subset have larger PGA than that of M7, but they have TA/TM ratios shorter than one (i.e. damaged period of vibration is longer than aftershock's dominant period). An additional example of the aforementioned observations is shown in Fig. 20 for the distribution along height of IDR corresponding to the 12-story frame (T1 ¼1.40 s). Therefore, findings of this study provide further explanation why RC buildings with periods of vibration from 0.5 s to 1.5 s in the lake-bed zone of Mexico City (with soil dominant periods around 2 s) that suffered structural damaged after the September 19, 1985 earthquake could increase their state of damage during the September 20 earthquake, although the latter seismic event had smaller intensity. Finally, Figs. 21 and 22 show the distribution along height of median RIDR demands corresponding to the family of case-study buildings, which takes into account low and high member's unloading stiffness, respectively. Although residual drift demands do not show a clear pattern, it can be seen that seismic sequences could increase, in general, their amplitude. However, only the 4- and 8-story buildings might experience RIDR demands that could lead to human discomfort. It should be noted that increment in RIDR demands for the 4-story building with respect to those triggered under seismic sequences derived from Set A is related, as previously discussed in this section, to the effect of the frequency content in the seismic sequences derived from Set B.

66


Sv (cm/s)

450 400 350 300 250 200 150 100 50 0 0.0

Sv (cm/s)

M2 PGA=301.9 cm/s2 M8 PGA=226.7 cm/s2

Tg=1.28s

Tg=2.06s 1.0

2.0

3.0

M7 PGA=171.0 cm/s2

450 400 350 300 250 200 150 100 50 0 0.0

4.0

M8 PGA=226.7 cm/s2

Tg=1.89s

Tg=2.06s 1.0

2.0

3.0

4.0

T (s)

T (s)

Sv (cm/s) M8 PGA=226.7 cm/s2

450 400 350 300 250 200 150 100 50 0 0.0

Tg=1.89s

M7 PGA=301.9 cm/s2

Tg=2.06s 1.0

2.0

3.0

4.0

T (s) Fig. 19. Comparison of velocity response spectra corresponding to ground motions included in three artificial sequences derived from Set B: (a) Sequence M2–M8, (b) sequence M7–M8, and (c) sequence M8–M7.

z/H

z/H

1.0

1.0

M2+A M2 Mean

0.9 0.8

0.8 0.7

0.6

0.6

0.4

0.3

1.09

0.5

1.61

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

M7+A M7 Mean

0.9

0.7 0.5

z/H

0.4 0.3

1.48

0.2

0.2

0.1

0.1 0.0

0.0 0.0

1.0

2.0

3.0

4.0

5.0

0.0

1.0

2.0

IDR [%]

3.0

IDR [%]

4.0

5.0

M8+A M8

0.0

1.0

2.0

3.0

4.0

5.0

IDR [%]

Fig. 20. Heightwise distribution of IDR demands for the 12-story building under three subsets of artificial seismic sequences generated from the mainshocks M2, M7 and M8 of Set B listed in Table 2.

4. Summary and conclusions Two strong earthquakes struck Mexico City on September 19 and 20, 1985 causing large damage to, mainly, RC buildings. In spite of these historical earthquake events, no investigation was conducted up to date to understand the effect of seismic sequences in the response of buildings located in soft soil sites. Perhaps, this lack of research was based on the insufficiency of ground motion recordings of mainshock– aftershock sequences. Therefore, this paper has summarized the results of an analytical study aimed at providing further understanding on the influence of seismic sequences on drift demands in regular existing RC moment-resisting frame buildings located in soft soil sites. This investigation focused on investigating whether aftershocks could increase peak (transient) and residual (permanent) drift demands in

frame models with different number of stories. Due to the insufficiency of real seismic sequences gathered in soft soil sites, two sets of artificial mainshock–aftershock sequences were generated using a randomized approach using real mainshock records gathered in soft soil sites. Under artificial sequences derived from Set A, it was shown that inter-story drift demands (IDR) in the case-study RC frames tends to increase as the ratio of the peak ground velocity of the aftershock with respect to the mainshock, VA/VM, increases. The effect of stiffness degradation is important since building models with low member's unloading stiffness degradation led to smaller IDR than their counterparts having high member's unloading stiffness degradation. On the contrary, low unloading stiffness degradation drives, in general, larger residual interstory drift demands.


z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

C-4N ( =0.1)

M M+A_100%

0.05

0.10

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

RIDR [%]

z/H

C-8N ( =0.1)

0.05

0.10

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

RIDR [%]

z/H

C-12N ( =0.1)

0.05

0.10

67

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

RIDR [%]

C=16N ( =0.1)

0.05

0.10

0.15

RIDR [%]

Fig. 21. Heightwise distribution of median RIDR demands for all buildings having low member's unloading stiffness degradation; sequences derived from Set B.

z/H 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

C-4N ( =0.5)

M M+A_100%

0.05

0.10

RIDR [%]

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

C-8N ( =0.5)

0.05

0.10

0.15

RIDR [%]

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

z/H

C-12N ( =0.5)

0.05

0.10

RIDR [%]

0.15

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

C-16N ( =0.5)

0.05

0.10

0.15

RIDR [%]

Fig. 22. Heightwise distribution of median RIDR for all buildings having high member's unloading stiffness degradation; sequences derived from Set B.

An important observation is that the building response strongly depends on the ratio of dominant period of the aftershock to the dominant period of the mainshock, TA/TM, and, most importantly, on the ratio of the damaged period of vibration (i.e. frame's period of vibration at the end of the mainshock) to the dominant period of the aftershock Td/TA. It has been shown that when the latter ratio is larger than one, IDR demands are not increased by the aftershock, independently of the fact that the peak ground acceleration of the aftershock could be larger than that of the mainshock.

Acknowledgments While the first and second authors would like to express their gratitude to Universidad Michoacana de San Nicolás de Hidalgo, the third author would like to thank Universidad Autonoma Metropolitana-Azcapotzalco. The authors acknowledge the National Council for Science and Technology, under grant CB-2008-102721, in Mexico for the financial support provided to develop the research reported in this paper. The authors are grateful for the comments and suggestions of three anonymous reviewers that helped to improve the final version of the paper.

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