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Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions
Engineering Structures 29 (2007) 2593–2601 www.elsevier.com/locate/engstruct
Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions Antonio B. Rigato, Ricardo A. Medina ∗ Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742, United States Received 16 November 2006; received in revised form 10 January 2007; accepted 12 January 2007 Available online 26 February 2007
Abstract This study examines the influence that the angle of incidence of the ground motion has on several engineering demand parameters (EDPs) for a single-storey structure subjected to bi-directional ground motions. The models in this work had various degrees of inelasticity, were subjected to a set of 39 ground motion pairs for which nonlinear time histories were conducted, and had fundamental periods that ranged from 0.2 to 2.0 s for both symmetrical and asymmetrical structures. It is demonstrated that applying bi-directional ground motions only along the principal axes of an inelastic building underestimates the inelastic peak deformation demands when compared to those obtained at other angles of incidence. Although an optimal building orientation that minimizes demands for all the EDPs considered for a given model cannot be determined explicitly, for a given degree of inelasticity, the average ratio of peak deformation responses based on all angles of incidence to the peak deformation response when the ground motions are applied at principal building orientations shows stable trends. Generally, these ratios increase with the fundamental period of vibration. These average ratios typically vary between 1.1 and 1.6; however, ratios for individual ground motions can be as high as 5 for the EDPs examined. Maximum responses for individual ground motions were found to occur for virtually any angle of incidence and varied with the degree of inelasticity, which implies that inaccurate estimates of structural performance and damage may result if based on ground motions applied at principal orientations alone. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Seismic response; Inelastic response; Bi-directional ground motions; Angle of incidence; Ductility demands; Drift demands
1. Introduction Currently it is believed that a significant portion of damage experienced by asymmetric structures is caused by torsion [1, 2]. The torsion generated within asymmetric structures originates from non-co-centric centers of mass, stiffness and strength. Several researchers [1,3] have investigated the capability of various code provisions to minimize seismic demands in these torsionally unbalanced structures (TUB). These studies examined single-storey structures with various configurations of structural elements that act in series or in parallel, with varying locations of mass, strength and stiffness. The ground motions utilized in these studies generally consisted ∗ Corresponding address: 1172 Glenn L. Martin Hall, University of Maryland, College Park, MD 20742, United States. Tel.: +1 301 4051948; fax: +1 301 4052585. E-mail addresses: [email protected] (A.B. Rigato), [email protected] (R.A. Medina).
c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.008
of a relatively small set of orthogonal horizontal seismic components pairs applied only along the principal axes of the structure, as was done in [1,3,4]. Other orientations at which these ground motions could be applied were not a focus of these studies, nor was the dependence of seismic demands on the degree of inelasticity of the building. It is still unclear as to how much of a role the angle of incidence has on the estimation of demands for asymmetric structures with varying degrees of inelasticity and for various fundamental periods of vibration. One of the earliest investigations into determining the critical angle of response due to seismic loads (i.e., the angle at which the quantity in consideration achieves a maximum) was done by Wilson [5] in an effort to display the shortcomings of the 100%–30% and 100%–40% combination rules. In this paper, the author provided a closed-form solution to determine the critical angle of response for an elastic, asymmetric structure for seismic loads that were based on the same ground motion spectrum with both components of ground motion assumed to be statistically independent. The proposed formula
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did not depend on angle of incidence when the two seismic component inputs had the same spectra, nor did it depend on the ratio of the spectral components if the second component was a fraction of the primary one. Cruz–Davila et al. [6] compared different methodologies such as the 100%–30% combination rule and the SRSS method in determining the maximum response of an elastic five-storey structure when one and two seismic components of ground motion were applied at various angles of incidence. Their results demonstrated that the SRSS and 100%–30% combination rule can underestimate the maximum response. Khoshnoudian and Poursha [7] obtained similar results for five-storey structures and evaluated the elastic and inelastic response of structures. They found that the angle at which the maximum inelastic response occurs for a given ground motion is not necessarily the same angle at which the maximum elastic response occurs. Other researchers [8–10] have improved upon the closedform solution developed by Wilson [5] by explicitly accounting for the statistical correlation between horizontal components of ground motion. Athanatopoulou [10] developed a formula for any given engineering demand parameter (EDP) (e.g., axial loads in columns) that allows the determination of the critical angle of response of an asymmetric structure provided that two analyses for each pair of accelerograms of a given ground motion are conducted. That is, one analysis done at the incident angle of α = 0◦ and a second analysis for the incident angle of α = 90◦ . The ground motions themselves may have any degree of correlation for the formula to be viable and it was only valid for elastic structures. MacRae [11] investigated the effect of angle of incidence on the seismic demands of a multi-storey steel structure exposed to near-fault ground motions, although only one degree of inelasticity was used and torsion within the structure was negligible. Tezcan and Alhan [12] also considered angle of incidence and examined three additional orientations beyond the principal building orientations for a given case, and used one degree of inelasticity. Despite some of the earliest work with regard to building orientation done by Franklin and Volker [13], who briefly highlighted the underestimation of seismic loads when considering only ground motions applied at the principal building orientations, to the best of the authors’ knowledge, systematic studies that address the influence of angle of incidence of ground motion input on the inelastic response of structures have not been conducted. This study examines the response of asymmetric and symmetric structures to the angle of incidence of ground motion input for structures with varying degrees of inelasticity and various fundamental periods of vibration. It is demonstrated that maximum values of ductility ratio, slab rotation and drift ratio demands often occur when the ground motion input is applied at orientations other than the principal orientations of the structure. The principal orientations of the structure are typically the orientations considered for the application of seismic loads in design and performance assessment procedures. On average, ductility and drift demands were underestimated by as much as 65% when compared to those obtained with ground motions applied parallel to the principal axis of the structure. Deformation demands for individual
ground motions were in some cases underestimated by as much as 500%. 2. Structural models Multiple single-storey models were examined and evaluated in this work. Two distinct types of model were developed. The first type was the TUB model, which was an asymmetric singlestorey shear type structure with four columns and off-centered mass as shown in Fig. 1(a). The columns were fixed at their base and had the potential to develop plastic hinges at both ends. A rigid diaphragm was used to model the roof of the structure. The location of the center of mass was chosen so as to induce torsion in the structure at any angle of incidence, and to avoid a situation where torsion may be absent from the analysis, i.e., which could occur if the resultant of the seismic forces passes concurrently through the center of strength, rigidity and mass. For a given mass, the column stiffness of the structures was tuned to achieve fundamental periods of 0.2, 0.3, 0.4, 0.5, 1.0 and 2.0 s. The second model (Fig. 1(b)) was the torsionally balanced system (TB), which had no torsion due to co-centric centers of strength, rigidity and mass. The fundamental period of vibration for the TB model was identical to its TUB counterpart. For the TB model, the off-centered mass defined in the TUB model was placed at the geometric center of the structure at roof level and represented a structure that had negligible torsional effects. In order to maintain the same fundamental period as that of the TUB counterpart, the columns of the TB models needed to be more flexible. Plastic hinges were located at the column ends as per the TUB system. These plastic hinges were used to model the material inelasticity of the members by means of a bilinear hysteretic model with a positive post-yield stiffness of 3%. Plastic moments were calculated using the modal analysis procedure in the IBC 2006 [14]. Equivalent lateral forces based on modal properties of the structure and the design spectrum shown in Fig. 2 were calculated and then applied to the center of mass of the structure to conduct a linear elastic static analysis. For this calculation only the first two modes were considered, for they corresponded to a cumulative mass participation of at least 90%. The design spectrum used was that of a location along coastal California with NEHRP site class D. The maximum moment observed in this linear elastic static analysis at the end of any column was the value assigned as the plastic moment to all plastic hinges in the model. Plastic moments determined in this manner pertained to a relative design intensity (Rd ) with a value equal to one. The parameter Rd is defined by the ratio of the ground motion intensity to the design lateral force (Vd ) in the structure divided by the weight of the structure (W ): Rd =
Sa (T1 ) g Vd W
.
(1)
In the analysis process, all records were scaled to the same 5%-damped pseudo-spectral acceleration at the fundamental period of the structure, Sa (T1 )/g. Thus, with Sa (T1 )/g of the
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(a) TUB model.
(b) TB model. Fig. 1. General configuration of torsionally unbalanced (TUB) and torsionally balanced (TB) models (CM, center of mass; CR, center of rigidity; CS, center of strength).
Fig. 2. Design spectrum for a coastal site in California, NEHRP site class D.
primary component always being a constant (C) value for a given model, Eq. (1) becomes: Rd = C ·
W . Vd
(2)
To produce various inelastic results from the aforementioned models, a fraction of the plastic moment value relative to the Rd of interest was used during inelastic analyses. Rd values of 1, 2, 4 and 6 were used for both the TB and TUB models for each fundamental period of interest. The authors recognize that the ductility demands obtained for short-period structures presented in the following section with Rd = 6 were very large; however, this relative design intensity value was presented to evaluate the behavior of systems that have the potential to exhibit excessive levels of inelasticity. In addition, an elastic case (i.e., with infinitely strong columns) was created for both the TB and TUB models for each fundamental period. A total of 60 models (6 periods × 5 strength values × 2 model types) were developed as part of this study. The effects of gravity load moments, P–M interaction, and P–delta effects were not taken into consideration. All models had 5% Rayleigh damping (ξ ) in the first and third modes. As an example of the model properties, the TUB model with a fundamental period of 0.5 s and Rd = 2 had a mass of 6500 kg, individual column stiffness
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Table 1 Ground motions used in this study Record ID
Event
Year
Moment magnitude
Station
Closest distance (km)
PGA (g) major component
PGA (g) minor component
IV79cal IV79chi IV79cmp IV79e01 IV79e12 IV79e13 IV79nil IV79pls IV79wsm LP89agw LP89cap LP89g03 LP89g04 LP89gmr LP89hch LP89hda LP89hvr LP89sjw LP89slc LP89svl NR94cen NR94cnp NR94far NR94fle NR94glp NR94hol NR94lh1 NR94lv2 NR94lv6 NR94nya NR94pic NR94stc NR94stn NR94ver SF71pel SH87bra SH87icc SH87pls SH87wsm
Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge San Fernando Superstition Hills Superstition Hills Superstition Hills Superstition Hills
1979 1979 1979 1979 1979 1979 1979 1979 1979 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1971 1987 1987 1987 1987
6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.6 6.7 6.7 6.7 6.7
Calipatria Fire Station Chihuahua Compuertas El Centro Array #1 El Centro Array #12 El Centro Array #13 Niland Fire Station Plaster City Westmorland Fire Station Agnews State Hospital Capitola Gilroy Array #3 Gilroy Array #4 Gilroy Array #7 Hollister City Hall Hollister Differential Array Halls Valley Salinas-John & Work Palo Alto-SLAC Lab. Sunnyvale-Colton Ave LA-Centinela St Canoga Park-Topanga Can. LA-N Faring Rd LA-Flecther Dr. Glendale-Las Palmas LA-Hollywood Stor FF Lake Highes #1 Leona Valley #2 Leona Valley #6 La Crescenta-New York LA-Pico & Sentous Northridge-17645 Saticoy St LA-Saturn St LA-E Vernon Ave LA-Hollywood Stor Lot Brawley El Cento Imp. Co. Cent Plaster City Westmorland Fire Station
23.8 28.7 32.6 15.5 18.2 21.9 35.9 23.6 15.1 28.2 14.5 14.4 16.1 24.2 28.2 25.8 31.6 32.6 36.6 28.8 30.9 15.8 23.9 29.5 25.4 25.5 36.3 37.7 38.5 22.3 32.7 13.3 30 39.3 21.2 18.2 13.9 21 13.3
0.128 0.27 0.186 0.139 0.143 0.139 0.109 0.057 0.11 0.172 0.529 0.555 0.417 0.323 0.247 0.279 0.134 0.112 0.278 0.209 0.465 0.42 0.273 0.24 0.357 0.358 0.087 0.091 0.178 0.178 0.186 0.477 0.474 0.153 0.21 0.156 0.358 0.186 0.211
0.078 0.254 0.147 0.134 0.116 0.117 0.069 0.042 0.074 0.159 0.443 0.367 0.212 0.226 0.215 0.269 0.103 0.091 0.194 0.207 0.322 0.356 0.242 0.162 0.206 0.231 0.077 0.063 0.131 0.159 0.103 0.368 0.439 0.12 0.174 0.116 0.258 0.121 0.172
of 354 kN/m for both the X and Y -directions and plastic moment values equal to 19.8 kN m. 3. Ground motions A suite of 39 pairs of horizontal ground motion records (Table 1) was obtained from the PEER strong motion database (http://peer.berkeley.edu/smcat/). Ground motions were recorded on stiff soil sites which were at least 13 km away from the fault rupture zone but occurred within 60 km of the site [15]. The ground motions had moment magnitudes between 6.5 and 6.9 and had similar frequency content. Additional information regarding the ground motions can be found in [15]. For a given ground motion, one of the two horizontal components was classified as either being a major component or a minor component based on its PGA value. The one with the highest PGA was considered to be the major component, while the other was labeled as the minor component. However, this definition did not ensure that the
Sa (T1 )/g value of the major component was larger than that of the minor component. For instance, at 0.5 s, 7 of the 39 records had larger Sa (T1 )/g values for the minor component than for the major component. The major component was applied at various angles of incidence, α, while the minor component was applied at an angle of α + 90◦ with respect to the X -axis as shown in Fig. 1. As stated before, all major components of the 39 pairs of ground motions were scaled to the same Sa (T1 )/g value (Fig. 2) at the fundamental period of the model, while the minor component was proportionally scaled relative to its corresponding major component. In this way, the frequency content of both components, as well as the intensity of one relative to the other, were maintained. 4. Methodology In an effort to evaluate the dependence of ductility ratios, slab rotations and drift ratios on the angle of incidence, time history analyses using the 39 pairs of orthogonal components were
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conducted at increasing 5◦ increments counter clockwise relative to the X -axis of the structure (see Fig. 1). The seismic loads were applied only from 0◦ to 180◦ since the remaining orientations from 185◦ to 360◦ would produce duplicate values due to symmetry. The results at 180◦ were identical to those produced at 0◦ but were included for completeness. Thus, a total of 37 orientations were considered per ground motion pair. In this context, the angle of incidence defines the orientation of the major and minor components of ground motion relative to the X -axis. With respect to the degree of inelastic behavior, the relative design intensity values examined in this paper were Rd = 1, 2, 4 and 6. As mentioned earlier, inelastic results were obtained in the analysis by keeping the ground motion intensity constant and reducing the value of the design plastic moments. Hence, plastic moments were proportionally scaled to the value of the Rd under consideration. For instance, Rd = 2 indicates that the plastic moment value in the model need only be half of the value required for Rd = 1. Column displacement ductility ratios, slab rotations and column drift ratios were recorded for each time history analysis in each direction (i.e., the X - or Y -direction). In this context, the column displacement ductility ratio (ductility) was defined as the displacement observed at the top of the column normalized by its yield displacement. Column drift ratios (drift) were defined as the displacement observed at the top of the column normalized by the column height. With respect to ductility, the maximum ductility observed at any particular column and the average ductility of all columns were used. The average ductility was defined as the sum of the individual column ductility ratios in one direction divided by four, and was considered to be a global measure of damage. 5. Dependence of inelastic response on angle of incidence For a given building and pair of bi-directional ground motions, inelastic demands (e.g., ductility, slab rotation and drift) were sensitive to the angle of incidence of the ground motion input. This is illustrated in Fig. 3(a) where the grey lines correspond to ductility ratios in the X -direction for individual ground motion pairs and the black line represents the average of the ductility ratios in the X -direction for a given angle of incidence. In this example, column 1 of the TUB 0.5 s model with Rd = 6 was utilized. From this figure it is evident that certain individual responses appear to be more sensitive to angle than others. In an effort to better understand why a pair of horizontal ground motion components may produce significant variations for a given EDP with respect to angle while another may not, the variation of ductility demands with respect to the ratio of Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) of a ground motion pair was investigated. For a given model and direction (i.e., the X - or Y -direction), variation was defined as the ratio of the maximum value of ductility observed for a given pair of records at any angle divided by the minimum value for the same pair of records. The dependence of the ductility variation with the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) at various periods was investigated. The ratios were calculated
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Fig. 3. (a) Dependence of ductility demands in the X -direction on angle of incidence and (b) dependence of the variation in column ductility demands in the X -direction on the ratio of spectral accelerations of major to minor ground motion component at the fundamental period.
at the fundamental period of vibration of the structures and at periods up to twice the fundamental period. These results failed to illustrate a significant correlation of the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) with ductility variation, as illustrated in the representative case shown in Fig. 3(b) for column 1 of the TUB structure with a fundamental period of 0.5 s. This indicates that the relationship between ductility demands and angle of incidence was not strongly dependent on the ground motion spectral ratios. For example, in Fig. 3(b), the correlation coefficient between ductility variation and the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) was equal to 0.33. The linear model fitted to these data is also shown in Fig. 3(b). The observed differences in the magnitude of ductility demands as a function of angle of incidence of ground motion input are attributed to differences in the frequency-content characteristics of the ground motion records. Because of the need to generalize behavioral trends and quantify seismic demands for inelastic buildings exposed to bidirectional ground motions at different angles, average values are utilized in the following subsections to evaluate ductility, drift and slab rotation demands, in addition to upper bounds for maximum values corresponding to individual ground motion pairs. It will be shown that critical angle not only depends on the EDP of interest, but also on the fundamental period and the level of inelastic behavior of the structure.
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Fig. 5. TUB model with a fundamental period of 1.0 s. (a) Angle of incidence at which ductility ratios in the X -direction are maximum. Fig. 4. Influence of angle of incidence on (a) average and (b) maximum ductility demands in the X -direction, for a TUB model with a fundamental period of 0.5 s.
5.1. Evaluation of ductility and drift demands For a given level of inelastic behavior, the magnitude of maximum and average ductility values was a function of the angle of incidence. Column ductility ratios did not exhibit a maximum at 0◦ or at 90◦ for the majority of Rd values studied. These two values, i.e., 0◦ and 90◦ , were the angles of incidence traditionally used to evaluate seismic demand parameters for TB and TUB systems [1,4,16,17]. These trends can be seen in Fig. 4 which displays, for a given Rd , the mean of the average ductility of all columns (Fig. 4(a)) and the upper bound of maximum ductility values of all columns in the X -direction, µx , (Fig. 4(b)) for the 0.5 s TUB model as a function of angle of incidence. An alternative way of illustrating this phenomenon is by examining histogram graphs of maximum ductility demands in both the X - and Y -directions, which are shown in Fig. 5 for a TUB model with a period of 1.0 s. As illustrated in Fig. 5, maxima can occur at any angle, a trend observed for all models. As expected, as Rd increased so did the average ductility. Similar observations were made for the column ductility in the Y -direction (µ y ); however, these latter results differed from those in the X -direction in that the magnitudes of the ductility ratios were slightly smaller and peaked at different angles for a given Rd . The behavioral pattern illustrated in Fig. 4 for the
X -direction, and that described for the Y -direction, was typical of what was found for all TUB models at all fundamental periods evaluated. In addition, the observations made in the previous paragraph for the TUB model also applied to the TB model for 0.5 s as can be seen in Fig. 6. A comparison of Figs. 4 and 6 demonstrates that given the fundamental period of vibration, TB models exhibit higher ductility demands than TUB models. This result is consistent with observations made by Humar and Kumar [16], who utilized TUB models in which the center of rigidity rather than mass was offset in order to induce torsion. The patterns presented in Fig. 6 are typical of what was found for all TB models at all fundamental periods. Regardless of whether or not torsional effects were significant, both maximum and average ductility demands were strongly dependent on the angle of incidence for a given Rd . Therefore, damage assessment based on either maximum or average storey ductility ratios will be dependent on the incident angle of the ground motion input. This observation is consistent with those presented in subsequent sections with regards to drift and slab rotation demands. Graphs of the mean of the average drift and upper bound of maximum drift produced very similar trends to those of ductility. Drift demands did not exhibit a maximum at 0◦ or at 90◦ for the majority of Rd values studied. In addition, peak inelastic drift normalized by peak elastic drift values for two separate lines of columns indicated that, on average, inelastic displacements relative to elastic ones did not always peak at
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Fig. 6. Influence of angle of incidence on (a) average and (b) maximum ductility demands in the X -direction, for a TB model with a fundamental period of 0.5 s.
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Fig. 7. Dependence of the ratio of inelastic to elastic drifts on angle of incidence: (a) column line 1–2, X -direction; (b) column line 3–4, X -direction.
angles 0◦ and 90◦ as shown in Fig. 7 for the X -direction, a trend which was also observed in the Y -direction. 5.2. Evaluation of slab rotation demands While the average slab rotations appeared to be weakly dependent on angle of incidence, the amount of torsion-induced damage and torsion itself may be reduced by allowing the building to experience higher levels of inelastic behavior (see Figs. 8 and 9). This observation presupposes that slab rotations are good indicators of torsion. In particular, Fig. 8 shows the average slab rotations as a function of angle of incidence and degree of inelasticity for the TUB model with fundamental period of 0.3 s, while Fig. 9 depicts the ratio of inelastic to elastic slab rotations for angles 0◦ and 90◦ for all TUB models. Other studies such as those in [17] found that elastic slab rotations were greater than inelastic slab rotations; however, this observation was made for only two fundamental periods (0.4 s and 0.8 s), different models and ground motions. Studies by De-La-Colina [18] also found slab rotations to decrease with increasing strength-reduction factor (R). Since the model characteristics in both [17,18] (such as mass and stiffness) are different from those in this work, it can be concluded that this phenomenon is controlled by the fundamental period and the level of inelastic behavior of the structural system.
Fig. 8. Dependence of slab rotation demands on angle of incidence, for a TUB model with a fundamental period of 0.3 s.
6. Implications for performance-based evaluation and design The results presented in previous sections demonstrated that seismic demands on inelastic buildings exposed to bidirectional ground motions can be underestimated if the pair of ground motion records is only applied at the principal orientations of the building. The implication is that inaccurate estimates of damage assessment, and hence direct dollar losses,
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Fig. 9. Dependence of the ratio of inelastic to elastic slab rotations on fundamental period, for angles of incidence equal to 0◦ and 90◦ .
Fig. 11. Ratio of maximum ductility for any angle of incidence to the maximum ductility with an angle of incidence of (a) 0◦ (X -direction) and (b) 90◦ (Y direction). Fig. 10. Empirical fragility curves for ductility demands in the X -direction as a function of angle of incidence for a TUB model with a fundamental period of 2 s.
could be obtained from performance evaluation conducted with bi-directional ground motions with the major and minor components applied at 0◦ and 90◦ , respectively. This is illustrated with the empirical ductility demand fragility curves shown in Fig. 10. It can be observed that certain angles were more likely to have greater ductility demands than others. Here, the probability that a given µx0 = 2 will be exceeded is plotted versus Sa (T1 )/g for a TUB model with a normalized base shear strength of the structure, γ , equal to 0.25. Fig. 10 indicates that for most Sa (T1 )/g values, results at different angles such as α = 45◦ and 90◦ for the 2.0 s TUB model had a greater likelihood of exceeding a ductility value of 2.0 as compared to the results at 0◦ . The exception is the range corresponding to Sa (T1 )/g > 0.75, in which differences between fragilities for 90◦ and 0◦ exhibit opposite trends. These observations imply that damage estimates based on fragilities for ground motions applied at principal orientations can be grossly inadequate. With maxima not always occurring when the major component of ground motion is applied at α = 0◦ or at 90◦ , it is important to develop simplified design procedures to quantify seismic demands as a function of the angle of incidence. The results from this study did not show consistent, stable trends that will allow the robust quantification of such
demands. However, on average, the ratio of the maximum ductility demand at any angle of incidence to the ductility demand at 0◦ (for ductility in the X -direction) and at 90◦ (for ductility in the Y -direction) varied between 1.1 and 1.6. This variation exhibited trends that indicate a significant dependence with respect to period and a weak dependence with respect to Rd (as shown in Fig. 11). Fig. 11(a) illustrated this behavior for maxima, where the mean of the ratio of the maximum value µx of all columns for any angle of incidence divided by the maximum µx any column experiences with an angle of incidence of 0◦ has been plotted. Fig. 11(b) is similar, with the exception that the maximum value of µY is examined and is normalized by the maximum µY experienced by any column when the angle of incidence is 90◦ . Due to the symmetry of the model, the results for the TB model were identical in both figures. The mean of the ratio of the peak average ductility in the X -direction at any angle of incidence to the average ductility demand at 0◦ produced very similar results. Although these graphs were for ductility, they can be interchanged for drift values because the properties of each column in any given model (i.e., stiffness and yield rotation) were the same. These results reinforce the notion that design and performance assessment procedures based on ground motions applied at the principal orientations of the building will tend to underestimate peak inelastic seismic demands along the principal orientations of the building, especially at longer periods (greater than 0.5 s).
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7. Conclusions Regardless of whether or not a building has a torsional irregularity, the maximum values for the EDPs examined in this study have the potential to occur when the pair of horizontal ground motion records is applied at any angle other than the traditional α = 0◦ , 90◦ , i.e., angles that coincide with the principal directions of the building under consideration. Results from this study demonstrate that, in general, peak inelastic deformation demands are underestimated when the horizontal components of ground motion are applied along the principal orientations of an inelastic structure. In addition, the information presented on slab rotations shows that the overall torsion in a building can be reduced by increasing the degree of inelasticity for most of the models studied. Two model types were examined; the torsionally balanced model which had co-centric mass, strength and stiffness and the torsionally unbalanced model which had co-centric strength and rigidity but offset mass. The magnitudes of the EDPs examined for both model types were quite different; nevertheless, general behavioral trends with respect to angle of incidence were similar. The observations and findings presented in this work are applicable to buildings and ground motions with characteristics consistent with those utilized in this study. The critical angle for a given EDP varies with fundamental period, model type and the level of inelastic behavior, and it is difficult to determine a priori like that of an elastic structure. This implies that performance evaluation and damage assessment based on inelastic deformation demands will be dependent on the angle of incidence of the ground motion input. Thus, performance assessment and design verification of structures designed to undergo inelastic deformations should ideally be conducted with bi-directional ground motions applied at various angles with respect to the principal orientations of the building. However, for the majority of structures, such an approach is impractical and methods to determine the required number of pairs of ground motion records and angles of incidence to estimate peak inelastic deformation demands with a given precision and level of confidence are needed. Results from this study demonstrate that for a given period, the results based solely on angles of incidence of 0◦ and 90◦ diverge by a consistent amount from the peak inelastic deformations observed at other angles regardless of the level of inelastic behavior. For instance, for a given structure and ground motions, the ratio of the peak of the maximum inelastic deformation obtained for a given angle of incidence to the maximum inelastic deformation with an angle of incidence of 0◦ (for the X -direction) and of 90◦ (for the Y direction) tends to increase with fundamental period and varies, on average, between 1.1 and 1.6 for both torsionally balanced and torsionally unbalanced models. This type of information
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is currently being used by the authors in a parallel study to develop simplified methodologies to efficiently estimate the magnitude of peak inelastic deformation demands for systems with different number of storeys, offset strength and rigidity, as well as mass in any combination thereof, which are exposed to ground motions with different frequency content. References [1] Tso WK, Smith RSH. Re-evaluation of seismic tensional provisions. Earthq Eng Struct Dyn 1999;28:899–917. [2] Aziminejad A, Moghadam AS. Behavior of asymmetric structures in near field ground motions. In: Conference on earthquake engineering. 2006. Paper 399. [3] Ghersi A, Rossi PP. Influence of bi-directional ground motions on the inelastic response of one-storey in-plan irregular systems. Eng Struct 2001;23:579–91. [4] Goel RK. Seismic response of asymmetric systems: Energy approach. J Struct Eng 1997;123(11):1444–53. [5] Wilson EL, Suharwardy I. A clarification of the orthogonal effects in a three-dimensional seismic analysis. Earthq Spectra 1995;11(4):659–66. [6] Fernandez-Davila I, Comeinetti S, Cruz EF. Considering the bidirectional effects and the seismic angle variations in building design. In: Proceedings of the twelfth world conference on earthquake engineering. 2000. [7] Khoshnoudian F, Poursha M. Responses of three dimensional buildings under bi-directional and unidirectional seismic excitations. In: Proceedings of the thirteenth world conference on earthquake engineering. 2004. [8] Lopez OA, Chopra AK, Hernandez JJ. Critical response of structures to multicomponent earthquake excitation. Earthq Eng Struct Dyn 2000;29: 1759–78. [9] Lopez OA, Torres R. The critical angle of seismic incidence and structural response. Earthq Eng Struct Dyn 1997;26:881–94. [10] Athanatopoulou AM. Critical orientation of three seismic components. Eng Struct 2005;27:301–12. [11] MacRae GA, Mattheis J. Three-dimensional steel building response to near-fault motions. ASCE J Struct Eng 2000;126(1):117–26. [12] Tezcan SS, Alhan C. Parametric analysis of irregular structures under seismic loading according to the new Turkish Earthquake Code. Eng Struct 2001;23:600–9. [13] Franklin CY, Volker JA. Effect of various 3-D seismic input directions on inelastic building systems based on INRESB-3D-82 Computer Program. In: Proceedings of the seventh European conference on earthquake engineering. 1982. [14] International code council. International building code (IBC-06). Whittier (CA); 2006. p. 663. [15] Medina RA, Krawinkler H. Seismic demands for non-deteriorating frame structures and their dependence on ground motions. PEER Report 2003/15. Berkeley (CA): Pacific Earthquake Engineering Research Center; 2004. [16] Humar JL, Kumar P. Effect of orthogonal inplane structural elements on inelastic torsional response. Earthq Eng Struct Dyn 1999;28:1071–97. [17] Perus I, Fajfar P. On the inelastic torsional response of single-storey structures under bi-axial excitation. Earthq Eng Struct Dyn 2005;34: 931–41. [18] De-La-Colina J. Effects of torsion factors on simple non-linear systems using fully bidirectional analyses. Earthq Eng Struct Dyn 1999;28: 691–706.
Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions Antonio B. Rigato, Ricardo A. Medina ∗ Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742, United States Received 16 November 2006; received in revised form 10 January 2007; accepted 12 January 2007 Available online 26 February 2007
Abstract This study examines the influence that the angle of incidence of the ground motion has on several engineering demand parameters (EDPs) for a single-storey structure subjected to bi-directional ground motions. The models in this work had various degrees of inelasticity, were subjected to a set of 39 ground motion pairs for which nonlinear time histories were conducted, and had fundamental periods that ranged from 0.2 to 2.0 s for both symmetrical and asymmetrical structures. It is demonstrated that applying bi-directional ground motions only along the principal axes of an inelastic building underestimates the inelastic peak deformation demands when compared to those obtained at other angles of incidence. Although an optimal building orientation that minimizes demands for all the EDPs considered for a given model cannot be determined explicitly, for a given degree of inelasticity, the average ratio of peak deformation responses based on all angles of incidence to the peak deformation response when the ground motions are applied at principal building orientations shows stable trends. Generally, these ratios increase with the fundamental period of vibration. These average ratios typically vary between 1.1 and 1.6; however, ratios for individual ground motions can be as high as 5 for the EDPs examined. Maximum responses for individual ground motions were found to occur for virtually any angle of incidence and varied with the degree of inelasticity, which implies that inaccurate estimates of structural performance and damage may result if based on ground motions applied at principal orientations alone. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Seismic response; Inelastic response; Bi-directional ground motions; Angle of incidence; Ductility demands; Drift demands
1. Introduction Currently it is believed that a significant portion of damage experienced by asymmetric structures is caused by torsion [1, 2]. The torsion generated within asymmetric structures originates from non-co-centric centers of mass, stiffness and strength. Several researchers [1,3] have investigated the capability of various code provisions to minimize seismic demands in these torsionally unbalanced structures (TUB). These studies examined single-storey structures with various configurations of structural elements that act in series or in parallel, with varying locations of mass, strength and stiffness. The ground motions utilized in these studies generally consisted ∗ Corresponding address: 1172 Glenn L. Martin Hall, University of Maryland, College Park, MD 20742, United States. Tel.: +1 301 4051948; fax: +1 301 4052585. E-mail addresses: [email protected] (A.B. Rigato), [email protected] (R.A. Medina).
c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.008
of a relatively small set of orthogonal horizontal seismic components pairs applied only along the principal axes of the structure, as was done in [1,3,4]. Other orientations at which these ground motions could be applied were not a focus of these studies, nor was the dependence of seismic demands on the degree of inelasticity of the building. It is still unclear as to how much of a role the angle of incidence has on the estimation of demands for asymmetric structures with varying degrees of inelasticity and for various fundamental periods of vibration. One of the earliest investigations into determining the critical angle of response due to seismic loads (i.e., the angle at which the quantity in consideration achieves a maximum) was done by Wilson [5] in an effort to display the shortcomings of the 100%–30% and 100%–40% combination rules. In this paper, the author provided a closed-form solution to determine the critical angle of response for an elastic, asymmetric structure for seismic loads that were based on the same ground motion spectrum with both components of ground motion assumed to be statistically independent. The proposed formula
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did not depend on angle of incidence when the two seismic component inputs had the same spectra, nor did it depend on the ratio of the spectral components if the second component was a fraction of the primary one. Cruz–Davila et al. [6] compared different methodologies such as the 100%–30% combination rule and the SRSS method in determining the maximum response of an elastic five-storey structure when one and two seismic components of ground motion were applied at various angles of incidence. Their results demonstrated that the SRSS and 100%–30% combination rule can underestimate the maximum response. Khoshnoudian and Poursha [7] obtained similar results for five-storey structures and evaluated the elastic and inelastic response of structures. They found that the angle at which the maximum inelastic response occurs for a given ground motion is not necessarily the same angle at which the maximum elastic response occurs. Other researchers [8–10] have improved upon the closedform solution developed by Wilson [5] by explicitly accounting for the statistical correlation between horizontal components of ground motion. Athanatopoulou [10] developed a formula for any given engineering demand parameter (EDP) (e.g., axial loads in columns) that allows the determination of the critical angle of response of an asymmetric structure provided that two analyses for each pair of accelerograms of a given ground motion are conducted. That is, one analysis done at the incident angle of α = 0◦ and a second analysis for the incident angle of α = 90◦ . The ground motions themselves may have any degree of correlation for the formula to be viable and it was only valid for elastic structures. MacRae [11] investigated the effect of angle of incidence on the seismic demands of a multi-storey steel structure exposed to near-fault ground motions, although only one degree of inelasticity was used and torsion within the structure was negligible. Tezcan and Alhan [12] also considered angle of incidence and examined three additional orientations beyond the principal building orientations for a given case, and used one degree of inelasticity. Despite some of the earliest work with regard to building orientation done by Franklin and Volker [13], who briefly highlighted the underestimation of seismic loads when considering only ground motions applied at the principal building orientations, to the best of the authors’ knowledge, systematic studies that address the influence of angle of incidence of ground motion input on the inelastic response of structures have not been conducted. This study examines the response of asymmetric and symmetric structures to the angle of incidence of ground motion input for structures with varying degrees of inelasticity and various fundamental periods of vibration. It is demonstrated that maximum values of ductility ratio, slab rotation and drift ratio demands often occur when the ground motion input is applied at orientations other than the principal orientations of the structure. The principal orientations of the structure are typically the orientations considered for the application of seismic loads in design and performance assessment procedures. On average, ductility and drift demands were underestimated by as much as 65% when compared to those obtained with ground motions applied parallel to the principal axis of the structure. Deformation demands for individual
ground motions were in some cases underestimated by as much as 500%. 2. Structural models Multiple single-storey models were examined and evaluated in this work. Two distinct types of model were developed. The first type was the TUB model, which was an asymmetric singlestorey shear type structure with four columns and off-centered mass as shown in Fig. 1(a). The columns were fixed at their base and had the potential to develop plastic hinges at both ends. A rigid diaphragm was used to model the roof of the structure. The location of the center of mass was chosen so as to induce torsion in the structure at any angle of incidence, and to avoid a situation where torsion may be absent from the analysis, i.e., which could occur if the resultant of the seismic forces passes concurrently through the center of strength, rigidity and mass. For a given mass, the column stiffness of the structures was tuned to achieve fundamental periods of 0.2, 0.3, 0.4, 0.5, 1.0 and 2.0 s. The second model (Fig. 1(b)) was the torsionally balanced system (TB), which had no torsion due to co-centric centers of strength, rigidity and mass. The fundamental period of vibration for the TB model was identical to its TUB counterpart. For the TB model, the off-centered mass defined in the TUB model was placed at the geometric center of the structure at roof level and represented a structure that had negligible torsional effects. In order to maintain the same fundamental period as that of the TUB counterpart, the columns of the TB models needed to be more flexible. Plastic hinges were located at the column ends as per the TUB system. These plastic hinges were used to model the material inelasticity of the members by means of a bilinear hysteretic model with a positive post-yield stiffness of 3%. Plastic moments were calculated using the modal analysis procedure in the IBC 2006 [14]. Equivalent lateral forces based on modal properties of the structure and the design spectrum shown in Fig. 2 were calculated and then applied to the center of mass of the structure to conduct a linear elastic static analysis. For this calculation only the first two modes were considered, for they corresponded to a cumulative mass participation of at least 90%. The design spectrum used was that of a location along coastal California with NEHRP site class D. The maximum moment observed in this linear elastic static analysis at the end of any column was the value assigned as the plastic moment to all plastic hinges in the model. Plastic moments determined in this manner pertained to a relative design intensity (Rd ) with a value equal to one. The parameter Rd is defined by the ratio of the ground motion intensity to the design lateral force (Vd ) in the structure divided by the weight of the structure (W ): Rd =
Sa (T1 ) g Vd W
.
(1)
In the analysis process, all records were scaled to the same 5%-damped pseudo-spectral acceleration at the fundamental period of the structure, Sa (T1 )/g. Thus, with Sa (T1 )/g of the
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(a) TUB model.
(b) TB model. Fig. 1. General configuration of torsionally unbalanced (TUB) and torsionally balanced (TB) models (CM, center of mass; CR, center of rigidity; CS, center of strength).
Fig. 2. Design spectrum for a coastal site in California, NEHRP site class D.
primary component always being a constant (C) value for a given model, Eq. (1) becomes: Rd = C ·
W . Vd
(2)
To produce various inelastic results from the aforementioned models, a fraction of the plastic moment value relative to the Rd of interest was used during inelastic analyses. Rd values of 1, 2, 4 and 6 were used for both the TB and TUB models for each fundamental period of interest. The authors recognize that the ductility demands obtained for short-period structures presented in the following section with Rd = 6 were very large; however, this relative design intensity value was presented to evaluate the behavior of systems that have the potential to exhibit excessive levels of inelasticity. In addition, an elastic case (i.e., with infinitely strong columns) was created for both the TB and TUB models for each fundamental period. A total of 60 models (6 periods × 5 strength values × 2 model types) were developed as part of this study. The effects of gravity load moments, P–M interaction, and P–delta effects were not taken into consideration. All models had 5% Rayleigh damping (ξ ) in the first and third modes. As an example of the model properties, the TUB model with a fundamental period of 0.5 s and Rd = 2 had a mass of 6500 kg, individual column stiffness
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Table 1 Ground motions used in this study Record ID
Event
Year
Moment magnitude
Station
Closest distance (km)
PGA (g) major component
PGA (g) minor component
IV79cal IV79chi IV79cmp IV79e01 IV79e12 IV79e13 IV79nil IV79pls IV79wsm LP89agw LP89cap LP89g03 LP89g04 LP89gmr LP89hch LP89hda LP89hvr LP89sjw LP89slc LP89svl NR94cen NR94cnp NR94far NR94fle NR94glp NR94hol NR94lh1 NR94lv2 NR94lv6 NR94nya NR94pic NR94stc NR94stn NR94ver SF71pel SH87bra SH87icc SH87pls SH87wsm
Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Imperial Valley Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Loma Prieta Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge Northridge San Fernando Superstition Hills Superstition Hills Superstition Hills Superstition Hills
1979 1979 1979 1979 1979 1979 1979 1979 1979 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1994 1971 1987 1987 1987 1987
6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.9 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.6 6.7 6.7 6.7 6.7
Calipatria Fire Station Chihuahua Compuertas El Centro Array #1 El Centro Array #12 El Centro Array #13 Niland Fire Station Plaster City Westmorland Fire Station Agnews State Hospital Capitola Gilroy Array #3 Gilroy Array #4 Gilroy Array #7 Hollister City Hall Hollister Differential Array Halls Valley Salinas-John & Work Palo Alto-SLAC Lab. Sunnyvale-Colton Ave LA-Centinela St Canoga Park-Topanga Can. LA-N Faring Rd LA-Flecther Dr. Glendale-Las Palmas LA-Hollywood Stor FF Lake Highes #1 Leona Valley #2 Leona Valley #6 La Crescenta-New York LA-Pico & Sentous Northridge-17645 Saticoy St LA-Saturn St LA-E Vernon Ave LA-Hollywood Stor Lot Brawley El Cento Imp. Co. Cent Plaster City Westmorland Fire Station
23.8 28.7 32.6 15.5 18.2 21.9 35.9 23.6 15.1 28.2 14.5 14.4 16.1 24.2 28.2 25.8 31.6 32.6 36.6 28.8 30.9 15.8 23.9 29.5 25.4 25.5 36.3 37.7 38.5 22.3 32.7 13.3 30 39.3 21.2 18.2 13.9 21 13.3
0.128 0.27 0.186 0.139 0.143 0.139 0.109 0.057 0.11 0.172 0.529 0.555 0.417 0.323 0.247 0.279 0.134 0.112 0.278 0.209 0.465 0.42 0.273 0.24 0.357 0.358 0.087 0.091 0.178 0.178 0.186 0.477 0.474 0.153 0.21 0.156 0.358 0.186 0.211
0.078 0.254 0.147 0.134 0.116 0.117 0.069 0.042 0.074 0.159 0.443 0.367 0.212 0.226 0.215 0.269 0.103 0.091 0.194 0.207 0.322 0.356 0.242 0.162 0.206 0.231 0.077 0.063 0.131 0.159 0.103 0.368 0.439 0.12 0.174 0.116 0.258 0.121 0.172
of 354 kN/m for both the X and Y -directions and plastic moment values equal to 19.8 kN m. 3. Ground motions A suite of 39 pairs of horizontal ground motion records (Table 1) was obtained from the PEER strong motion database (http://peer.berkeley.edu/smcat/). Ground motions were recorded on stiff soil sites which were at least 13 km away from the fault rupture zone but occurred within 60 km of the site [15]. The ground motions had moment magnitudes between 6.5 and 6.9 and had similar frequency content. Additional information regarding the ground motions can be found in [15]. For a given ground motion, one of the two horizontal components was classified as either being a major component or a minor component based on its PGA value. The one with the highest PGA was considered to be the major component, while the other was labeled as the minor component. However, this definition did not ensure that the
Sa (T1 )/g value of the major component was larger than that of the minor component. For instance, at 0.5 s, 7 of the 39 records had larger Sa (T1 )/g values for the minor component than for the major component. The major component was applied at various angles of incidence, α, while the minor component was applied at an angle of α + 90◦ with respect to the X -axis as shown in Fig. 1. As stated before, all major components of the 39 pairs of ground motions were scaled to the same Sa (T1 )/g value (Fig. 2) at the fundamental period of the model, while the minor component was proportionally scaled relative to its corresponding major component. In this way, the frequency content of both components, as well as the intensity of one relative to the other, were maintained. 4. Methodology In an effort to evaluate the dependence of ductility ratios, slab rotations and drift ratios on the angle of incidence, time history analyses using the 39 pairs of orthogonal components were
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conducted at increasing 5◦ increments counter clockwise relative to the X -axis of the structure (see Fig. 1). The seismic loads were applied only from 0◦ to 180◦ since the remaining orientations from 185◦ to 360◦ would produce duplicate values due to symmetry. The results at 180◦ were identical to those produced at 0◦ but were included for completeness. Thus, a total of 37 orientations were considered per ground motion pair. In this context, the angle of incidence defines the orientation of the major and minor components of ground motion relative to the X -axis. With respect to the degree of inelastic behavior, the relative design intensity values examined in this paper were Rd = 1, 2, 4 and 6. As mentioned earlier, inelastic results were obtained in the analysis by keeping the ground motion intensity constant and reducing the value of the design plastic moments. Hence, plastic moments were proportionally scaled to the value of the Rd under consideration. For instance, Rd = 2 indicates that the plastic moment value in the model need only be half of the value required for Rd = 1. Column displacement ductility ratios, slab rotations and column drift ratios were recorded for each time history analysis in each direction (i.e., the X - or Y -direction). In this context, the column displacement ductility ratio (ductility) was defined as the displacement observed at the top of the column normalized by its yield displacement. Column drift ratios (drift) were defined as the displacement observed at the top of the column normalized by the column height. With respect to ductility, the maximum ductility observed at any particular column and the average ductility of all columns were used. The average ductility was defined as the sum of the individual column ductility ratios in one direction divided by four, and was considered to be a global measure of damage. 5. Dependence of inelastic response on angle of incidence For a given building and pair of bi-directional ground motions, inelastic demands (e.g., ductility, slab rotation and drift) were sensitive to the angle of incidence of the ground motion input. This is illustrated in Fig. 3(a) where the grey lines correspond to ductility ratios in the X -direction for individual ground motion pairs and the black line represents the average of the ductility ratios in the X -direction for a given angle of incidence. In this example, column 1 of the TUB 0.5 s model with Rd = 6 was utilized. From this figure it is evident that certain individual responses appear to be more sensitive to angle than others. In an effort to better understand why a pair of horizontal ground motion components may produce significant variations for a given EDP with respect to angle while another may not, the variation of ductility demands with respect to the ratio of Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) of a ground motion pair was investigated. For a given model and direction (i.e., the X - or Y -direction), variation was defined as the ratio of the maximum value of ductility observed for a given pair of records at any angle divided by the minimum value for the same pair of records. The dependence of the ductility variation with the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) at various periods was investigated. The ratios were calculated
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Fig. 3. (a) Dependence of ductility demands in the X -direction on angle of incidence and (b) dependence of the variation in column ductility demands in the X -direction on the ratio of spectral accelerations of major to minor ground motion component at the fundamental period.
at the fundamental period of vibration of the structures and at periods up to twice the fundamental period. These results failed to illustrate a significant correlation of the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) with ductility variation, as illustrated in the representative case shown in Fig. 3(b) for column 1 of the TUB structure with a fundamental period of 0.5 s. This indicates that the relationship between ductility demands and angle of incidence was not strongly dependent on the ground motion spectral ratios. For example, in Fig. 3(b), the correlation coefficient between ductility variation and the ratio Sa (T1 )/g (major component)/Sa (T1 )/g (minor component) was equal to 0.33. The linear model fitted to these data is also shown in Fig. 3(b). The observed differences in the magnitude of ductility demands as a function of angle of incidence of ground motion input are attributed to differences in the frequency-content characteristics of the ground motion records. Because of the need to generalize behavioral trends and quantify seismic demands for inelastic buildings exposed to bidirectional ground motions at different angles, average values are utilized in the following subsections to evaluate ductility, drift and slab rotation demands, in addition to upper bounds for maximum values corresponding to individual ground motion pairs. It will be shown that critical angle not only depends on the EDP of interest, but also on the fundamental period and the level of inelastic behavior of the structure.
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Fig. 5. TUB model with a fundamental period of 1.0 s. (a) Angle of incidence at which ductility ratios in the X -direction are maximum. Fig. 4. Influence of angle of incidence on (a) average and (b) maximum ductility demands in the X -direction, for a TUB model with a fundamental period of 0.5 s.
5.1. Evaluation of ductility and drift demands For a given level of inelastic behavior, the magnitude of maximum and average ductility values was a function of the angle of incidence. Column ductility ratios did not exhibit a maximum at 0◦ or at 90◦ for the majority of Rd values studied. These two values, i.e., 0◦ and 90◦ , were the angles of incidence traditionally used to evaluate seismic demand parameters for TB and TUB systems [1,4,16,17]. These trends can be seen in Fig. 4 which displays, for a given Rd , the mean of the average ductility of all columns (Fig. 4(a)) and the upper bound of maximum ductility values of all columns in the X -direction, µx , (Fig. 4(b)) for the 0.5 s TUB model as a function of angle of incidence. An alternative way of illustrating this phenomenon is by examining histogram graphs of maximum ductility demands in both the X - and Y -directions, which are shown in Fig. 5 for a TUB model with a period of 1.0 s. As illustrated in Fig. 5, maxima can occur at any angle, a trend observed for all models. As expected, as Rd increased so did the average ductility. Similar observations were made for the column ductility in the Y -direction (µ y ); however, these latter results differed from those in the X -direction in that the magnitudes of the ductility ratios were slightly smaller and peaked at different angles for a given Rd . The behavioral pattern illustrated in Fig. 4 for the
X -direction, and that described for the Y -direction, was typical of what was found for all TUB models at all fundamental periods evaluated. In addition, the observations made in the previous paragraph for the TUB model also applied to the TB model for 0.5 s as can be seen in Fig. 6. A comparison of Figs. 4 and 6 demonstrates that given the fundamental period of vibration, TB models exhibit higher ductility demands than TUB models. This result is consistent with observations made by Humar and Kumar [16], who utilized TUB models in which the center of rigidity rather than mass was offset in order to induce torsion. The patterns presented in Fig. 6 are typical of what was found for all TB models at all fundamental periods. Regardless of whether or not torsional effects were significant, both maximum and average ductility demands were strongly dependent on the angle of incidence for a given Rd . Therefore, damage assessment based on either maximum or average storey ductility ratios will be dependent on the incident angle of the ground motion input. This observation is consistent with those presented in subsequent sections with regards to drift and slab rotation demands. Graphs of the mean of the average drift and upper bound of maximum drift produced very similar trends to those of ductility. Drift demands did not exhibit a maximum at 0◦ or at 90◦ for the majority of Rd values studied. In addition, peak inelastic drift normalized by peak elastic drift values for two separate lines of columns indicated that, on average, inelastic displacements relative to elastic ones did not always peak at
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Fig. 6. Influence of angle of incidence on (a) average and (b) maximum ductility demands in the X -direction, for a TB model with a fundamental period of 0.5 s.
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Fig. 7. Dependence of the ratio of inelastic to elastic drifts on angle of incidence: (a) column line 1–2, X -direction; (b) column line 3–4, X -direction.
angles 0◦ and 90◦ as shown in Fig. 7 for the X -direction, a trend which was also observed in the Y -direction. 5.2. Evaluation of slab rotation demands While the average slab rotations appeared to be weakly dependent on angle of incidence, the amount of torsion-induced damage and torsion itself may be reduced by allowing the building to experience higher levels of inelastic behavior (see Figs. 8 and 9). This observation presupposes that slab rotations are good indicators of torsion. In particular, Fig. 8 shows the average slab rotations as a function of angle of incidence and degree of inelasticity for the TUB model with fundamental period of 0.3 s, while Fig. 9 depicts the ratio of inelastic to elastic slab rotations for angles 0◦ and 90◦ for all TUB models. Other studies such as those in [17] found that elastic slab rotations were greater than inelastic slab rotations; however, this observation was made for only two fundamental periods (0.4 s and 0.8 s), different models and ground motions. Studies by De-La-Colina [18] also found slab rotations to decrease with increasing strength-reduction factor (R). Since the model characteristics in both [17,18] (such as mass and stiffness) are different from those in this work, it can be concluded that this phenomenon is controlled by the fundamental period and the level of inelastic behavior of the structural system.
Fig. 8. Dependence of slab rotation demands on angle of incidence, for a TUB model with a fundamental period of 0.3 s.
6. Implications for performance-based evaluation and design The results presented in previous sections demonstrated that seismic demands on inelastic buildings exposed to bidirectional ground motions can be underestimated if the pair of ground motion records is only applied at the principal orientations of the building. The implication is that inaccurate estimates of damage assessment, and hence direct dollar losses,
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Fig. 9. Dependence of the ratio of inelastic to elastic slab rotations on fundamental period, for angles of incidence equal to 0◦ and 90◦ .
Fig. 11. Ratio of maximum ductility for any angle of incidence to the maximum ductility with an angle of incidence of (a) 0◦ (X -direction) and (b) 90◦ (Y direction). Fig. 10. Empirical fragility curves for ductility demands in the X -direction as a function of angle of incidence for a TUB model with a fundamental period of 2 s.
could be obtained from performance evaluation conducted with bi-directional ground motions with the major and minor components applied at 0◦ and 90◦ , respectively. This is illustrated with the empirical ductility demand fragility curves shown in Fig. 10. It can be observed that certain angles were more likely to have greater ductility demands than others. Here, the probability that a given µx0 = 2 will be exceeded is plotted versus Sa (T1 )/g for a TUB model with a normalized base shear strength of the structure, γ , equal to 0.25. Fig. 10 indicates that for most Sa (T1 )/g values, results at different angles such as α = 45◦ and 90◦ for the 2.0 s TUB model had a greater likelihood of exceeding a ductility value of 2.0 as compared to the results at 0◦ . The exception is the range corresponding to Sa (T1 )/g > 0.75, in which differences between fragilities for 90◦ and 0◦ exhibit opposite trends. These observations imply that damage estimates based on fragilities for ground motions applied at principal orientations can be grossly inadequate. With maxima not always occurring when the major component of ground motion is applied at α = 0◦ or at 90◦ , it is important to develop simplified design procedures to quantify seismic demands as a function of the angle of incidence. The results from this study did not show consistent, stable trends that will allow the robust quantification of such
demands. However, on average, the ratio of the maximum ductility demand at any angle of incidence to the ductility demand at 0◦ (for ductility in the X -direction) and at 90◦ (for ductility in the Y -direction) varied between 1.1 and 1.6. This variation exhibited trends that indicate a significant dependence with respect to period and a weak dependence with respect to Rd (as shown in Fig. 11). Fig. 11(a) illustrated this behavior for maxima, where the mean of the ratio of the maximum value µx of all columns for any angle of incidence divided by the maximum µx any column experiences with an angle of incidence of 0◦ has been plotted. Fig. 11(b) is similar, with the exception that the maximum value of µY is examined and is normalized by the maximum µY experienced by any column when the angle of incidence is 90◦ . Due to the symmetry of the model, the results for the TB model were identical in both figures. The mean of the ratio of the peak average ductility in the X -direction at any angle of incidence to the average ductility demand at 0◦ produced very similar results. Although these graphs were for ductility, they can be interchanged for drift values because the properties of each column in any given model (i.e., stiffness and yield rotation) were the same. These results reinforce the notion that design and performance assessment procedures based on ground motions applied at the principal orientations of the building will tend to underestimate peak inelastic seismic demands along the principal orientations of the building, especially at longer periods (greater than 0.5 s).
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7. Conclusions Regardless of whether or not a building has a torsional irregularity, the maximum values for the EDPs examined in this study have the potential to occur when the pair of horizontal ground motion records is applied at any angle other than the traditional α = 0◦ , 90◦ , i.e., angles that coincide with the principal directions of the building under consideration. Results from this study demonstrate that, in general, peak inelastic deformation demands are underestimated when the horizontal components of ground motion are applied along the principal orientations of an inelastic structure. In addition, the information presented on slab rotations shows that the overall torsion in a building can be reduced by increasing the degree of inelasticity for most of the models studied. Two model types were examined; the torsionally balanced model which had co-centric mass, strength and stiffness and the torsionally unbalanced model which had co-centric strength and rigidity but offset mass. The magnitudes of the EDPs examined for both model types were quite different; nevertheless, general behavioral trends with respect to angle of incidence were similar. The observations and findings presented in this work are applicable to buildings and ground motions with characteristics consistent with those utilized in this study. The critical angle for a given EDP varies with fundamental period, model type and the level of inelastic behavior, and it is difficult to determine a priori like that of an elastic structure. This implies that performance evaluation and damage assessment based on inelastic deformation demands will be dependent on the angle of incidence of the ground motion input. Thus, performance assessment and design verification of structures designed to undergo inelastic deformations should ideally be conducted with bi-directional ground motions applied at various angles with respect to the principal orientations of the building. However, for the majority of structures, such an approach is impractical and methods to determine the required number of pairs of ground motion records and angles of incidence to estimate peak inelastic deformation demands with a given precision and level of confidence are needed. Results from this study demonstrate that for a given period, the results based solely on angles of incidence of 0◦ and 90◦ diverge by a consistent amount from the peak inelastic deformations observed at other angles regardless of the level of inelastic behavior. For instance, for a given structure and ground motions, the ratio of the peak of the maximum inelastic deformation obtained for a given angle of incidence to the maximum inelastic deformation with an angle of incidence of 0◦ (for the X -direction) and of 90◦ (for the Y direction) tends to increase with fundamental period and varies, on average, between 1.1 and 1.6 for both torsionally balanced and torsionally unbalanced models. This type of information
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is currently being used by the authors in a parallel study to develop simplified methodologies to efficiently estimate the magnitude of peak inelastic deformation demands for systems with different number of storeys, offset strength and rigidity, as well as mass in any combination thereof, which are exposed to ground motions with different frequency content. References [1] Tso WK, Smith RSH. Re-evaluation of seismic tensional provisions. Earthq Eng Struct Dyn 1999;28:899–917. [2] Aziminejad A, Moghadam AS. Behavior of asymmetric structures in near field ground motions. In: Conference on earthquake engineering. 2006. Paper 399. [3] Ghersi A, Rossi PP. Influence of bi-directional ground motions on the inelastic response of one-storey in-plan irregular systems. Eng Struct 2001;23:579–91. [4] Goel RK. Seismic response of asymmetric systems: Energy approach. J Struct Eng 1997;123(11):1444–53. [5] Wilson EL, Suharwardy I. A clarification of the orthogonal effects in a three-dimensional seismic analysis. Earthq Spectra 1995;11(4):659–66. [6] Fernandez-Davila I, Comeinetti S, Cruz EF. Considering the bidirectional effects and the seismic angle variations in building design. In: Proceedings of the twelfth world conference on earthquake engineering. 2000. [7] Khoshnoudian F, Poursha M. Responses of three dimensional buildings under bi-directional and unidirectional seismic excitations. In: Proceedings of the thirteenth world conference on earthquake engineering. 2004. [8] Lopez OA, Chopra AK, Hernandez JJ. Critical response of structures to multicomponent earthquake excitation. Earthq Eng Struct Dyn 2000;29: 1759–78. [9] Lopez OA, Torres R. The critical angle of seismic incidence and structural response. Earthq Eng Struct Dyn 1997;26:881–94. [10] Athanatopoulou AM. Critical orientation of three seismic components. Eng Struct 2005;27:301–12. [11] MacRae GA, Mattheis J. Three-dimensional steel building response to near-fault motions. ASCE J Struct Eng 2000;126(1):117–26. [12] Tezcan SS, Alhan C. Parametric analysis of irregular structures under seismic loading according to the new Turkish Earthquake Code. Eng Struct 2001;23:600–9. [13] Franklin CY, Volker JA. Effect of various 3-D seismic input directions on inelastic building systems based on INRESB-3D-82 Computer Program. In: Proceedings of the seventh European conference on earthquake engineering. 1982. [14] International code council. International building code (IBC-06). Whittier (CA); 2006. p. 663. [15] Medina RA, Krawinkler H. Seismic demands for non-deteriorating frame structures and their dependence on ground motions. PEER Report 2003/15. Berkeley (CA): Pacific Earthquake Engineering Research Center; 2004. [16] Humar JL, Kumar P. Effect of orthogonal inplane structural elements on inelastic torsional response. Earthq Eng Struct Dyn 1999;28:1071–97. [17] Perus I, Fajfar P. On the inelastic torsional response of single-storey structures under bi-axial excitation. Earthq Eng Struct Dyn 2005;34: 931–41. [18] De-La-Colina J. Effects of torsion factors on simple non-linear systems using fully bidirectional analyses. Earthq Eng Struct Dyn 1999;28: 691–706.