UNIVERSITY OF SOUTHERN CALIFORNIA DEPARTMENT OF CIVIL ENGINEERING ANALYSIS OF DRIFTS IN A SEVEN-STORY REINFORCED CONCRETE STRUCTURE

Transcription

1 UNIVERSITY OF SOUTHERN CALIFORNIA DEPARTMENT OF CIVIL ENGINEERING ANALYSIS OF DRIFTS IN A SEVEN-STORY REINFORCED CONCRETE STRUCTURE by M.D. Trifunac and S.S. Ivanović Report No. CE 03-0 August, 2003

2 TABLE OF CONTENTS ABSTRACT INTRODUCTION 2 DESCRIPTION OF THE BUILDING 5 EARTHQUAKE DAMAGE 7 EARTHQUAKE RECORDINGS 9 RELATIVE DISPLACEMENTS, DRIFT ANGLES AND TOTAL RESPONSE 2 Two-Dimensional NS Response 5 Averaged (One-Dimensional) NS Response 37 EW Response 37 Torsional Response 58 RELATED STUDIES OF VN7SH BUILDING 66 Isalm (996) 66 Li and Jirsa (998) 66 Browning et al. (2000) 70 De la Llera et al. (2002) 7 DISCUSSION AND CONCLUSIONS 73 Significance for Verification of Soil-structure Models 77 Significance for Structural Health Monitoring 78 Design Considerations 79 ACKNOWLEDGEMENTS 8 REFERENCES 82 i

3 ABSTRACT The anatomy of interstory drifts in a seven-story, reinforced concrete structure is investigated, based on multiple earthquake recordings. It is shown that for accurate estimation of relative interstory drifts the contribution to drifts caused by rocking during soil-structure interaction must be subtracted from the recorded data. Unless this rocking contribution is eliminated prior to fitting the response of structural models to the recorded data, erroneous (non-conservative) inferences about overall structural properties, and particularly about the structure s actual capacity to withstand strong-motion demands, will be inevitable.

4 INTRODUCTION Earthquake-resistant design must be based on realistic models of the structurefoundation-soil systems and must include all system degrees of freedom and all aspects of nonlinear response. Because such analyses require solution of a complicated system of equations and boundary conditions, it is common to make simplifications. Therefore, it becomes important to evaluate the accuracy of such approximations and to define the range of the model properties and parameters for which the approximations are realistic. This evaluation is best accomplished by careful experimental verification using full-scale tests of actual structures (Trifunac and Todorovska, 200a; Trifunac et al., 200a,b). In this report, an instrumented seven-story hotel building (Blume and Assoc., 973) in Van Nuys, California, is studied (Ivanović et al., 2000a,b; 200; Todorovska et al., 200a,b; Trifunac et al., 2002). Records of twelve earthquakes are available for the analysis, (Trifunac et al., 999a), but in this work we examine the data for only the four largest events: 987 Whittier Narrows (M L = 5.9, = 4 km), 992 Landers (M L = 7.5, = 86 km), 992 Big Bear (M L = 6.5, = 49 km), and 994 Northridge (M L = 6.4, = 4 km) (Fig. ). The 994 Northridge earthquake caused severe damage, and the building was declared unsafe. The damage was most severe at the fifth floor, where many columns failed in shear just below the spandrel beam (Trifunac et al., 999a). The specific aspects of the response, which caused this type of failure have been studied (De la Llera et al., 200; Islam, 996; Li and Jirsa, 998; Browning et al., 2000), but have not been deciphered uniquely thus far. One plausible group of causes can be sought in the large relative deformations of the foundation system (pile caps connected by grade beams) (Trifunac, 997; Trifunac et al., 999b), or in excessive torsional excitation (Trifunac et al., 999b), but the limited number of accelerographs in this building that recorded the Northridge event is not sufficient to verify these hypotheses. 2

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6 Dynamic analyses of structural response must consider the consequences of the flexibility of the soil that supports the structure. The overall soil-structure system period, T %, can be significantly longer than the fundamental period of the same structure, T, fixed at its base. Denoting by T r the period associated with the rocking of a rigid structure on a flexible base and by T h the period of horizontal vibrations of a rigid structure on flexible soil, it can be shown that T% = ( T + T + T ). For large amplitudes of excitation, non /2 r h linear response can lengthen some or all of the above periods. The lengthening of T %, caused by earthquake excitation, has been documented by analysis of data recorded in buildings (Udwadia and Trifunac, 974; Trifunac et al., 200a,b). However, because it has been common in engineering design analyses to neglect the effects of soil-structure interaction (this is equivalent to setting T r = T h = 0), it has been common also to assume that the observed changes in T % result from changes in T alone (e.g., Islam, 996; Li and Jirsa, 998; De la Llera et al., 200). Because most changes in T % actively result from changes in T r and T h, such analyses overestimate the non linear response in the structure and erroneously interpret the observed ductility and drifts and the energy absorption capacity of the structural system. The purpose of this work is first to describe and interpret relative and total drifts, inferred from recorded strong-motion in the building, and then to compare those inferences with field observations and with several published analyses of the response. As stated above, thus far most investigators have chosen to ignore soil-structure interaction in their interpretations of the recorded strong-motion response (De la Llera et al., 200; Islam 996; Li and Jirsa, 998; Browning et al., 2003; Shakal et al., 996) in the Van Nuys seven-story hotel (VN7SH), and therefore their estimates of drift include the contribution from rocking associated with soil-structure interaction. DESCRIPTION OF THE BUILDING The building analyzed in this work is a seven-story, reinforced concrete structure in the city of Van Nuys (Fig. ). It will be referred to as VN7SH for short. It was designed in 4

7 (a) D C N 62' - 8'' B A 8 8' - 9'' = 50' - 0'' (b) D C 62' - 8'' B A N 8 8' - 9'' = 50' - 0'' Fig. 2. (a) Typical floor plan, (b) foundation plan. 5

8 D C B A roof 3' - 6'' 5 x 8' - 8.5'' 8' - 8'' 7th 6th 5th 4th 3rd 2nd st N 20' -0'' 20' -0'' 20' -0'' 62' - 8'' 8'' slab 8.5'' slab 8.5'' slab 8.5'' slab 65' - 8.5'' 8.5'' slab 8.5'' slab 0'' slab 4'' slab (c) (d) Depth in feet % % % % - 98 % * 8* 8* 0* SM ML SM ML SM ML SM Elev ' Silty fine sand - brown; some gravel; dry Fine sandy silt - light brown; slightly damp Becoming more damp; includes small pores Silty fine sand - buff; damp Fine sandy silt - buff, some gravel and clay Silty very fine sand - light buff Very fine sandy silt - brown; slightly moist Silty fine sand - buff; damp No water, no caving Boring completed 7/9/65 LEGEND : A Field moisture expressed as a percentage of the dry weight soil. B Dry density expressed in pounds per cubic foot. C Blows per foot of penetration using a 2000-pound hammer dropping 2". * a 200 pound hammer was used. Depth at which undisturbed sample was extracted. Fig. 2. (c) Typical elevation, and (d) boring log. 6

9 965 and constructed in 966 (Blume et al., 973). Its plan dimensions are 62 by 50 feet (Fig. 2a), and the typical framing consists of columns spaced on 20-foot centers in the transverse direction and 9-foot centers in the longitudinal direction. Spandrel beams surround the perimeter of the structure. Lateral forces in each direction are resisted by the interior column slab frames and the exterior column spandrel beam frames. The added stiffness associated with the spandrel beams creates exterior frames that are roughly twice as stiff as interior frames. The structure is essentially symmetric. Except for two small areas at the ground floor level covered by one-story canopies, the plan configurations of the floors, including the roof, are the same. The floor system is a reinforced concrete flat slab 0 inches thick at the second floor, 8.5 inches thick at the third to seventh floors, and 8 inches thick at the roof. The north side of the building, along column line D (Fig. 2a), has four bays of brick masonry walls located between the ground and the second floor at the east end of the structure. The site lies on recent alluvium. A typical boring log (Fig. 2d) shows the underlying soil to be primarily fine sandy silts and silty fine sands. The average shear-wave velocity in the top 30 m is 300 m/s. The foundation system (Fig. 2b) consists of 38- inch deep pile caps supported by groups of two to four poured-in-place 24-inchdiameter reinforced concrete friction piles. These are centered under the main building columns. All pile caps are connected by a grid of the beams, and each pile is roughly 40 feet long with a design capacity of over 00 kips vertical load and up to 20 kips lateral load. The structure is constructed of regular-weight reinforced concrete (Blume et al., 973). EARTHQUAKE DAMAGE The Feb. 9, 97, San Fernando earthquake caused minor structural damage. Epoxy was used to repair the spalled concrete of the second floor beam column joints on the north side and east end of the building. The nonstructural damage, however, was extensive, and about 80 percent of all repair cost was used to fix the drywall partitions, bathroom tiles, and plumbing fixtures. 7

10 (a) roof FRAME D (North view) th 6th 4 " x " shear cracks <0.5 cm 5th " x " shear cracks <0.5 cm cracks through the beam cm 4th 3rd 2nd " x " shear cracks <0.5 cm 6 8 diagonal cracks along the column, 0.5 cm " x " shear cracks <0.5 cm diagonal cracks in beams <0.5 cm cracks along the column, 0.5 cm cracks through the beam cm 'short column' cracks, cm 5 7 st 3 cracks between bricks cracks between bricks (b) roof FRAME A (South view) th 6th "x" - shear cracks ~ 5.0 cm, bending of long. reinforcement " x " shear craks ~5.0 cm "x" - shear cracks ~ 5.0 cm, bending of long. and trans. reinforcement 4 5th 4th 5 "x" - shear cracks > 5.0 cm, bending of long. reinforcement "x" - shear cracks ~ 5.0 cm, bending of long. reinforcement cracks in the beam <.0 cm 6 3rd 2nd 7 cracks in the beam <.0 cm 8 st complete first floor on the south side was covered - there may have been some damage there 3 Fig. 3. Schematic representation of damage following the 994 Northridge earthquake: (a) Nort view, and (b) south view. 8

11 The building was damaged for the second time by the January 7, 994 Northridge earthquake. The structural damage was extensive in the exterior north (D) and south (A) frames, which were designed to take most of the lateral load in the longitudinal direction. Severe shear cracks occurred at the middle columns of frame A near the contact with the spandrel beam of the fifth floor (Fig. 3; Trifunac et al., 999a; Trifunac and Hao, 200). Those cracks significantly decreased the axial, moment, and shear capacity of the columns. The shear cracks that appeared in the north (D) frame on the third and fourth floors and the damage to columns D2, D3, and D4 on the first floor caused minor to moderate changes in the capacity of these structural elements. No major damage to the interior longitudinal (B and C) frames was noticed. There was no visible damage in the slabs around foundations. The nonstructural damage was significant. Almost every guest room suffered considerable damage, and severe cracks were noticed in the masonry brick walls and in the exterior cement plaster. EARTHQUAKE RECORDINGS The first digitized strong-motion accelerograms in the building are of the Feb. 9, 97 San Fernando earthquake (Fig. ; Trifunac et al., 973). The sensors (self- contained, triaxial AR-240 accelerographs) were at ground level, on the fourth floor, and at the roof. In the mid-970s a Central Recording System (CR-) was installed, with sensor locations as shown in Fig. 4. It recorded the 987 Whittier Narrows, the 992 Landers, and the 992 Big Bear earthquakes, which occurred at epicentral distances of 4, 86, and 49 km, respectively, and caused strong-motion arrivals from E 27 5, East, and E.5 S. During the 994 Northridge earthquake, the first motions started to arrive from the West, with the last arrivals coming from N 42 W about 7 to 0 seconds later. Table summarizes selected parameters of all the earthquakes for which digitized and processed data are available for this building (Trifuanc et al., 999a). The San Fernando accelerograms (Trifunac et al., 973) will not be considered here because the number and location of the recorders were not suitable for the analysis 9

12 3.24 m m 7.95 m Fig. 4. Location and orientation of 3 transducers of the CR- analog recording system that was in operation from the mid 970s onward. Transducers 4, 5, and 6 belong to the triaxial, self-contained, SMA- accelerograph. The CR- and SMA- recorders are interconnected so as to have a common trigger time. 0

13 Table. Changes of apparent frequency of response during strong shaking (before, after, minimum, f max largest difference, v-absolute difference between peak velocity recorded on roof and at ground level, and v max the ground velocity). Apparent NS rocking frequency (Hz) NS (cm/s) Apparent EW rocking frequency (Hz) Earthquake Date M km Before After Min f max v v max Before After Min f max v v max. S. Fernando 9 Feb * 0.95 (0.7) WhittierNarrows Oct W.N. aftershock 4 Oct (Oct. 4, 87) 4. Pasadena 3 Dec Malibu 9 Jan Montebello 2 June Sierra Madre 28 June Landers 28 June Big Bear 28 June Northridge 7 Jan * N. Aft 20 Mar (March 20, 94) 2. N. Aft (Dec. 6, 94) 6 Dec *Horizontal projection of the closet distance to fault surface.. From Trifunac et al. (200a). EW (cm/s)

14 of drifts. The accelerograms of the Whittier Narrows, Landers, Big Bear, and Northridge earthquakes (Shakal et al., 994) were processed by the California Division of Mines and Geology (CDMG). Records of other earthquakes listed in Table were processed by LeAuto software at USC (Lee and Trifunac, 990) from xerox copies of original recordings supplied by CDMG. RELATIVE DISPLACEMENTS, DRIFT ANGLES AND TOTAL RESPONSE The relative flexibility of the structure-foundation-soil system can have profound influence upon the three-dimensional deformation of structures excited by earthquakes. In Fig. 5a and b, we illustrate this by showing deformation during forced vibrations of a nine-story (plus roof) reinforced concrete library. Lateral stiffness for NS response (Fig. 5a) is provided by two symmetric shear walls at the east and west ends of the building. These shear walls also provide sufficient vertical stiffness for the foundation slab, so that it can be modeled by a rigid plate (Foutch et al., 975; Luco et al., 986; Moslem and Trifunac, 986). In contrast, Fig. 5b shows the EW response of the same building, which is resisted only by the elevator core in the center third of the EW cross section. It is evident that the foundation slab is deforming out of its plane and that this results in large bending and shear deformations along the entire height of the structure. For modeling the soil-structure interaction effects in this case, the rigid foundation approximation is not only too rough, but is also difficult to implement (Luco et al., 986). For both NS and EW responses in this example, it should be clear that the total response, and therefore the overall as well as the individual story drifts, contain significant components of rigid body rotation (rocking), so that the recorded motions in this structure during earthquake excitation cannot be used directly to compute relative inter-story drifts. It is clear that meaningful measurements of actual drifts may be feasible only after the rigid body translations and rotations associated with soilstructure interaction have been identified and eliminated. In general, this is not possible without recording rotational components of strong-motion (Trifunac and Todorovska, 200b), and therefore indiscriminate presentations of building drifts and their comparison with drift limits in the building codes can only lead to misleading 2

15 (a) (b) N E Fig.5. Deformation of a nine-story, reinforced concrete building, excited at the roof by a shaker, with two counter-rotating masses (a) along the west shear wall and (b) along the section through the elevator core. 3

16 Z U TN U T j HΦ 0 U N NORTH U T0 =U g+ U 0 9 2,3 R H-h j h j Φ 0 U j 5 4 5, ,8 2 h j 6 5,3 Φ 0 U g U o.. U g Fig. 5c NS deformation of VN7SH due to relative deformation (U j), horizontal deforamtion of soil (U o ), and rocking (Φ o ) of foundation soil. 4

17 interpretations (Shakal et al., 996). In the following it will be assumed that the motion of the foundation can be approximated by translation and rotation of a rigid plate bonded to the surface of the soil. Figure 5c, then, shows the approximate breakdown of the contributions to the recorded motions in the VN7SH seven-story (with roof) structure on flexible soil. The total horizontal motion at j-th floor, U Tj, results from:. U g, horizontal displacement of the ground caused by passage of earthquake waves 2. U 0, local horizontal deformation of soil due to soil structure interaction 3. h j Φ 0, motion caused by rocking of the foundation by angle Φ 0 4. U j, deformation of j-th floor relative to a moving coordinate system attached to the foundation. With horizontal transducers on st, 2 nd, 3 rd, 6 th, and 8 th (roof) floors, and with only one vertical transducer (near the southeastern corner of the building, Fig. 4), we cannot separate contributions to the total recorded motion from rocking Φ 0 and from the relative displacement response U j. At least two vertical transducers near the north and south and east and west ends of the first floor would have provided this information for NS and EW responses, respectively (Moslem and Trifunac, 986). For analysis of the dynamic response, it is necessary to separate these contributions. In absence of such such records, we can only attempt to do this approximately, as described below. Two-Dimensional NS Response Transverse (NS) vibrations of the VN7SH building can be described by a twodimensional model, with torsional modes contributing significantly to the overall response (Ivanović et al., 2000a,b; Todorovska et al., 200a,b; Trifunac et al., 999b, 2003). In the following, we describe the nature of this response, with emphasis on drift angles. In the next section we average out the torsional contribution to this response and describe average NS response together with EW response, again with emphasis on the drift angles. 5

18 First, we assume that the relative deformations of the structure U j can be described by the mode shapes of the fixed-base building. For NS motions, the lowest (first-mode) natural frequency is close to.4 Hz. During earthquake response, depending upon the amplitudes of response, the transfer function of NS motion is dominated by a peak in the range from 0.5 Hz to 0.9 Hz, which we interpret as resulting mainly from the contribution of rocking motion, Φ 0 (Trifunac et al., 999b; 200a,b). To approximately separate this rocking from relative displacements, we low-pass filter the data to obtain predominantly the contributions from Φ 0 and high-pass filter the data to obtain mainly the contributions from relative deformations U j. For NS response, we choose the transition zone (ramp) for those filters to be 0.8 to.0 Hz. For an elastic building with a rigid foundation on elastic soil, this approximation may be a reasonable one. However, with wave passage effects (Todorovska et al. 200a,b) and a flexible foundation (Trifuanc et al., 999b) in the presence of exponentially decaying motions from the foundation level, up into the building, the above approach represents only a rough approximation (Todorovska et al., 988; Todorovska and Trifunac 989; 990a,b). Figures 6 through 9 present drifts (tangents of drift angles), computed from pairs of recording stations and for low-pass-filtered displacement (below an 0.8 to Hz ramp). These results include mainly the effects of rocking and torsional deformation at long periods (>.25 s). Lateral deflection limits for tall buildings (Branson, 977), limit the story drift to 0.5 in. (4 mm) or D max = h, where h is the story height. In Los Angeles, for buildings less than 65 feet in height, calculated drift may not exceed 0.04/ R w (= for R w = 2) nor times the story height (R w = 2 is for moment-resistant frame buildings; Int. Conf. Building Officials, 994). It can be seen from Figs. 6 through 9 that the low-pass-filtered "drift" occasionally exceeds in VN7SH 6

19 Drift Angles from N-S Displacements Low-pass-filtered data from to Hz Whittier Narrows Earthquake (987) D 4, D 5,2 T 5,6 4 D 6, D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 6. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for filtered ( to Hz) displacements. 7

20 Drift Angles From N-S Displacements Low-pass-filtered data from to Hz 3 Landers Earthquake (992) D 4, D 5,2 T 5,6 4 D 6, D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 7. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for filtered ( to Hz) displacements. 8

21 Drift Angles From N-S Displacements Low-pass-filtered data from to Hz 3 Big Bear Earthquake (992) D 4, D 5,2 T 5,6 4 D 6, D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 8. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for filtered ( to Hz) displacements. 9

22 Drift Angles From N-S Displacements Low-pass-filtered data from to Hz Northridge Earthquake (994) D 4,3 Drift Angle -0.0 D ,2 T 5,6 4 D 6, D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Fig. 9. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for filtered ( to Hz) displacements. Roof 6 th 3 rd 2 nd st 20

23 during the Whittier-Narrows earthquake, everywhere during the Landers earthquake, and significantly during the Northridge earthquake. We have placed quotation marks on "drift" because we believe that most of it comes from HΦ 0 (see Fig. 5c) and not from relative deformation of the building. In Figures 0 through 3, we show high-pass-filtered (beyond 0.8 to.0 Hz) drift angles and rotations θ Z, approximated by average torsional angles, T i,j, computed from the difference between the motions at two ends of the building and divided by their separation distance. For frequencies higher than.0 Hz (the first NS translational mode is near.4 Hz) these results now contain deformations associated mainly with U j (see Fig.5c) and are more representative of "true" drifts. Except for the Northridge earthquake, it can be seen that all drifts are smaller than During Northridge earthquake the drifts between 3rd floor and the roof reached and exceeded the limit of In this general area, and particularly at 5th floor, the building suffered serious damage (Fig. 3 and Trifunac et al., 999a). Analysis of Figures 6 through 9 suggests that the NS vibrations of the thevn7sh building can be viewed as predominantly rocking of a rigid body on an inflexible foundation. To illustrate this quantitatively, we calculated the displacements for channel 2 (NS motion at the western end of VN7SH on the roof; see Fig. 4) in terms of ground translation ( u( t) + u3( t))/2, overall rocking u2() t + u3() t u() t + u3() t θ y () t =, and relative torsion 2 2 H θ z ( t) = ( u ( t) u ( t)) / L ( u ( t)) u ( t)) / L. H is height of the building, and L is distance between instruments 2 and 3 and between 3 and. The results are shown in Figures 4 through 7. In Figures 8 through 2 we repeat this representation for displacement at the site of recording channel 7 (see Fig. 4b). It can be seen that θ y (t) is the main contributor to the observed displacements. 2

24 Drift Angles From N-S Displacements High-pass-filtered data from to Hz Whittier Narrows Earthquake (987) 3 D 4, D 5,2 T 5,6 4 D 6,4 D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 0. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for high-pass-filtered displacements beyond Hz. 22

25 Drift Angles From N-S Displacements High-pass-filtered data from to Hz 3 Landers Earthquake (992) D 4, D 5,2 T 5,6 4 D 6,4 D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig.. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for high-pass-filtered displacements beyond Hz. 23

26 Drift Angle From N-S Displacements High-pass-filtered data from to Hz 3 Big Bear Earthquake (992) D 4, D 5,2 T 5,6 4 D 6,4 D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 2. Inter-story "drift" angles (left and right) and floor rotations θ z (center) for high-pass-filtered displacements beyond Hz. 24

27 Drift Angles From N-S Displacements High-pass-filterred data from to Hz Northridge Earthquake (994) 3 D 4,3-0.0 D 5,2 T ,6 4 D 6,4 D 7,5 T 7,8 D 5 6 8,6 7 8 D,7 West East T,3 D 3, Time (s) Time (s) Time (s) Roof 6 th 3 rd 2 nd st Drift Angle Fig. 3. Inter-story "drift" angles (left and right) and flor rotations θ z (center) for high-pass-filtered displacements beyond Hz. 25

28 2 0 - z 3 θ z 3 x 2 y N θ y Whittier Narrows Earthquake, Roof (Ch. 2) ground + rocking+ torsion ground + rocking ground Displacement (cm) Time (s) Fig. 4. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 2. 26

29 0 z θ z y θ y x N ground Landers Earthquake, Roof (Ch. 2) ground + rocking+ torsion ground + rocking Displacement (cm) Time (s) Fig. 5. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 2. 27

30 Big Bear Earthquake, Roof (Ch. 2) 2 ground + rocking+ torsion ground + rocking 0 - z ground -2 θ z Displacement (cm) y x N θ y Time (s) Fig. 6. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 2. 28

31 z θ z 2 3 y x 3 N θ y Northridge Earthquake, Roof (Ch. 2) ground + rocking+ torsion ground + rocking ground Displacement (cm) Time (s) Fig. 7. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 2. 29

32 2 0 - z 8 3 θ z x 7 y N θ y Whittier Narrows Earthquake, 2nd Story (Ch. 7) ground + rocking+ torsion ground + rocking Displacement (cm) -2 ground Time (s) Fig. 8. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 7. 30

33 z 8 3 θ z x 7 y N θ y Landers Earthquake, 2nd story (Ch. 7) ground + rocking+ torsion ground + rocking ground Displacement (cm) Time (s) Fig. 9. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 7. 3

34 Big Bear Earthquake, 2nd Story (Ch. 7) ground + rocking ground + rocking+ torsion ground z 8 3 θ z Displacement (cm) x 7 y N θ y Time (s) Fig. 20. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 7. 32

35 Northridge Earthquake, 2nd Story (Ch. 7) 20 ground + rocking+ torsion ground + rocking z 8 3 θ z x 7 y Displacement (cm) ground N θ y Time (s) Fig. 2. Contributions from ground translation, rocking θ y, and torsion θ z to computed displacement at the site of recording channel 7. 33

36 NS DRIFT RATIOS West End Whittier R 7 East End Landers R Height (m) Big Bear R 7 Northridge R Height (m) Observed drift ratio (percent) drift ratio (percent) Fig. 22. NS drift ratios ( tangents of drift angles) at East end ( solid lines) and West end (dashed lines) of VN7SH building. Total ( heavy lines) and high-pass-filtered ratios (light lines) are shown. 34

37 Table 2a Drift Angles, East Side Broad band to Hz Event Roof-6th 6 th -3rd 3 rd -2nd 2 nd -st Averaged Whittier Narrows, Landers, Big Bear Northridge, Low-Pass-Filtered Data High-pass-Filtered Data Event Roof-6th 6th-3rd 3rd-2nd 2nd-st Averaged Roof-6 th 6th-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge,

38 Table 2b Drift Angels West Side Broad band to Hz Event Roof-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge, Low-Pass-Filtered data High-Pass-Filtered data Event Roof-3rd 3rd-2nd 2nd-st Averaged Roof-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge,

39 Figure 22 summarizes the vertical distribution of the maximum NS drift ratios (tangents of drift angles) computed for the response data during the four earthquakes. To help with relative comparisons, the drift angles for total and for low-pass-filtered data are shown together, with heavy and light lines respectively. Drift angles along the East end of the VN7SH building are shown by continuous lines (using recordings at stations 3, 4, 6, 8, and 3; see Fig. 4), while those at the west end of the building (using data recorded at stations 2, 5, 7, and ; see Fig. 4), are shown by dashed lines. Numerical values of all the drifts are shown in Tables 2a and 2b. Averaged (One-Dimensional) NS Response By averaging the recorded motion for station pairs (2,3), (5,6), (7,8), and (, 3), the contributions from torsional response can be eliminated, which emphasized the contribution of NS rocking angles θ y. Figures 23 through 26 show low-pass-filtered drifts from the above-stated four pairs of recording channels. Figures 27 through 30 show the corresponding high-pass-filtered drift angles. The pass-band cut off frequencies are shown in each figure. Figures 23 through 26 correspond to Figures 6 through 9, and Figures 27 through 30 correspond to Figures 0 through 3. Figure 3 shows the average drift angles for NS response. Heavy continuous lines show peak drift angles computed from high-pass-filtered and averaged NS displacements, while light dashed lines show the amplitudes of the low-pass-filtered maxima of averaged drifts. These and the total average drift ratios (using broad-band displacement data) are summarized in Table 2c. EW Response EW is the longitudinal direction for the VN7SH building (L = 50 ft, W = ft, and L/W = 2.39; see Fig. 2). The centrally located (with respect to the building width) transducers at the east end of the building (channels 9, 0,, and 2 of the Central Recording System and channel 6 of the SMA- recorder on the ground floor) allow an estimation of one-dimensional view of the predominantly EW 37

40 Drift Angles From Averaged N-S Displacements Low-pass-filtered data from to Hz Whittier Narrows Earthquake (987) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 23. Averaged NS inter-story "drift" angles for low-pass-filtered displacements. 38

41 Drift Angles From Averaged N-S Displacements Low-pass-filtered data from to Hz Landers Earthquake (992) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 24. Averaged NS inter-story "drift" angles for low-pass-filtered displacements. 39

42 Drift Angles From Averaged N-S Displacements Low-pass-filtered data from to Hz Big Bear Earthquake (992) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 25. Averaged NS inter-story "drift" angles for low-pass-filtered displacements. 40

43 Drift Angles From Averaged N-S Displacements Low-pass-filtered data from to Hz 0.0 Northridge Earthquake (994) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 26. Averaged NS inter-story "drift" angles for low-pass-filtered displacements. 4

44 Drift Angles From Averaged N-S Displacements Hig-pass-filtered data from to Hz Whittier Narrows Earthquake (987) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 27. Averaged NS inter-story "drift" angles for high-pass-filtered displacements. 42

45 Drift Angles From Averaged N-S Displacements High-pass-filtered data from to Hz Landers Earthquake (992) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 28. Averaged NS inter-story "drift" angles for high-pass-filtered displacements. 43

46 Drift Angles From Averaged N-S Displacements High-pass-filtered data from to Hz Big Bear Earthquake (992) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 29. Averaged NS inter-story "drift" angles for high-pass-filtered displacements. 44

47 Drift Angle From Averaged N-S Displacements High-pass-filtered data from to Hz 0.0 Northridge Earthquake (994) Drift Angle 0 2,3 Roof ,6 3 rd 7,8 2 nd 3, st Time (s) Fig. 30. Averaged NS inter-story "drift" angles for high-pass-filtered displacements. 45

48 AVERAGED NS DRIFT RATIOS Whittier 987 Landers R 20.0 R Height (m) 0.0 low-pass data Big Bear R Northridge R Height (m) Observed Drift Ratio (percent) Drift Ratio (percent) Fig. 3. Averaged NS drift ratios ( tangents of drift angles) for high-pass-filtered (heavy lines) and low-pass-filtered (light-dashed lines) displacements. 46

49 Table 2c. NS Averaged Drift Low-Pass-Filtered data High-Pass-Filtered data Event Roof-3rd 3rd-2nd 2nd-st Averaged roof-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge, Broad Band to Hz Event Roof-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge,

50 Drift Angles From E-W Displacements Low-pass-filtered data from to Hz Whittier Narrows Earthquake (987) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 32. EW inter-story drift angles for low-pass-filtered displacements. 48

51 Drift Angles From E-W Displacements Low-pass-filtered data from to Hz Landers Earthquake (992) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 33. EW inter-story drift angles for low-pass-filtered displacements. 49

52 Drift Angles From E-W Displacements Low-pass-filtered data from to Hz Big Bear Earthquake (992) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 34. EW inter-story drift angles for low-pass-filtered displacements. 50

53 Drift Angles From E-W Displacements Low-pass-filterred data from to Hz 0.0 Northridge Earthquake (994) Drift Angle D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 35. EW inter-story drift angles for low-pass-filtered displacements. 5

54 Drift Angles From E-W Displacements High-pass-filtered data from to Hz Whittier Narrows Earthquake (987) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 36. EW inter-story drift angles for high-pass-filtered displacements. 52

55 Drift Angles From E-W Displacements High-pass-filtered data from to Hz Landers Earthquake (992) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 37. EW inter-story drift angles for high-pass-filtered displacements. 53

56 Drift Angles From E-W Displacements High-pass-filtered data from to Hz Big Bear Earthquake (992) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 38. EW inter-story drift angles for high-pass-filtered displacements. 54

57 Drift Angles From E-W Displacements High-pass-filtered data from to Hz 0.0 Northridge Earthquake (994) Drift Angle 0 9 D 0,9 Roof th D,0 D 2, 3 rd 2 2 nd 6 D 6,2 st Time (s) Fig. 39. EW inter-story drift angles for high-pass-filtered displacements. 55

58 EW DRIFT RATIOS Whittier 987 Landers R 20.0 R Height (m) Big Bear R 7 Northridge R Height (m) Observed Drift Ratio (percent) Drift Ratio (percent) Fig. 40. Maxima of EW drift ratios ( tangent of drift angles) for low-pass-filtered (heavy lines) and high-pass-filtered (light lines) displacements. 56

59 Table 3. EW Drift Angles Broad Band to Hz Event Roof-6th 6th-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge, Low-Pass-Filtered data High-Pass-Filtered Data Event Roof-6th 6th-3rd 3rd-2nd 2nd-st Averaged Roof-6th 6th-3rd 3rd-2nd 2nd-st Averaged Whittier Narrows, Landers, Big Bear Northridge,

60 EW response (Fig. 4). In Figures 32 through 35 we show the time dependence of EW drift angles computed from low-pass-filtered displacements. In Figures 36 through 39 we show the corresponding EW drift angles from the high-pass-filtered displacements. Figure 40 shows the maxima in time of these low-pass (heavy continuous lines) and high-pass-filtered (light continuous lines) responses. For the Whittier Narrows 987 event no data were recorded on channel 6, and so, in Fig. 40 the maximum drift angles could only be calculated above the second floor. Table 3 shows the peak values of EW drift ratios for low-pass-, high-pass-, and broad-bandfiltered displacements. Torsional Response By calculating the differences between the motions at station pairs (2,3), (5,6) (7,8), and (, 3) and dividing those differences by L 2,3 = 45.7 m for stations 2 and 3 and by L 5,6 = L 7,8 = L,3 = 40 m for the other three pairs, angles of rotation θ z versus time can be calculated at the roof and at the 3 rd, 2 nd, and st floors (see Fig. 4), assuming that the floor slabs are rigid in their own plane. These rotations then can be used to calculate relative torsion between the roof and third, third and second, and second and first floors. Peak amplitudes of this relative torsion, in time, are shown in Table 4 for the four earthquakes considered in this work. By dividing relative torsion by the vertical separation of the floors where the measurements were taken (h 3,roof = 3.3 m, h 2,3 = 2.7 m, and h,2 = 4. m, for Roof 3 rd floor, 3 rd 2 nd floors, and 2 nd st floors respectively), one can calculate torsional drift angles, T, per unit of height. For example, between the 3 rd floor and the roof this angle per unit height is T 3,roof u u u u = L2,3 L5, and likewise for T,2 and T 2,3. The variation of these torsional drift angles (per unit height in centimeters) over time is shown in Figures 4 through 44. The peak amplitudes are listed in the first three h 3,roof, 58

61 Drift Angles From Torsional Displacements Broad-band-filtered data from to Hz Whittier Narrows Earthquake (987).5 x0-6 Drift Angle Per Unit Heiht 0 (rad/cm) 2 T 3,roof 3 Roof -.5 x0-6 5 T 2,3 6 3 rd 7 T,2 8 2 nd 3 st Time (s) Fig. 4. Drift angles (per unit height) computed from relative rotation Θ z between two floors. 59

62 Drift Angles From Torsional Displacements Broad-band-filtered data from to Hz Landers Earthquake (992).5 x0-6 Drift angle Per Unit Heiht 0 (rad/cm) 2 T 3,roof 3 Roof -.5 x0-6 5 T 2,3 6 3 rd 7 T,2 8 2 nd 3 st Time (s) Fig. 42. Drift angles (per unit height) computed from relative rotation Θ z between two floors. 60

63 Drift Angles From Torsional Displacements Broad-band-filtered data from to Hz Big Bear Earthquake (992).5 x0-6 Drift Angle Per Unit Heiht 0 (rad/cm) -.5 x0-6 2 T 3,roof 3 Roof 5 T 2,3 6 3 rd 7 T,2 8 2 nd 3 st Time (s) Fig. 43. Drift angles (per unit height) computed from relative rotation Θ z between two floors. 6

64 Drift Angles From Torsional Displacements Broad-band-filtered data from to Hz Northridge Earthquake (994) 2.5 x0-5 Drift Angle Per Unit Heiht (rad/cm) 0 2 T 3,roof 3 Roof -2.5 x0-5 5 T 2,3 6 3 rd 7 T,2 8 2 nd 3 st Time (s) Fig. 44. Drift angles (per unit height) computed from relative rotation Θ z between two floors. 62

65 Table 4 Torsion Drift Angles Per Unit Length And Relative Torsion Drift Angle x 0-9 Relative Torsion x 0-6 Event Roof-3rd 3rd-2nd 2nd-st Roof-3 rd 3rd-2nd 2nd-st 63 Whittier Narrows, Landers, Big Bear Northridge,

66 TORSIONAL DRIFT ANGLES Whittier 987 Landers R 20.0 R Height (m) Big Bear R 7 Northridge R 7 Height (m) De la Llera et al Drift Angle X Drift Angle X 0-6 Fig. 45. Maxima of drift angles (per unit height) computed from relative rotation Θ z between two floors. 64

67 columns of Table 4 and are plotted in Fig. 45. For the Northridge 994 earthquake, this figure also shows the estimates of the torsional drift angles by De la Llera et al. (200). RELATED STUDIES OF VN7SH BUILDING Since 994, the VN7SH building has been studied by many investigators and has emerged as a useful benchmark for comparison of different analysis methods. Trifunac et al. (999a) and Trifunac and Hao (200) presented photographs of the damage following the Northridge earthquake of 994 and collected all of the strongmotion data digitized so far for the period between 97 and 994. Following the earthquake, two full-scale ambient vibration tests were performed, one on February 4-5, 994, and the second on April 9-20, 994 (Ivanović et al., 999; 2000a,b). During the second ambient vibration survey, detailed measurements of wave motion through the building foundation showed that the foundation is flexible and deforms with the passage of micro-tremor waves, which indicates that for studies of soilstructure interaction the rigid foundation assumption may not be appropriate (Trifunac et al., 999b). The apparent period of the soil-structure-system and the dependence of this period on the response amplitudes in VN7SH were described by Trifunac et al. (200a,b). An application of off-line and on-line identification techniques to the building response data in VN7SH was presented by Loh and Lin (996). A continuum mechanics representation of VN7SH was considered in terms of isotropic and anisotropic two-dimensional models and their response to incident wave motion by Todorovska et al. (200a,b). The feasibility of identifying the observed damage through wave propagation studies using recorded earthquake responses was explored in Ivanović et al. (200) and Trifunac et al. (2003). The accounting of incident-wave energy, its redistribution among different response energies, and the power of incident-wave motion and its capacity to damage the VN7SH structure has been described in Trifuanc et al. (200c,d). 65

68 So far, engineering studies of VN7SH have focused mainly on its longitudinal (EW) response. Without exception, these studies have neglected the effect of soil-structure interaction, and thus they have implicitly assumed that all nonlinearities in the observed response are associated with the building structure. Islam (996) considered two two-dimensional models for the building, one with and one without the brick infill walls in the four bays of the northern perimeter frame D (see Figs. 2 and 3). Assuming the building to be fixed at the ground floor level, he used the triangular distributed horizontal load to perform a push-over analysis. Figure 46 shows V/W, the resulting horizontal force versus roof displacement, assuming that the south perimeter frame (A) resists one third of the lateral load. Figure 47 shows the story drifts calculated by Islam, using elastic time history analysis, at the center of mass (COM), the NE corner, and the SW corner, for the two models with and without the infill walls in the frame A. For comparison, we also show in this figure the observed drift amplitudes, evaluated on February 4, 994, based on detailed analysis of the cracks, the scratches caused by relative motion on the partition walls, and the marks on the interior appearance of the perimeter walls. Islam concluded that many of the structural elements may have exceeded their elastic limit state at approximately 4 seconds into the earthquake. However, the most severe damage-e.g., breakdown of the entire load path in the south perimeter frame columns immediately below the 5th floor level-may have actually occurred at approximately 9 second, which coincides with the time of the peak ground acceleration in the longitudinal direction. He also notes that a push-over analysis performed on the longitudinal frame with a triangular load pattern was unable to predict the damage observed in the building. Li and Jirsa (998) performed a nonlinear time history analysis of VN7SH in the longitudinal (EW) direction only because most of the damage occurred in this direction. Acceleration time histories recorded at ground level were used as input 66

69 UBC-94, V/W=0.54 First column yield South perimeter frame (Islam, 996) (triangular load; V is assumed to resist W/3) Exterior column failure Model A First column yielding First column shear failure 0.08 V/W First beam yield Formation of mechanism Li and Jirsa (998) Uniform Triangular Islam (996) Model B First beam yielding Maximum recorded displacement during the 994 Northridge earthquake 6 Browning (2000) Model A Model B 8 0 Roof Displacement (in.) Fig. 46. Base shear (V) coefficient normalized by total building weight, W, versus EW roof displacement of VN7SH (after Islam, 996; Li and Jirsa, 998; and Browning et al, 2000: Models A and B). 67