Dynamic Analysis of a Forty‐Four Story Building

DYNAMIC ANALYSIS OF A FORTY-FOUR STORY BUILDING

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By Bruce F. Maison 1 and Carl F. Neuss, 2 M e m b e r s , ASCE ABSTRACT: Extensive computer analysis of an existing 44 story steel frame highrise building is performed to study the influence of various modeling aspects on the predicted dynamic properties and computed seismic response behaviors. The predicted dynamic properties are compared to the building's true properties as previously determined from experimental testing. The seismic response behaviors are computed using the response spectrum (Newmark and ATC spectra) and equivalent static load methods (ATC and UBC). Interpretations of the analysis results are provided. Conclusions are drawn regarding general results that are relevant to the analysis of other high-rise buildings.

INTRODUCTION

In the design of tall high-rise buildings, it is generally recognized that seismic design based exclusively on the common equivalent static lateral load techniques may be inadequate because these structures are not well suited to the simplifying assumptions inherent to such methods. For this reason, dynamic seismic analysis by computer is often used to assist in the design process. It has the advantages of being able to provide useful insights into a building's true dynamic properties as well as the ability to generate design quantities based on the dynamic properties and estimates of possible future seismic events. The application of dynamic analysis involves the formulation of an analytical model that incorporates the building's physical characteristics. The model formulation criteria has an important effect on the computed dynamic properties and resulting seismic response behavior. Therefore, the development of an appropriate building model requires considerable engineering judgement because the model must be able to capture the significant dynamic properties and yet be manageable in terms of engineering labor and computer costs. The study reported herein deals with several aspects of dynamic analysis. Its purposes are as follows: (1) To investigate the degree to which practical mathematical models of limited complexity can accurately reflect the true dynamic properties of an actual high-rise building; (2) to assess the influence of various detailed modeling aspects on the prediction of the dynamic properties and computed seismic response; (3) to identify the general dynamic characteristics of high-rise buildings and show how these contribute to response induced by earthquakes; and, (4) to compare earthquake response based on dynamic theory with the static response based on code type lateral load provisions. 'Struct. Engr., 10458 So. St. Louis Ave., Chicago, 111. 60655. 2 Civ. Engr., One Soldiers Field Park, No. 401, Boston, Mass. 02163. Note.—Discussion open until December 1, 1985. To extend the closing date one month, a written request must be filed with the ASCE Manager of lournals. The manuscript for this paper was submitted for review and possible publication on November 23, 1983. This paper is part of the Journal of Structural Engineering, Vol. Ill, No. 7, July, 1985. ©ASCE, ISSN 0733-9445/85/0007-1559/$01.00. Paper No. 19869. 1559

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ELEVATION

PLAN

FIG. 1.—Building Elevation and Plan Views

The south building of the Century City Theme Towers located in Los Angeles, California, is used as an example study building. The building is 44 stories (570 ft) in height above the plaza level and has six underground parking levels (Fig. 1). It has a novel equilateral triangular floor plan having side dimensions of 254 ft-6 in. This structure has been the subject of a previous study in which the building's actual dynamic properties were determined by experimental testing (8). The test results serve as a data base to be used in the evaluation of various analytical models. The analytical investigation includes the calculation of periods and mode shapes as well as seismic response behavior computed by the response spectrum and equivalent static load techniques. The present investigation represents one of five such studies performed on different actual high-rise buildings (6). DESCRIPTION OF STRUCTURAL SYSTEMS

The south building has an exterior wall system, and an interior core framing system. The exterior walls are steel moment resistant frames that are designed to resist vertical and lateral loads. Each exterior frame has 23 bays that have columns located at 10 ft-2 in. centers. Typical story heights are 12 ft-7 in. The columns are W21 shapes or built-up sections (21 in. deep), and the spandrel beams are built-up sections having a depth of 48 in. The corner columns are built-up sections that vary according to the height along the building. Plate girders are located at the building top (28 ft—1-1/2 in. deep) and at the second floor (7 ft deep) levels. The exterior frames terminate at the second floor level, whereas the corner columns extend below to the B level (Fig. 1). Located at the second floor level is a horizontal truss that connects the exterior wall and interior core systems. Its function is to transfer lateral loads from the exterior walls to the interior core. Below the second floor level, the vertical loads associated with the exterior frames are resisted entirely by the corner columns. The interior core system is primarily a vertical load 1560

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resisting system above the second floor, and a vertical and lateral load resisting system below the second floor level. The interior core framing consists of steel W shape columns (W14 section type) and beams (up to W30 section type). Between the second floor and the B level, the interior core perimeter framing is interconnected by steel plates, thus forming a steel shear wall system to resist lateral loads. The floor system typically consists of a 4-3/4 in. thick concrete slab placed on a 1-1/2 in. steel deck that is supported by steel floor beams. Shear connectors are provided for slab composite action with the floor framing, including the exterior wall spandrel beams. From the B level down to the foundation (F level), the structural system consists of reinforced concrete elements. The building is situated upon a concrete mat foundation. DESCRIPTION OF ANALYTICAL MODELS

Due to the symmetrical triangular structural arrangement, the building's lateral mode shapes do not have unique principal directions. The building has pairs of lateral modes having identical periods and modal components at right angles to each other. This property is exploited to reduce the model size. The lateral modes in a single direction are extracted. Dynamic analyses of the south building are performed using a modified version (5) of the ETABS (9) computer program. Planar symmetry exists about the North-South (N-S) building axis, thus only one half of the building is modeled thereby reducing the model size. Structural elements which intersect the plane of symmetry are connected to fictitious columns located in the plane of symmetry that have stiffness properties which simulate the appropriate boundary conditions. For calculation of the N-S dynamic properties, boundary conditions allowing symmetric behavior are imposed at the plane of symmetry. All models are completely fixed at the B level. The model floor masses are calculated from assumed floor weights that average about 97 psf. Torsional mass moment of inertias are based on the assumption that the mass is uniformly distributed over each floor. The models are developed by the progressive addition of various features. Each model is described in the following. Model 1.—Model 1 is formulated to represent the lateral behavior of the building in the N-S direction. The model is composed of the exterior frame that is oriented in the N-S direction, and the interior core shear walls that are located below the second floor level [Fig. 2(a)]. The exterior frame is modeled with 3-D beam-column, and 2-D beam elements that have stiffness properties based on the center-to-center member lengths and the steel section properties (composite action ignored). The interior core steel shear walls are modeled by a single 3-D beam-column element having the equivalent lateral stiffness. Model 2.—Model 2 accounts for the stiffening that results from the inherent rigidity of the joint regions at the column-spandrel beam intersections. It is the same as Model 1 except that the exterior frame beamcolumn connection regions are assumed to be infinitely rigid, whereby the element stiffnesses are calculated using the clear spans of the columns and beams [Fig. 2(a)]. The element ends are connected to the nodes 1561

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Interior Core Framing (Model 5}

a

ib-Spandrel Composite Hon (Model 3!

Two Exterior Frames

FIG. 2.—Building Model Idealizations: (a) Models 1, 2, and 3; (b) Models 4 and 5

by rigid links. The rigid joint zones reduce the effective column heights and beam lengths by 32% and 17%, respectively. Model 3.—Model 3 is developed from Model 2 by incorporating the effects of slab-spandrel beam composite action [Fig. 2(a)]. The spandrel beam section properties are calculated according to AISC (1) recommendations in which an effective slab width of b + 6i is assumed (b is flange width and t is slab thickness). The increase in the spandrel beam moment of inertias is about 40% on the average over those neglecting composite action. Model 4.—Model 4 includes the exterior E-W as well as the N-S frames [Fig. 2(b)]. Like Model 3, both rigid joint zones and composite action are incorporated into the exterior wall model formulation. The N-S and EW frames are defined as a single substructure, therefore the displacement compatibility at the corner column that is common to both frames is properly modeled. The steel shear walls are modeled by a single element having equivalent N-S lateral, E-W lateral, and torsional stiffnesses. For the calculation of the torsional dynamic properties, antisymmetric behavior boundary conditions are imposed on the plane of symmetry. Model 5.—Model 5 is developed from Model 4 by including the interior core framing system that is located above the steel shear walls and extending to the roof [Fig. 2(b)]. The peripheral beams and columns of the interior core (along column lines parallel to the exterior walls) are included in the model. Although not designed as a lateral moment-resisting frame system, fully rigid connection regions and slab-beam composite action are assumed. COMPARISON OF ANALYTICAL AND EXPERIMENTAL DYNAMIC PROPERTIES

The natural periods of the analytical models, and the experimental periods obtained from the forced vibration tests are presented in Table 1. For each model, the percentage difference between the analytical and experimental N-S periods is somewhat independent of mode number. Therefore, the ratios of the higher modal periods to the fundamental period (ratio Ti/T,, in which T,- = the z'th period) in each model are similar to those from the experimental data. Models 1 and 4 have N-S period ratios of 1.00, 2.85, 4.93, 7.18 and 1.00, 2.92, 5.12, 7.30, respectively. 1562

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TABLE 1.—Experimental versus Analytical Natural Periods N-S Translational Periods

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Evaluation method

0)

1st, sec (2)

(3)

Experimental Model 1 Model 2 Model 3 Model 4 UBC Eq. a ATC Eq. a

3.75 5.67 4.16 4.05 3.94 4.60 4.19

51.2 10.9 8.0 5.1 22.7 11.7

% —

2nd, sec (4)

% (5)

3rd, sec (6)

% (7)

4th, sec (8)

Torsional Periods /o

(9)

1st, sec (10)

2nd, sec (11) (12)

%

% (13)

1.28 — 0.73 — 0.51 — 2.80 — 1.01 — 1.99 55.5 1.15 57.5 0.79 54.9 — — — — 1.43 11.7 0.82 12.3 0.56 9.8 — — — — 1.39 8.6 0.80 9.6 0.55 7.8 — — — — 1.35 5.5 0.77 5.5 0.54 5.9 2.76 - 1 . 4 0.96 - 5 . 1

a Code periods calculated using a 46-story height as follows: UBC (Ref. 4, Eq. 12-3B), T = 0.10 N = 0.10 x 46 = 4.60 sec; ATC (Ref. 2, Eq. 4-4), T = CThT = 0.035 x (590)3/4 = 4.19 sec. Note: % = percentage variation from experimental period.

These compare closely with the experimental period ratios of 1.00, 2.93, 5.14, and 7.35. This indicates that the relative spacing of the periods is correctly accounted for in each model. The trend of smaller periods according to increasing model number (Table 1) reflects the fact that the modeling refinements involve the progressive addition of features which increase stiffness. Model 1 is the most flexible building idealization, therefore its natural periods are the largest. It has a fundamental period of 5.67 sec, which is 51.2% greater than the experimental value of 3.75 sec. Model 4, representing a refined model, has a fundamental period of 3.94 sec which is only 5.1% larger than the experimental period. It has an average increase in modal stiffness of 115% (as determined from period shifts) when compared to Model 1. A second analysis of Model 4 was performed with antisymmetric behavior boundary conditions to determine the torsional natural periods. As shown in Table 1, the torsional periods compare well to the experimental values. The fundamental torsional period of 2.76 sec is within 1.4% of the experimental value of 2.80 sec. Although not presented in the tables or figures, the N-S dynamic properties of Model 5 were computed, and the resulting natural periods and mode shapes agree closely to those from Model 4. The N-S fundamental period of Model 5 is 3.92 sec, which is within 1% of the Model 4 fundamental period of 3.94 sec. This indicates that the interior core framing above the second floor has only a very small effect on the building's lateral stiffness. Also contained in Table 1 are the fundamental period values as calculated by the approximate UBC (4) and ATC (2) formulas. The UBC value of 4.6 sec is 22.7% larger, and the ATC value of 4.19 sec is 11.7% larger than the experimental fundamental period. Note that Models 2, 3, and 4 provide better estimates than either of the code formulas. The first four N-S translational mode shapes from Models 1 and 4 are compared with the experimental forced vibration results in Fig. 3. Model 1 is the simplest, whereas Model 4 represents a refined model. The striking similarity of the mode shapes among the models is because the var1563

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-orced Vibration Model 1 Model 4

0

u

44 40

40

A

35

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THIRD MODE

SECOND MODE

0.0

0.5

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FOURTH MODE

FIG. 3.—Comparison of Experimental and Analytical N-S Mode Shapes JZDNE B , . ZONE C ..ZONE D, o Forced Vibration — Mode! l*

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UNDAMPED NATURAL PERIOD ISECONDSI

FIG. 4.—Comparison of Experimental and Analytical Torsional Mode Shapes

FIG. 5.—Tripartite Plot of Newmark Design Spectrum (Peak Gr. Ace. 0.4 g, Peak Gr. Vel. 1.60 ft/sec, Peak Gr. Displ. 1.20 ft)

ious modeling features incorporated produce a roughly uniform change (i.e. scaling) in lateral stiffness along the building height which does not modify the mode shapes (6). The analytical mode shapes agree well with the experimental results. The first two torsional mode shapes from Model 4 are compared with the experimental results in Fig. 4. The analytical torsional mode shapes agree favorably with the experimental results. INFLUENCE OF MODELING APPROACH ON COMPUTED SEISMIC RESPONSE

In this section, the results from response spectrum dynamic analyses of Models 1-4 are presented to show the influence of the different modeling approaches on the calculated seismic response. The Newmark design spectrum (7) scaled to 0.05 g peak ground acceleration is used for the earthquake excitation (Fig. 5). The first six N-S translational modes are included in the analyses and the total peak responses are estimated by using a square root sum of the squares (SRSS) combination of the individual peak modal responses. 1564

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44-i 4035302520

15-1

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10J

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Modal Modal Model Model

1 2 3 4

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FIG. 6.—Story Force Quantities from Analytical Models: (a) Shears; (b) Overturning Moments

Peak story shear envelopes plotted along the height of the building are shown in Fig. 6(a). The shape of the envelope curves are similar for all models. This is because of the mode shapes from the various models are somewhat invariant (Fig. 3). Therefore, the distribution of the response quantities among the models can be expected to be similar. The amplitudes of the shear envelopes increase from Models 1—4. This results from the increasing spectral accelerations which are a consequence of the decreasing natural periods from Models 1-4 (Table 1, Fig. 5). The story inertia forces (thus, resulting story shears) are directly proportional to the spectral accelerations which are a function of the natural periods. Shown in Table 2 are the values of base shear from the various models and the percentage change with respect to Model 1. Model 1 composed of the bare steel N-S exterior frame and interior core shear walls has the largest periods, and therefore, the smallest story shears. The base shear progressively increases from 1,030 kips in Model 1 to 1,886 kips in Model 4, representing an 83% increase. Peak story overturning moment envelopes are shown in Fig. 6(b). The overturning moment envelopes show the same trends as the story shear envelopes; that is, the envelopes have similar characteristic shapes but different amplitudes. Model 1 has the smallest base overturning moment of 3,656 X 103 kip-in., whereas Model 4 has the largest value of 7,506 x 103 kip-in., representing a 105% increase (Table 2). Peak story deflection envelopes for Models 1 and 4 (other models have similar values) are shown in Fig. 7(a). Note that the deflections are virtually identical for all models. This is a consequence of general seismic TABLE 2.—Influence of Modeling Features on Building Seismic Response Model number

(D 1 2 3 4

Base shear (kip) (2) 1,030 1,704 1,782 1,886

% (3)

— 65 73 83

Base OTM (k-in. x 103) (4) 3,656 6,764 7,160 7,506

% (5)

— 85 96 105

Roof deflection (in.) (6) 3.45 3.41 3.41 3.42

Note: % = incremental percentage fr am Miidel 1.

1565

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% (7)

— -1 -1 -1

30th level drift (in.) (8) 0.111 0.096 0.094 0.098

% (9)

— -14 -15 -12

Model 1

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