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Seismic analysis of the world’s tallest building
Journal of Constructional Steel Research 65 (2009) 1206–1215
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Seismic analysis of the world’s tallest building Hong Fan a,b , Q.S. Li a,∗ , Alex Y. Tuan c , Lihua Xu d a
Department of Building and Construction, City University of Hong Kong, Hong Kong
b
China Nuclear Power Design Company, Shenzhen 518026, China
c
Department of Civil Engineering, Tam Kang University, Taipei, Taiwan
d
School of Civil Engineering, Wuhan University, Hubei Wuhan 430072, China
article
info
Article history: Received 13 March 2008 Accepted 8 October 2008 Keywords: Super-tall building Mega-frame structure Finite element modeling Seismic analysis Dynamics response Shaking table test
a b s t r a c t Taipei 101 (officially known as the Taipei Financial Center) with 101 stories and 508 m height, located in Taipei where earthquakes and strong typhoons are common occurrences, is currently the tallest building in the world. The great height of the building, the special geographic and environmental conditions, not surprisingly, presented one of the greatest challenges for structural engineers. In particular, its dynamic performance under earthquake or wind actions requires intensive research. The structure of the building is a mega-frame system composed of concrete filled steel tube (CFT) columns, steel brace core and belt trusses which are combined to resist vertical and lateral loads. In this study, a shaking table test was conducted to determine the constitutive relationships and finite element types for the CFT columns and steel members for establishing the finite element (FE) model of the tall building. Then, the seismic responses of the super-tall building were numerically investigated. An earthquake spectrum generated for Taipei Basin was adopted to calculate the lateral displacements and distributions of interior column forces. Furthermore, time-history analyses of elastic and inelastic seismic response were carried out using scaled accelerograms representing earthquake events with return periods of 50-year, 100-year, and 950-year, respectively. The computational results indicate that the super-tall building with the mega-frame system possesses substantial reserve strength, and the high-rise structure would satisfy the design requirements under severe seismic events. The output of this study is expected to be of considerable interest and practical use to professionals and researchers involved in the design of super-tall buildings. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Owing to the growing use of high-strength materials and advanced construction techniques, building structures have become more and more flexible and taller. The increasing height of modern tall buildings posed a series of challenges for structural engineers. In the design of such a tall building, the structural system must meet three major requirements: strength, rigidity, and stability [1]. As is well known, the strength requirement is the dominant factor in the design of low-rise structures. However, as building height increases, the rigidity and stability requirements become more important, and they are often the dominant factors in the structural design. Especially under lateral loads, interior forces are quite variable and increase rapidly with increases in height, and lateral deflection may vary as the fourth power of the height of a building [2], and structural dynamic behavior is thus one of the most important design considerations in the design of a modern tall building.
∗
Corresponding author. Tel.: +852 27844677; fax: +852 27887612. E-mail address: [email protected] (Q.S. Li).
0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.10.005
Taipei 101, rising 508 m above the city of Taipei, earns the title of the tallest building in the world. Its dynamic responses due to wind, earthquake and other extraordinary loads are of great concern. As Taiwan is located in one of the most active seismic regions in the world, this super-tall building may be susceptible to damage caused by strong earthquakes. These features make a detailed study on the structural performance of the world’s highest tall building under earthquake excitations of particular importance and necessity. Numerous investigations on seismic behavior of tall buildings have been carried out in the past; in particular shaking table tests play an important role in earthquake-resistant design of structures, analysis of seismic responses and failure mechanisms [3–5]. On the other hand, the finite element method (FEM) is a powerful tool for structural analysis of tall buildings. Fan and Long [6] adopted spline elements in the analysis of tall buildings. In their method, the element displacements are interpolated with spline functions and accurate results could be achieved with lower-order functions and a few degrees of freedom. Takebatake et al. [7] presented a simplified analytical method for the preliminary design of doubly symmetric single and double frame-tubes in high-rise structures by replacing a tube with an equivalent rod,
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
with consideration of the effect of bending, transverse shear deformation, shear-lag and torsion. Li et al. [8,9] proposed finite segment approaches for estimating the dynamic characteristics of tall buildings. Recently, Li and Wu [10] established seven 3D FE models for a 78-story super-tall building, and numerical results of the structural dynamic characteristics were compared with their field measurements to identify the FEM modeling errors for the purpose of updating the FEM models. Ventura and Schuster [11] presented a numerical study on estimation of dynamic characteristic of a 30-storey RC building. A reducedorder continuum model was proposed by Chajes et al. [12] to conduct dynamic analysis of a 47-storey steel-framed building and correlate the numerical results with those from measured responses during an earthquake. Pan et al. [13] and Brownjohn et al. [14] presented numerical studies on dynamic responses of the tallest building in Singapore with correlation with their field measurements. Qi et al. [15] employed the finite element method to study the seismic performance of a tall building and their results illustrated that the building is likely to perform satisfactorily under severe seismic events. Rahimian and Romero [16] established a finite element model to study the seismic response of the tallest building in Mexico City by time-history analysis and spectral method. These investigations indicated that numerical simulation is an effective tool to determine the dynamic characteristics and seismic responses of tall buildings. However, literature review reveals that comprehensive research studies on seismic effects on a super-tall building (building height > 500 m) have rarely been reported in the literature. So, a detailed analysis is presented in this paper to investigate the dynamic characteristics and seismic responses of the super-tall building. The objective of this study is to investigate the seismic effects on the world’s tallest building in order to provide valuable information for the design and construction of other similar structures in the future.
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Fig. 1. Elevation view of Taipei 101.
2. The structural system of Taipei 101 Taipei 101, a 508-m high office tower, is located at the east district of downtown Taipei City, and the elevation view of the building is shown in Fig. 1. The structure is symmetrical with a 62.4 m by 62.4 m square footprint [17]. Two sloping rectangular mega-columns with a maximum cross-sectional dimension of 2400 mm × 3000 mm, are positioned one at each side of the building extending to the 90th floor, and finally the cross-sectional dimension of the mega-column is reduced to 1600 mm × 2000 mm. All perimeter columns are sloping from the ground floor to 25th floor, and the sloping angle is 4.4. The core columns are square and rectangular concrete-filled steel tubes (CFT). The compressive strength of the concrete is 70 MPa, which provide extra stiffness and strength to the steel tubes, from bottom of the basement to the 62nd floor. The section of the core columns reduce from 1200 mm × 1200 mm to 900 mm × 900 mm, which are smaller than the mega-column. The composite metal deck-slabs built at each floor are 135 mm thick, however, those at mechanical floors are 200 mm thick. The primary girders are composed of H-steel beams with moment-resisting connections at the beam-column joints. Dog-bone connections are also provided at locations where ductility is required while beams are pinned to the primary girders, as shown in Fig. 2. Belt-trusses, one or two-stories high, are placed every 8-story interval at the perimeter frame, and the brace core is connected to mega-columns via belt-trusses consisting of in-floor braces and vertical trusses. The locations of the belt-trusses in the 8th floor are shown in Fig. 3 [17]. When the space of the column is 10.5 m, the shape of the steel braces are ‘‘V’’ or reverse ‘‘V’’, and when the space is 6 m, the shape of steel braces are acclivitous braces Fig. 4 presents an elevation view and the locations of the belttrusses in axes M9 and P1 [17]. The belt-trusses and mega-columns
Fig. 2. Sketch of the dog-bone connection.
help to stabilize the building core in the same way that the brace core helps to balance the perimeter frame. This mega-frame design maximizes the spaces inside the building and carries perimeter gravity loads at selected columns. The bracing, outriggers, and belts that link the columns would also redistribute loads if some members are damaged by unforeseen circumstances. The structure is a dual system: the external structure is composed of the megacolumns and external columns providing the lateral rigidity to the seismic and wind loads, and the internal structure, also designated as substructures, provides the utilizable space and allows for significant amount of energy dissipation. Belt trusses composed of a transfer floor system are placed at every eighth or tenth floor, so the interior columns only carry the gravity loads from a limited number of floors. As a result, their sizes are substantially smaller than those in a conventional structural system, in which they would rise from foundation level to the building top. A FE model was established in this study based on the design drawings of the super-tall building. The dead loads for building elements were determined by a commercial FE program ANSYS 10.0 [18] and the design live loads were calculated according to the data found from the design documents [17].
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Fig. 3. Plan of belt-trusses in the 8th floor.
Fig. 5. Stress condition for steel and concrete in a CFT column under axial compression.
analysis has become a routine design tool of tall buildings. Four kinds of elements are employed in establishing the FE model of Taipei 101 structure: 3-D beam elements, suitable for nonlinear large rotations and large strains, are employed to model the columns and beams. Link elements are used to model the brace. Mass elements are employed to model the live loads and nonstructural components. The floors are modeled with shell elements. The connection between the structure and its foundation is treated to be fixed. 3.2. Constitutive relationships of rectangular CFT columns Concrete-filled steel tube (CFT) columns are widely used due to their good earthquake resistant behavior such as improved strength and high ductility capacity. When a short CFT column is under an axial load, as shown in Fig. 4(b), there is a basic assumption that steel and concrete have the same longitudinal strain ε3 , then the hoop stains of steel ε1s and concrete ε1c can be calculated by:
ε1s = µs ε3 ,
Fig. 4. Elevation and the location of the belt-trusses.
3. Structural analysis 3.1. Finite element modeling With rapid development in computer technology and computational mechanics algorithms, three-dimensional finite element
ε1c = µc ε3
(1)
where µs , µc are the Poisson’s ratio of steel and concrete, respectively. Generally, at low stress conditions, concrete has a lower value of Poisson’s ratio than steel, which may result in occurrence of separation between the two materials in a CFT column. At high compressive stresses, internal micro-cracking in concrete causes it to swell. Its outwards movement is restrained by steel, and the strength of concrete is increased due to this lateral restraint. Thus the concrete and steel are stressed triaxially, as shown in Fig. 5(a). Zhong [19,20] proposed a unified theory to model CFT columns based on extensive experimental and FE analysis results of CFT columns under axial loading. According to the theory, a CFT column is regarded as a new composite column or material instead of separate components of concrete and steel. The properties of the composite column depend on those of steel and concrete and their dimensions (e.g., tube diameter and steel wall thickness). The ultimate strength and other property parameters of a CFT column can then be determined based on the mechanical and geometrical properties of the composite material. The following formulas for rectangular CFT columns are adopted in this study based on the unified theory [19,20]:
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
(a) Stress–strain curve for a concrete-filled steel tube.
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(b) Stress–strain curve for steel.
Fig. 6. Schematic graph of material constitutive relationships.
The yield strength of the composite column is fscy = (1.212 + Bξ + C ξ 2 )fck
(2)
where, B and C are coefficients. They depend on the cross-section geometry. For a rectangular cross-section, one has: B = 0.1381(fy /235) + 0.7646, C = −0.0727(fck /20) + 0.2016, where ξ is the confining factor which is expressed as
ξ=
fy A s fck Ac
in which fy , fck , As and Ac are the yield strength of steel, the unconfined strength of concrete and the areas of steel and concrete components in the column, respectively. The elastic modulus Esc of the composite column can be expressed as Esc = fscp /εscp
(3)
where fscp , εscp are the proportional stress and strain of the composite column, respectively. For a rectangular CFT column, one has fscp = [0.192(fy /235) + 0.488] fscy
(4)
εscp = 0.67fy / Es .
(5)
The tangent module of the composite column can be calculated as Esct
(fscy − σ¯ )σ¯ Esc = (fscy − fscp )fscp
(6)
where σ¯ = N /Asc , N is the axial load on the column and Asc is the total area of the column section. The hardening modulus of the composite column can be determined by 0 Esc = 400ξ − 150.
(7)
In this paper, the load–deformation (stress–strain) relation of a CFT column was determined based on experimental measurements from CFT columns under axial compressions [19], which was simplified as a tri-linear stress–strain model including proportional, yield and hardening stages, as shown in Fig. 6(a). The tangent module is substituted by the module of a straight line connected the proportional point and the yield point. According to Eqs. (2)–(5) and (7), the related parameters for a CFT column can thus be determined. For structural analysis of the steel beams and brace members in Taipei 101 building, a bilinear stress–strain curve with 2% postyield hardening (see Fig. 6(b)) was adopted to model the inelastic behavior of these structural members, with Young’s modulus of 420 MPa and Poisson’s ratio of 0.3, respectively. Von Mises yield criterion with kinematic hardening rule was employed in the numerical analysis.
Fig. 7. Test structure.
3.3. Verification of the constitutive relationships of CFT columns For verification of the adequacy of the constitutive relationships of CFT columns and steel members discussed above as well as the selected finite element types for modeling the structural members of Taipei 101 structural system, a shaking table test and the associated FEM analysis were conducted in this study for a frame structure model composed of rectangular CFT columns and steel members by comparing the numerical results with the experimental data. The test model and its finite element model are shown in Fig. 7 and 8, respectively. The scaled model was tested on the shaking table adopting three representative earthquake records as inputs: (1) An artificial seismic accelerogram (made according to the design code of China [21]); (2) El-Centro earthquake record; and (3) Tianjin earthquake record. The peak ground accelerations (PGA) in the three accelerogams were scaled to 0.05g and 0.1g to represent the design earthquake actions with intensity 6 and 7 degree as stipulated in the design code of China [21], respectively. The design code [21] classifies regions of different seismicity in terms of seismic intensity which is usually regarded to be an equivalent of peak ground acceleration. Table 1 shows the relationship between the seismic intensity and peak ground acceleration [21]. Each accelerogram duration was reduced to 1/5 of its original duration according to the scale factor listed in Table 2. Table 3 lists the first four natural frequencies of the model obtained from the test and the numerical analysis of the FE model. Furthermore, acceleration dynamic amplification factors of the
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modeling strategies presented above in the establishment of the FE model of Taipei 101 structural system. 4. Dynamic characteristics of the super-tall building
Fig. 8. FE model.
Table 1 Relation between the seismic intensity and horizontal peak ground acceleration [20]. Seismic intensity (degree)
Horizontal peak ground acceleration (g)
6 7 8 9
0.05 0.1 0.2 0.4
A three-dimensional FE model of Taipei 101 structural system was established for numerical analysis of the super-tall building, as shown in Fig. 10, based on the constitutive relationships for rectangular CFT columns and steel members as well as the selected finite element types which were verified above. The FE model of the super-tall building contains 20 532 beam elements, 24 048 shell elements, and 3496 link elements. In addition to the main structural elements, non-structural components were modeled with mass elements. Fig. 11 shows the first six mode shapes of the FE model including two for translational motions in each horizontal direction and two for torsional motions about the vertical axis. Modes 1 and 2 are the translational modes in x and y directions, respectively. Mode 3 is the fundamental torsional mode. The fundamental periods of the building are 6.21 s. in the x direction, 6.19 s in the y direction, and 3.62 s in torsion. The modal participation coefficient for each mode [22] is defined as [Ei ][M][uj ] ηji = , which shows the modes that contribute most T [uj ] [M][uj ] to the dynamic response, where [M ] is the mass matrices, [u] is the vibration mode vector of the structure, and [E ] is the uniform matrix. The model participation ratio in every direction for each mode is defined as ϕic = P
ηic
j=1
ηjc
(c = x, y, z), while the cumulative Pi
mc
j=1 modal participation mass ratios is defined as ψic = Pn mjc (c = t =1
t
Physical parameters
Scale factors
Ratio (model/prototype)
Length Density Elastic modulus Strain
Sl Sρ SE Sε
1/5 1.0 1.0 1.0
Time
St = Sl Sρ2 SE
1/5
x, y, z), where m is the effective mass participating the dynamic response of each mode. Table 4 shows the modal participation ratios and the cumulative modal participation mass ratios of the first 30 modes. The vibration modal participation ratios for the first two modes in the x and y directions reach 1. Such ratios for other modes decrease as the mode number increases. The cumulative modal participation mass ratios for the first 30 modes reach 1 in the horizontal directions. Therefore, it was decided to use the first 30 modes in the response spectrum analysis of the tall building, which will be described below.
Sf = Su = Sl Sa = Sl /St2
5 1/5 5
5. Response spectrum analysis
Table 2 The similarity relationship between the model and the prototype structure.
1
− 12
−1 1 Sl−1 Sρ 2 SE2
Frequency Displacement Acceleration
Table 3 Natural frequencies of the scaled mode obtained by the shaking table test and the numerical analysis. Mode
1
2
3
4
Test (Hz) FE analysis (Hz) Differences (%)
17.48 16.50 −5.61
26.68 29.43 10.27
40.00 54.44 36.1
72.50 58.14 −19.8
structural model under seismic excitation with PGA of 0.1g, which were determined by the test and the numerical computation, are shown in Fig. 9. It is shown in Table 3 that the fundamental natural frequencies of the model obtained from the test and the numerical analysis agree fairly well. It is observed from Fig. 9 that the two sets of the results of acceleration dynamic amplification factors under the three earthquake records were also in good agreement. It is thus expected that the constitutive relationships for rectangular CFT columns and steel members as well as the selected finite element types for modeling these structural components are adequate, since the numerical results match the experimental data reasonably well. Hence, it was decided to adopt the numerical
For earthquake-resistant designs, a structure should meet performance requirements at two different levels, depending upon the magnitude of earthquake actions. The first level of performance essentially requires structural response in the elastic range without significant structural damage under a moderate earthquake action, and the second level of performance requires that the structure does not collapse under a severe earthquake event with rare occurrence. Taipei 101 is located in the Taipei Basin, where there are deep and soft soil deposits with a long predominant period. According to the Taiwanese Building Technology Standards (BTS) [23], the building site is in a region of moderate seismicity. A response spectrum, originally developed by the National Taiwan University and other research institutions for the design of Taipei 101, has been modified based on the response spectra stipulated in the local seismic code (BST), as shown in Fig. 12 [17]. The first 30 translational–torsional coupling modes were used to calculate the seismic response of the super-tall building by the Complete Quadratic Combination (CQC) method on the basis of the response spectrum analysis. The damping ratio of the high-rise structure was assumed to be 5% for all the 30 modes, and the seismic responses in the two orthogonal directions were calculated accordingly.
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
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Table 4 Modal participation ratios and participation mass ratios. Mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Period (s)
6.21 6.19 3.62 2.52 2.47 1.65 1.64 1.42 1.41 1.37 1.166 1.161 0.929 0.922 0.898 0.740 0.733 0.632 0.624 0.576 0.557 0.532 0.489 0.394 0.379 0.299 0.285 0.244 0.235 0.153
(a) Experimental results.
Model participation ratio
Cumulative participation mass ratio
x direction
y direction
x direction
y direction
0.000704 1 0.000002 0.001392 0.778882 0.000383 0.414047 0.000099 0.000036 0.000007 0.000044 0.000087 0.12753 0.000068 0.000011 0.07398 0.000006 0.000015 0.054531 0.000006 0.000003 0.000003 0.023607 0.000018 0.000077 0.04295 0.000006 0.001874 0.000001 0.015782
1 0.000692 0.000359 0.775207 0.001382 0.420086 0.000395 0.000007 0.000004 0.000002 0.132226 0.000133 0.000053 0.000704 0.075824 0.000019 0 0.056465 0.000022 0.000006 0.000001 0.000001 0.000021 0.025123 0.042725 0.000079 0.000007 0.000001 0.002004 0.000006
2.70E-07 0.545056 0.545056 0.545057 0.875625 0.875625 0.96938 0.96938 0.96938 0.96938 0.96938 0.96938 0.985388 0.985388 0.985388 0.991815 0.991815 0.991815 0.995633 0.995633 0.995633 0.995633 0.996521 0.996521 0.996521 0.999566 0.999566 0.999572 0.999572 1
0.544991 0.544991 0.544991 0.872428 0.872429 0.968316 0.968317 0.968317 0.968317 0.968317 0.985225 0.985225 0.985225 0.985225 0.991897 0.991897 0.991897 0.995977 0.995977 0.995977 0.995977 0.995977 0.995977 0.996983 0.999993 0.999993 0.999993 0.999993 1 1
(b) Numerical results.
Fig. 9. Dynamic amplification factors from the test and numerical analysis.
5.1. Displacement analysis The results of the response spectrum analysis indicate that the deformations in the x direction are approximately equal to those in the y direction (see Fig. 13). This can be attributed to the symmetric structural system and shape. The maximum interstory drift ratios in the x direction and y direction are 1/281.7 and 1/261.1, respectively, both under the criteria stipulated in the local design code (BST). Thus, the structure can be regarded to satisfy the first level performance requirement. Due to the two outrigger trusses forming a cell at every eight or ten stories, the inter-story drift ratio of each cell is in the shape of an arc, as shown in Fig. 14. The drift ratios are small in the top and bottom stories and large in the middle inter-stories. Such an outrigger truss functions like a ring, which controls the lateral structural deformation. The storey
Fig. 10. Finite element model.
displacement curves take the shape of a deforming cantilever and are relatively smooth without obvious inflexions, which imply that the distribution of equivalent rigidities along the height of the structure is well proportioned. 5.2. Interior force analysis Three representative columns were selected to analyze the interior forces in order to study the structural deformation mechanisms. The selected columns are the exterior corner column
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Fig. 11. Mode shapes of the first six modes obtained from numerical analysis. Fig. 13. Lateral deformation curves.
Fig. 12. Design response spectrum.
C9, exterior side column C1, and interior corner column C5, as shown in Fig. 3. It can be seen from Fig. 15 that the axial force is the dominant interior force in each column and other interior forces are relatively small. The mega-frame is a dual system, where the steel frames are composed of the mega-columns and perimeter columns along each sloping face of the building, working in parallel with the braced core which is the secondary resistant system. Therefore, the exterior side column C1 carries the heaviest vertical loads on its huge section. Since the structure is symmetrical, the torsional moment in each column is small. C9 and C1 are sloping below the 27th floor, and the torsional moments are great in some lower floors. Abrupt increases in shear forces and bending moment in these columns are observed near the outrigger belts, especially near the 26th floor and 27th floor. This is because the stiffness of the outrigger belt is generally much larger than that between adjoining stories. Consequently, there tends to be a weak story near an outrigger belt. This issue should be adequately dealt with in earthquake-resistant design of outrigger-braced tall buildings.
Fig. 14. Inter-story drift ratios curves.
materials, a mega-frame provides lateral load resistance to prevent instability of the global structure under compressive loads and moments (i.e., p–∆ effect). As shown in Fig. 16, the MEI curves are wavy with discontinuities near the outrigger belts. Since the outrigger belts transfer large lateral forces to the columns under lateral loading, the flexural deformations of the columns are increased and the percentage of bending stress are also increased, leading to a reduction of MEI.
5.3. Material efficiency index 6. Time history analysis Material efficiency index (MEI) is usually used to estimate the efficiency of concrete-filled steel tubular columns, which is defined as the ratio of average axial stress to the maximum principal stress, MEI = σn /σmax × 100%. Values of the MEI of the three representative columns are listed in Table 5 and discussed below. The exterior corner column C9 and exterior side column C1, as shown as Fig. 3, are the main load-resistant components, and the values of MEI of the two columns are above 80%. Due to a large cross-section, a mega-column provides stiffness to resist lateral forces and reduce lateral deformation. Due to efficient utilization of
The response spectrum analysis conducted previously only considered the maximum amplitude of seismic effect, but dynamic analysis in the time domain can provide more information for earthquake-resistant design of structures. The latter approach consists of a step-by-step direct integration in which the time domain is discretized into a large number of small increments, and for each time interval the equations of motion are solved to obtain the structural responses such as displacements. Four recorded accelerograms were selected as inputs in the present
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
(a) Axial force.
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(b) Shear force.
(c) Bending-moment.
(d) Torsional-moment.
Fig. 15. Interior forces of the columns.
Table 5 MEI of the columns. Floor
C1
C5
C9
Floor
C1
C5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.81 0.84 0.90 0.94 0.98 0.99 0.98 0.98 0.95 0.95 0.96 0.98 0.99 0.97 0.96 0.95 0.95 0.97 0.96 0.96 0.98 0.97 0.94 0.93 0.91 0.92 0.78 0.81 0.85 0.99
0.59 0.21 0.30 0.43 0.67 0.76 0.63 0.68 0.41 0.42 0.48 0.61 0.71 0.56 0.43 0.37 0.39 0.59 0.50 0.53 0.63 0.57 0.42 0.35 0.33 0.44 0.21 0.29 0.35 0.59
0.65 0.72 0.84 0.89 0.95 0.95 0.82 0.98 0.78 0.87 0.94 0.96 0.96 0.95 0.92 0.87 0.79 0.96 0.81 0.88 0.92 0.92 0.91 0.87 0.65 0.60
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0.90 0.95 0.97 0.90 0.87 0.77 0.82 0.85 0.91 0.96 0.96 0.88 0.85 0.75 0.81 0.85 0.91 0.97 0.95 0.87 0.83 0.73 0.79 0.84 0.91 0.96 0.95 0.86 0.81 0.72
0.45 0.65 0.70 0.45 0.32 0.22 0.27 0.38 0.51 0.73 0.69 0.47 0.34 0.24 0.31 0.43 0.57 0.78 0.70 0.49 0.32 0.23 0.30 0.43 0.56 0.75 0.70 0.46 0.30 0.23
time-history analysis: the El-Centro (N-S ), Taft (E-W ), Chi-Chi and Taiwan1115. Each accelerogram is composed of two horizontal and one vertical components of ground acceleration excitation. For each accelerogram, the magnitude of the ground acceleration was scaled. The scaled ground accelerations used in the present analysis have the peak ground accelerations of 70 gal, 195 gal and 390
C9
Floor
C1
C5
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
0.82 0.91 0.97 0.98 0.99 0.95 0.90 0.86 0.93 0.96 0.97 0.98 0.98 0.95 0.88 0.79 0.92 0.97 0.98 0.98 0.97 0.93 0.86 0.67 0.90 0.93 0.93 0.92 0.91 0.84
0.37 0.58 0.79 0.84 0.88 0.67 0.34 0.33 0.51 0.70 0.75 0.77 0.74 0.45 0.11 0.13 0.34 0.76 0.75 0.65 0.41 0.23 0.15 0.07 0.27 0.55 0.38 0.26 0.49 0.55
C9
gal, representing an earthquake event with 50-year, 100-year and 950-year return period, respectively. The maximum lateral displacement and inter-story drift ratio of the structure under the four recorded accelerograms are listed in Table 6. It is illustrated that the structure still remains elastic after the action of the earthquake excitation with 50-year return
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Table 6 Maximum displacement and inter-story drift radio. Earthquake action
50-yr return period 100-yr return period 950-yr return period
Maximum displacement/mm
∆/H
x
y
x
y
x
y
286 697 1127
357 851 1320
1/1491 1/613 1/379
1/1196 1/502 1/324
1/656 1/224 1/136
1/525 1/182 1/119
Maximum inter-story drift ratio
Fig. 16. MEI of the columns.
Fig. 18. Displacement envelope curves for the four records with 950-year return period.
return period. The dynamic magnification factors of the middle floors are close to 1, which suggest that the equivalent rigidities along the height of the structure is distributed well proportionally. Since the base of the finite element model was assumed to be rigid, there is a discontinuity between the first floor and the ground. The floor area from the storeys 91st to 101st is smaller than that of a standard floor, so the rigidities of these floors are smaller, resulting in larger dynamic amplification factors. Fig. 17. Displacement envelope curves for Taiwan1115 record.
7. Conclusions period and satisfies the first lever performance requirement. It is also revealed that the structure deforms plastically under seismic action with a 950-year return period; however, the maximum inter-story drift ratio is 1/119, which is less than the corresponding criteria (1/70) stipulated in the local design code (BTS) [23]. Fig. 17 shows the curves of the displacement envelope in the y direction under the excitation of Taiwan1115 record with different return periods, it is noted that under the seismic action with 70 gal peak ground acceleration, the deformation curve is smooth without inflexions, similar to a deforming cantilever, indicating that the structure remains in an elastic stage. When the peak ground acceleration is increased to 390 gal, the curve has an inflexion, which implies that some structural components have deformed plastically. The curves of the displacement envelope in the y direction subjected to the four accelerograms with a 950-year return period are shown in Fig. 18 and the seismic responses of the lateral displacement at the top storey are shown in Fig. 19. Since the spectral compositions of the four earthquake records are different, there are differences in the curves of timehistory responses of the lateral displacements. However, it is observed from Fig. 18 that the curves of the displacement envelope have a similar shape, showing that the mega-frame has a shear deformation mode. Fig. 20 plots the dynamic amplification factors for the case of earthquake events with a 50-year return period and a 950-year
A detailed study on the dynamic characteristics and seismic responses of Taipei 101, the world’s tallest building, was presented in this paper. The constitutive relationships for rectangular CFT columns were established based on the unified theory, and then were verified through comparison between the shaking table test data and numerical analysis results. A 3-D finite element model of Taipei 101 structure was established based on the verified constitutive relationships for the rectangular CFT columns and selected finite element types for the structural members. The seismic analysis results of the super-tall building indicated that the structural system, with belt trusses at every eighth or tenth story, provides equal stiffness along the height of the building, which can decrease the lateral deformation efficiently. Meanwhile, for such a mega-frame structural system with a central braced core connected to perimeter columns on each building face, the total dead and live loads at every floor are transferred to the sloping exterior columns, thereby the structural capacity to withstand lateral loading is enhanced. The results of this research also showed that Taipei 101 has relatively high earthquake resistance and could guarantee the structural safety under a seismic action with a moderate seismic fortification, as stipulated in the local seismic design code. However, it was revealed that there are abrupt changes in the shear force in the columns near the floors with outrigger belts. This issue should be adequately dealt with in the earthquake-resistant design of high-rise structures of this type.
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Fig. 19. Time-history responses of the top storey lateral displacement for the four records with a 950-year return period.
(a) The results with a 50-year return period.
(b) The results with a 950-year return period.
Fig. 20. Dynamic magnification factors.
Acknowledgement The work described in this paper was fully supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project No: CityU 116906). Appendix The tested model with 0.58 m storey height is a 1/5 scale model of a 5-storey steel frame. The cross-sections of the columns are 60 mm × 60 mm × 2 mm, and the cross-sections of I-shape steel beams are 40 mm × 60 mm × 2 mm × 2 mm. The steel tube columns were filled with concrete, and the concrete compressive strength is 40 MPa. Braces were used in the perimeter bays parallel with the direction of earthquake excitation. The model, instrumented with an accelerometer and a displacement-meter at the basement and each floor, was bolted to the shaking table (shown in Fig. 6). References [1] Bungale ST. Structural analysis and design of tall building. New York: The William Byrd Press; 1988.
[2] Li GQ, Li QS. Theory and its application of time-dependent reliability of engineering structures. Beijing: Science Press; 2001. [3] Li QS, Li ZN, Li GQ, Meng JF, Tang J. Experimental and numerical seismic investigations of the Three Gorges Dam. Engineering Structures 2005;27: 501–13. [4] Lv XL, Zou Y, Lu WS. Shaking table model test on Shanghai World Financial Center Tower. Earthquake Engineering & Structural Dynamics 2007;36: 439–47. [5] Ou JP, Long X, Li QS, Xiao YQ. Vibration control of steel jacket offshore platform structures with damping isolation systems. Engineering Structures 2007;29: 1525–38. [6] Fan Z, Long Y. Linear analysis of tall buildings using spline elements. Engineering Structures 1991;13:27–33. [7] Takabatake H, Takesako R, Kobayashi M. Simplified analysis of doubly symmetric tube structures. Structural Design of Tall Buildings 1995;4:137–53. [8] Li QS, Cao H, Li GQ. Analysis of free vibrations of tall buildings. Journal of Engineering Mechanics, ASCE 1994;120(9):1861–76. [9] Li QS, Cao H, Li GQ. Static and dynamic analysis of straight bars with variable cross-section. Computers and Structures 1994;59:1185–91. [10] Li QS, Wu JR. Correlation of dynamic characteristics of a super tall building from full-scale measurements and numerical analysis with various finite element models. Earthquake Engineering & Structural Dynamics 2004;33: 1311–36. [11] Ventura CE, Schuster ND. Structural dynamic properties of a reinforced concrete high-rise building during construction. Canadian Journal of Civil Engineering 1996;23:950–72. [12] Chajes MJ, Zhang L, Kirby JT. Dynamic analysis of tall building using reducedorder continuum model. Journal of Structural Engineering, ASCE 1996;122: 1284–91. [13] Pan TC, Brownjohn JMW, You XT. Correlating measured and simulated dynamic responses of a tall building to long-distance earthquakes. Earthquake Engineering & Structural Dynamics 2004;33:611–32. [14] Brownjohn JMW, Pan TC, Deng XY. Correlating dynamic characteristics from field measurements and numerical analysis of a high-rise building. Earthquake Engineering & Structural Dynamics 2000;29:523–43. [15] Qi X, Chang KL, Tsai KC. Seismic design of eccentrically braced space frame. Journal of Structural Engineering, ASCE 1997;74(11):977–85. [16] Rahimian A, Romero EM. The tallest building in Mexico city Torre Mayor Mexico city, Mexico. In: Proceedings of the second international conference on advances in steel structures. 1999. p. 755–73. [17] Research Institute of Building & Construction. Report on the structural design scheme of Taipei 101. Evengreen Consulting Engineering, Inc., Taipei; 2003. [18] Guide of Operation and Meshing for ANSYS. The ANSYS Company; 2002. [19] Zhong ST. Concrete-filled steel tubular structure. Beijing: Tsinghua University Press; 2003. [20] Design code for concrete filled steel tubes (DL/T5085-1999). Beijing: Electric Power Publishing House; 1999. [21] China Ministry of Construction. Code for seismic design of buildings (GB50011-2001). Beijing: China Architecture & Building Press; 2003. [22] Clough WR, Penzien J. Dynamics of structures. New York: McGraw Hill; 1993. [23] Ministry of the Interior of the Republic of China. Building Technology Standards. Construction Bureau. Taipei.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Seismic analysis of the world’s tallest building Hong Fan a,b , Q.S. Li a,∗ , Alex Y. Tuan c , Lihua Xu d a
Department of Building and Construction, City University of Hong Kong, Hong Kong
b
China Nuclear Power Design Company, Shenzhen 518026, China
c
Department of Civil Engineering, Tam Kang University, Taipei, Taiwan
d
School of Civil Engineering, Wuhan University, Hubei Wuhan 430072, China
article
info
Article history: Received 13 March 2008 Accepted 8 October 2008 Keywords: Super-tall building Mega-frame structure Finite element modeling Seismic analysis Dynamics response Shaking table test
a b s t r a c t Taipei 101 (officially known as the Taipei Financial Center) with 101 stories and 508 m height, located in Taipei where earthquakes and strong typhoons are common occurrences, is currently the tallest building in the world. The great height of the building, the special geographic and environmental conditions, not surprisingly, presented one of the greatest challenges for structural engineers. In particular, its dynamic performance under earthquake or wind actions requires intensive research. The structure of the building is a mega-frame system composed of concrete filled steel tube (CFT) columns, steel brace core and belt trusses which are combined to resist vertical and lateral loads. In this study, a shaking table test was conducted to determine the constitutive relationships and finite element types for the CFT columns and steel members for establishing the finite element (FE) model of the tall building. Then, the seismic responses of the super-tall building were numerically investigated. An earthquake spectrum generated for Taipei Basin was adopted to calculate the lateral displacements and distributions of interior column forces. Furthermore, time-history analyses of elastic and inelastic seismic response were carried out using scaled accelerograms representing earthquake events with return periods of 50-year, 100-year, and 950-year, respectively. The computational results indicate that the super-tall building with the mega-frame system possesses substantial reserve strength, and the high-rise structure would satisfy the design requirements under severe seismic events. The output of this study is expected to be of considerable interest and practical use to professionals and researchers involved in the design of super-tall buildings. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Owing to the growing use of high-strength materials and advanced construction techniques, building structures have become more and more flexible and taller. The increasing height of modern tall buildings posed a series of challenges for structural engineers. In the design of such a tall building, the structural system must meet three major requirements: strength, rigidity, and stability [1]. As is well known, the strength requirement is the dominant factor in the design of low-rise structures. However, as building height increases, the rigidity and stability requirements become more important, and they are often the dominant factors in the structural design. Especially under lateral loads, interior forces are quite variable and increase rapidly with increases in height, and lateral deflection may vary as the fourth power of the height of a building [2], and structural dynamic behavior is thus one of the most important design considerations in the design of a modern tall building.
∗
Corresponding author. Tel.: +852 27844677; fax: +852 27887612. E-mail address: [email protected] (Q.S. Li).
0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.10.005
Taipei 101, rising 508 m above the city of Taipei, earns the title of the tallest building in the world. Its dynamic responses due to wind, earthquake and other extraordinary loads are of great concern. As Taiwan is located in one of the most active seismic regions in the world, this super-tall building may be susceptible to damage caused by strong earthquakes. These features make a detailed study on the structural performance of the world’s highest tall building under earthquake excitations of particular importance and necessity. Numerous investigations on seismic behavior of tall buildings have been carried out in the past; in particular shaking table tests play an important role in earthquake-resistant design of structures, analysis of seismic responses and failure mechanisms [3–5]. On the other hand, the finite element method (FEM) is a powerful tool for structural analysis of tall buildings. Fan and Long [6] adopted spline elements in the analysis of tall buildings. In their method, the element displacements are interpolated with spline functions and accurate results could be achieved with lower-order functions and a few degrees of freedom. Takebatake et al. [7] presented a simplified analytical method for the preliminary design of doubly symmetric single and double frame-tubes in high-rise structures by replacing a tube with an equivalent rod,
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
with consideration of the effect of bending, transverse shear deformation, shear-lag and torsion. Li et al. [8,9] proposed finite segment approaches for estimating the dynamic characteristics of tall buildings. Recently, Li and Wu [10] established seven 3D FE models for a 78-story super-tall building, and numerical results of the structural dynamic characteristics were compared with their field measurements to identify the FEM modeling errors for the purpose of updating the FEM models. Ventura and Schuster [11] presented a numerical study on estimation of dynamic characteristic of a 30-storey RC building. A reducedorder continuum model was proposed by Chajes et al. [12] to conduct dynamic analysis of a 47-storey steel-framed building and correlate the numerical results with those from measured responses during an earthquake. Pan et al. [13] and Brownjohn et al. [14] presented numerical studies on dynamic responses of the tallest building in Singapore with correlation with their field measurements. Qi et al. [15] employed the finite element method to study the seismic performance of a tall building and their results illustrated that the building is likely to perform satisfactorily under severe seismic events. Rahimian and Romero [16] established a finite element model to study the seismic response of the tallest building in Mexico City by time-history analysis and spectral method. These investigations indicated that numerical simulation is an effective tool to determine the dynamic characteristics and seismic responses of tall buildings. However, literature review reveals that comprehensive research studies on seismic effects on a super-tall building (building height > 500 m) have rarely been reported in the literature. So, a detailed analysis is presented in this paper to investigate the dynamic characteristics and seismic responses of the super-tall building. The objective of this study is to investigate the seismic effects on the world’s tallest building in order to provide valuable information for the design and construction of other similar structures in the future.
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Fig. 1. Elevation view of Taipei 101.
2. The structural system of Taipei 101 Taipei 101, a 508-m high office tower, is located at the east district of downtown Taipei City, and the elevation view of the building is shown in Fig. 1. The structure is symmetrical with a 62.4 m by 62.4 m square footprint [17]. Two sloping rectangular mega-columns with a maximum cross-sectional dimension of 2400 mm × 3000 mm, are positioned one at each side of the building extending to the 90th floor, and finally the cross-sectional dimension of the mega-column is reduced to 1600 mm × 2000 mm. All perimeter columns are sloping from the ground floor to 25th floor, and the sloping angle is 4.4. The core columns are square and rectangular concrete-filled steel tubes (CFT). The compressive strength of the concrete is 70 MPa, which provide extra stiffness and strength to the steel tubes, from bottom of the basement to the 62nd floor. The section of the core columns reduce from 1200 mm × 1200 mm to 900 mm × 900 mm, which are smaller than the mega-column. The composite metal deck-slabs built at each floor are 135 mm thick, however, those at mechanical floors are 200 mm thick. The primary girders are composed of H-steel beams with moment-resisting connections at the beam-column joints. Dog-bone connections are also provided at locations where ductility is required while beams are pinned to the primary girders, as shown in Fig. 2. Belt-trusses, one or two-stories high, are placed every 8-story interval at the perimeter frame, and the brace core is connected to mega-columns via belt-trusses consisting of in-floor braces and vertical trusses. The locations of the belt-trusses in the 8th floor are shown in Fig. 3 [17]. When the space of the column is 10.5 m, the shape of the steel braces are ‘‘V’’ or reverse ‘‘V’’, and when the space is 6 m, the shape of steel braces are acclivitous braces Fig. 4 presents an elevation view and the locations of the belttrusses in axes M9 and P1 [17]. The belt-trusses and mega-columns
Fig. 2. Sketch of the dog-bone connection.
help to stabilize the building core in the same way that the brace core helps to balance the perimeter frame. This mega-frame design maximizes the spaces inside the building and carries perimeter gravity loads at selected columns. The bracing, outriggers, and belts that link the columns would also redistribute loads if some members are damaged by unforeseen circumstances. The structure is a dual system: the external structure is composed of the megacolumns and external columns providing the lateral rigidity to the seismic and wind loads, and the internal structure, also designated as substructures, provides the utilizable space and allows for significant amount of energy dissipation. Belt trusses composed of a transfer floor system are placed at every eighth or tenth floor, so the interior columns only carry the gravity loads from a limited number of floors. As a result, their sizes are substantially smaller than those in a conventional structural system, in which they would rise from foundation level to the building top. A FE model was established in this study based on the design drawings of the super-tall building. The dead loads for building elements were determined by a commercial FE program ANSYS 10.0 [18] and the design live loads were calculated according to the data found from the design documents [17].
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Fig. 3. Plan of belt-trusses in the 8th floor.
Fig. 5. Stress condition for steel and concrete in a CFT column under axial compression.
analysis has become a routine design tool of tall buildings. Four kinds of elements are employed in establishing the FE model of Taipei 101 structure: 3-D beam elements, suitable for nonlinear large rotations and large strains, are employed to model the columns and beams. Link elements are used to model the brace. Mass elements are employed to model the live loads and nonstructural components. The floors are modeled with shell elements. The connection between the structure and its foundation is treated to be fixed. 3.2. Constitutive relationships of rectangular CFT columns Concrete-filled steel tube (CFT) columns are widely used due to their good earthquake resistant behavior such as improved strength and high ductility capacity. When a short CFT column is under an axial load, as shown in Fig. 4(b), there is a basic assumption that steel and concrete have the same longitudinal strain ε3 , then the hoop stains of steel ε1s and concrete ε1c can be calculated by:
ε1s = µs ε3 ,
Fig. 4. Elevation and the location of the belt-trusses.
3. Structural analysis 3.1. Finite element modeling With rapid development in computer technology and computational mechanics algorithms, three-dimensional finite element
ε1c = µc ε3
(1)
where µs , µc are the Poisson’s ratio of steel and concrete, respectively. Generally, at low stress conditions, concrete has a lower value of Poisson’s ratio than steel, which may result in occurrence of separation between the two materials in a CFT column. At high compressive stresses, internal micro-cracking in concrete causes it to swell. Its outwards movement is restrained by steel, and the strength of concrete is increased due to this lateral restraint. Thus the concrete and steel are stressed triaxially, as shown in Fig. 5(a). Zhong [19,20] proposed a unified theory to model CFT columns based on extensive experimental and FE analysis results of CFT columns under axial loading. According to the theory, a CFT column is regarded as a new composite column or material instead of separate components of concrete and steel. The properties of the composite column depend on those of steel and concrete and their dimensions (e.g., tube diameter and steel wall thickness). The ultimate strength and other property parameters of a CFT column can then be determined based on the mechanical and geometrical properties of the composite material. The following formulas for rectangular CFT columns are adopted in this study based on the unified theory [19,20]:
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
(a) Stress–strain curve for a concrete-filled steel tube.
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(b) Stress–strain curve for steel.
Fig. 6. Schematic graph of material constitutive relationships.
The yield strength of the composite column is fscy = (1.212 + Bξ + C ξ 2 )fck
(2)
where, B and C are coefficients. They depend on the cross-section geometry. For a rectangular cross-section, one has: B = 0.1381(fy /235) + 0.7646, C = −0.0727(fck /20) + 0.2016, where ξ is the confining factor which is expressed as
ξ=
fy A s fck Ac
in which fy , fck , As and Ac are the yield strength of steel, the unconfined strength of concrete and the areas of steel and concrete components in the column, respectively. The elastic modulus Esc of the composite column can be expressed as Esc = fscp /εscp
(3)
where fscp , εscp are the proportional stress and strain of the composite column, respectively. For a rectangular CFT column, one has fscp = [0.192(fy /235) + 0.488] fscy
(4)
εscp = 0.67fy / Es .
(5)
The tangent module of the composite column can be calculated as Esct
(fscy − σ¯ )σ¯ Esc = (fscy − fscp )fscp
(6)
where σ¯ = N /Asc , N is the axial load on the column and Asc is the total area of the column section. The hardening modulus of the composite column can be determined by 0 Esc = 400ξ − 150.
(7)
In this paper, the load–deformation (stress–strain) relation of a CFT column was determined based on experimental measurements from CFT columns under axial compressions [19], which was simplified as a tri-linear stress–strain model including proportional, yield and hardening stages, as shown in Fig. 6(a). The tangent module is substituted by the module of a straight line connected the proportional point and the yield point. According to Eqs. (2)–(5) and (7), the related parameters for a CFT column can thus be determined. For structural analysis of the steel beams and brace members in Taipei 101 building, a bilinear stress–strain curve with 2% postyield hardening (see Fig. 6(b)) was adopted to model the inelastic behavior of these structural members, with Young’s modulus of 420 MPa and Poisson’s ratio of 0.3, respectively. Von Mises yield criterion with kinematic hardening rule was employed in the numerical analysis.
Fig. 7. Test structure.
3.3. Verification of the constitutive relationships of CFT columns For verification of the adequacy of the constitutive relationships of CFT columns and steel members discussed above as well as the selected finite element types for modeling the structural members of Taipei 101 structural system, a shaking table test and the associated FEM analysis were conducted in this study for a frame structure model composed of rectangular CFT columns and steel members by comparing the numerical results with the experimental data. The test model and its finite element model are shown in Fig. 7 and 8, respectively. The scaled model was tested on the shaking table adopting three representative earthquake records as inputs: (1) An artificial seismic accelerogram (made according to the design code of China [21]); (2) El-Centro earthquake record; and (3) Tianjin earthquake record. The peak ground accelerations (PGA) in the three accelerogams were scaled to 0.05g and 0.1g to represent the design earthquake actions with intensity 6 and 7 degree as stipulated in the design code of China [21], respectively. The design code [21] classifies regions of different seismicity in terms of seismic intensity which is usually regarded to be an equivalent of peak ground acceleration. Table 1 shows the relationship between the seismic intensity and peak ground acceleration [21]. Each accelerogram duration was reduced to 1/5 of its original duration according to the scale factor listed in Table 2. Table 3 lists the first four natural frequencies of the model obtained from the test and the numerical analysis of the FE model. Furthermore, acceleration dynamic amplification factors of the
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modeling strategies presented above in the establishment of the FE model of Taipei 101 structural system. 4. Dynamic characteristics of the super-tall building
Fig. 8. FE model.
Table 1 Relation between the seismic intensity and horizontal peak ground acceleration [20]. Seismic intensity (degree)
Horizontal peak ground acceleration (g)
6 7 8 9
0.05 0.1 0.2 0.4
A three-dimensional FE model of Taipei 101 structural system was established for numerical analysis of the super-tall building, as shown in Fig. 10, based on the constitutive relationships for rectangular CFT columns and steel members as well as the selected finite element types which were verified above. The FE model of the super-tall building contains 20 532 beam elements, 24 048 shell elements, and 3496 link elements. In addition to the main structural elements, non-structural components were modeled with mass elements. Fig. 11 shows the first six mode shapes of the FE model including two for translational motions in each horizontal direction and two for torsional motions about the vertical axis. Modes 1 and 2 are the translational modes in x and y directions, respectively. Mode 3 is the fundamental torsional mode. The fundamental periods of the building are 6.21 s. in the x direction, 6.19 s in the y direction, and 3.62 s in torsion. The modal participation coefficient for each mode [22] is defined as [Ei ][M][uj ] ηji = , which shows the modes that contribute most T [uj ] [M][uj ] to the dynamic response, where [M ] is the mass matrices, [u] is the vibration mode vector of the structure, and [E ] is the uniform matrix. The model participation ratio in every direction for each mode is defined as ϕic = P
ηic
j=1
ηjc
(c = x, y, z), while the cumulative Pi
mc
j=1 modal participation mass ratios is defined as ψic = Pn mjc (c = t =1
t
Physical parameters
Scale factors
Ratio (model/prototype)
Length Density Elastic modulus Strain
Sl Sρ SE Sε
1/5 1.0 1.0 1.0
Time
St = Sl Sρ2 SE
1/5
x, y, z), where m is the effective mass participating the dynamic response of each mode. Table 4 shows the modal participation ratios and the cumulative modal participation mass ratios of the first 30 modes. The vibration modal participation ratios for the first two modes in the x and y directions reach 1. Such ratios for other modes decrease as the mode number increases. The cumulative modal participation mass ratios for the first 30 modes reach 1 in the horizontal directions. Therefore, it was decided to use the first 30 modes in the response spectrum analysis of the tall building, which will be described below.
Sf = Su = Sl Sa = Sl /St2
5 1/5 5
5. Response spectrum analysis
Table 2 The similarity relationship between the model and the prototype structure.
1
− 12
−1 1 Sl−1 Sρ 2 SE2
Frequency Displacement Acceleration
Table 3 Natural frequencies of the scaled mode obtained by the shaking table test and the numerical analysis. Mode
1
2
3
4
Test (Hz) FE analysis (Hz) Differences (%)
17.48 16.50 −5.61
26.68 29.43 10.27
40.00 54.44 36.1
72.50 58.14 −19.8
structural model under seismic excitation with PGA of 0.1g, which were determined by the test and the numerical computation, are shown in Fig. 9. It is shown in Table 3 that the fundamental natural frequencies of the model obtained from the test and the numerical analysis agree fairly well. It is observed from Fig. 9 that the two sets of the results of acceleration dynamic amplification factors under the three earthquake records were also in good agreement. It is thus expected that the constitutive relationships for rectangular CFT columns and steel members as well as the selected finite element types for modeling these structural components are adequate, since the numerical results match the experimental data reasonably well. Hence, it was decided to adopt the numerical
For earthquake-resistant designs, a structure should meet performance requirements at two different levels, depending upon the magnitude of earthquake actions. The first level of performance essentially requires structural response in the elastic range without significant structural damage under a moderate earthquake action, and the second level of performance requires that the structure does not collapse under a severe earthquake event with rare occurrence. Taipei 101 is located in the Taipei Basin, where there are deep and soft soil deposits with a long predominant period. According to the Taiwanese Building Technology Standards (BTS) [23], the building site is in a region of moderate seismicity. A response spectrum, originally developed by the National Taiwan University and other research institutions for the design of Taipei 101, has been modified based on the response spectra stipulated in the local seismic code (BST), as shown in Fig. 12 [17]. The first 30 translational–torsional coupling modes were used to calculate the seismic response of the super-tall building by the Complete Quadratic Combination (CQC) method on the basis of the response spectrum analysis. The damping ratio of the high-rise structure was assumed to be 5% for all the 30 modes, and the seismic responses in the two orthogonal directions were calculated accordingly.
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Table 4 Modal participation ratios and participation mass ratios. Mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Period (s)
6.21 6.19 3.62 2.52 2.47 1.65 1.64 1.42 1.41 1.37 1.166 1.161 0.929 0.922 0.898 0.740 0.733 0.632 0.624 0.576 0.557 0.532 0.489 0.394 0.379 0.299 0.285 0.244 0.235 0.153
(a) Experimental results.
Model participation ratio
Cumulative participation mass ratio
x direction
y direction
x direction
y direction
0.000704 1 0.000002 0.001392 0.778882 0.000383 0.414047 0.000099 0.000036 0.000007 0.000044 0.000087 0.12753 0.000068 0.000011 0.07398 0.000006 0.000015 0.054531 0.000006 0.000003 0.000003 0.023607 0.000018 0.000077 0.04295 0.000006 0.001874 0.000001 0.015782
1 0.000692 0.000359 0.775207 0.001382 0.420086 0.000395 0.000007 0.000004 0.000002 0.132226 0.000133 0.000053 0.000704 0.075824 0.000019 0 0.056465 0.000022 0.000006 0.000001 0.000001 0.000021 0.025123 0.042725 0.000079 0.000007 0.000001 0.002004 0.000006
2.70E-07 0.545056 0.545056 0.545057 0.875625 0.875625 0.96938 0.96938 0.96938 0.96938 0.96938 0.96938 0.985388 0.985388 0.985388 0.991815 0.991815 0.991815 0.995633 0.995633 0.995633 0.995633 0.996521 0.996521 0.996521 0.999566 0.999566 0.999572 0.999572 1
0.544991 0.544991 0.544991 0.872428 0.872429 0.968316 0.968317 0.968317 0.968317 0.968317 0.985225 0.985225 0.985225 0.985225 0.991897 0.991897 0.991897 0.995977 0.995977 0.995977 0.995977 0.995977 0.995977 0.996983 0.999993 0.999993 0.999993 0.999993 1 1
(b) Numerical results.
Fig. 9. Dynamic amplification factors from the test and numerical analysis.
5.1. Displacement analysis The results of the response spectrum analysis indicate that the deformations in the x direction are approximately equal to those in the y direction (see Fig. 13). This can be attributed to the symmetric structural system and shape. The maximum interstory drift ratios in the x direction and y direction are 1/281.7 and 1/261.1, respectively, both under the criteria stipulated in the local design code (BST). Thus, the structure can be regarded to satisfy the first level performance requirement. Due to the two outrigger trusses forming a cell at every eight or ten stories, the inter-story drift ratio of each cell is in the shape of an arc, as shown in Fig. 14. The drift ratios are small in the top and bottom stories and large in the middle inter-stories. Such an outrigger truss functions like a ring, which controls the lateral structural deformation. The storey
Fig. 10. Finite element model.
displacement curves take the shape of a deforming cantilever and are relatively smooth without obvious inflexions, which imply that the distribution of equivalent rigidities along the height of the structure is well proportioned. 5.2. Interior force analysis Three representative columns were selected to analyze the interior forces in order to study the structural deformation mechanisms. The selected columns are the exterior corner column
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Fig. 11. Mode shapes of the first six modes obtained from numerical analysis. Fig. 13. Lateral deformation curves.
Fig. 12. Design response spectrum.
C9, exterior side column C1, and interior corner column C5, as shown in Fig. 3. It can be seen from Fig. 15 that the axial force is the dominant interior force in each column and other interior forces are relatively small. The mega-frame is a dual system, where the steel frames are composed of the mega-columns and perimeter columns along each sloping face of the building, working in parallel with the braced core which is the secondary resistant system. Therefore, the exterior side column C1 carries the heaviest vertical loads on its huge section. Since the structure is symmetrical, the torsional moment in each column is small. C9 and C1 are sloping below the 27th floor, and the torsional moments are great in some lower floors. Abrupt increases in shear forces and bending moment in these columns are observed near the outrigger belts, especially near the 26th floor and 27th floor. This is because the stiffness of the outrigger belt is generally much larger than that between adjoining stories. Consequently, there tends to be a weak story near an outrigger belt. This issue should be adequately dealt with in earthquake-resistant design of outrigger-braced tall buildings.
Fig. 14. Inter-story drift ratios curves.
materials, a mega-frame provides lateral load resistance to prevent instability of the global structure under compressive loads and moments (i.e., p–∆ effect). As shown in Fig. 16, the MEI curves are wavy with discontinuities near the outrigger belts. Since the outrigger belts transfer large lateral forces to the columns under lateral loading, the flexural deformations of the columns are increased and the percentage of bending stress are also increased, leading to a reduction of MEI.
5.3. Material efficiency index 6. Time history analysis Material efficiency index (MEI) is usually used to estimate the efficiency of concrete-filled steel tubular columns, which is defined as the ratio of average axial stress to the maximum principal stress, MEI = σn /σmax × 100%. Values of the MEI of the three representative columns are listed in Table 5 and discussed below. The exterior corner column C9 and exterior side column C1, as shown as Fig. 3, are the main load-resistant components, and the values of MEI of the two columns are above 80%. Due to a large cross-section, a mega-column provides stiffness to resist lateral forces and reduce lateral deformation. Due to efficient utilization of
The response spectrum analysis conducted previously only considered the maximum amplitude of seismic effect, but dynamic analysis in the time domain can provide more information for earthquake-resistant design of structures. The latter approach consists of a step-by-step direct integration in which the time domain is discretized into a large number of small increments, and for each time interval the equations of motion are solved to obtain the structural responses such as displacements. Four recorded accelerograms were selected as inputs in the present
H. Fan et al. / Journal of Constructional Steel Research 65 (2009) 1206–1215
(a) Axial force.
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(b) Shear force.
(c) Bending-moment.
(d) Torsional-moment.
Fig. 15. Interior forces of the columns.
Table 5 MEI of the columns. Floor
C1
C5
C9
Floor
C1
C5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.81 0.84 0.90 0.94 0.98 0.99 0.98 0.98 0.95 0.95 0.96 0.98 0.99 0.97 0.96 0.95 0.95 0.97 0.96 0.96 0.98 0.97 0.94 0.93 0.91 0.92 0.78 0.81 0.85 0.99
0.59 0.21 0.30 0.43 0.67 0.76 0.63 0.68 0.41 0.42 0.48 0.61 0.71 0.56 0.43 0.37 0.39 0.59 0.50 0.53 0.63 0.57 0.42 0.35 0.33 0.44 0.21 0.29 0.35 0.59
0.65 0.72 0.84 0.89 0.95 0.95 0.82 0.98 0.78 0.87 0.94 0.96 0.96 0.95 0.92 0.87 0.79 0.96 0.81 0.88 0.92 0.92 0.91 0.87 0.65 0.60
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
0.90 0.95 0.97 0.90 0.87 0.77 0.82 0.85 0.91 0.96 0.96 0.88 0.85 0.75 0.81 0.85 0.91 0.97 0.95 0.87 0.83 0.73 0.79 0.84 0.91 0.96 0.95 0.86 0.81 0.72
0.45 0.65 0.70 0.45 0.32 0.22 0.27 0.38 0.51 0.73 0.69 0.47 0.34 0.24 0.31 0.43 0.57 0.78 0.70 0.49 0.32 0.23 0.30 0.43 0.56 0.75 0.70 0.46 0.30 0.23
time-history analysis: the El-Centro (N-S ), Taft (E-W ), Chi-Chi and Taiwan1115. Each accelerogram is composed of two horizontal and one vertical components of ground acceleration excitation. For each accelerogram, the magnitude of the ground acceleration was scaled. The scaled ground accelerations used in the present analysis have the peak ground accelerations of 70 gal, 195 gal and 390
C9
Floor
C1
C5
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
0.82 0.91 0.97 0.98 0.99 0.95 0.90 0.86 0.93 0.96 0.97 0.98 0.98 0.95 0.88 0.79 0.92 0.97 0.98 0.98 0.97 0.93 0.86 0.67 0.90 0.93 0.93 0.92 0.91 0.84
0.37 0.58 0.79 0.84 0.88 0.67 0.34 0.33 0.51 0.70 0.75 0.77 0.74 0.45 0.11 0.13 0.34 0.76 0.75 0.65 0.41 0.23 0.15 0.07 0.27 0.55 0.38 0.26 0.49 0.55
C9
gal, representing an earthquake event with 50-year, 100-year and 950-year return period, respectively. The maximum lateral displacement and inter-story drift ratio of the structure under the four recorded accelerograms are listed in Table 6. It is illustrated that the structure still remains elastic after the action of the earthquake excitation with 50-year return
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Table 6 Maximum displacement and inter-story drift radio. Earthquake action
50-yr return period 100-yr return period 950-yr return period
Maximum displacement/mm
∆/H
x
y
x
y
x
y
286 697 1127
357 851 1320
1/1491 1/613 1/379
1/1196 1/502 1/324
1/656 1/224 1/136
1/525 1/182 1/119
Maximum inter-story drift ratio
Fig. 16. MEI of the columns.
Fig. 18. Displacement envelope curves for the four records with 950-year return period.
return period. The dynamic magnification factors of the middle floors are close to 1, which suggest that the equivalent rigidities along the height of the structure is distributed well proportionally. Since the base of the finite element model was assumed to be rigid, there is a discontinuity between the first floor and the ground. The floor area from the storeys 91st to 101st is smaller than that of a standard floor, so the rigidities of these floors are smaller, resulting in larger dynamic amplification factors. Fig. 17. Displacement envelope curves for Taiwan1115 record.
7. Conclusions period and satisfies the first lever performance requirement. It is also revealed that the structure deforms plastically under seismic action with a 950-year return period; however, the maximum inter-story drift ratio is 1/119, which is less than the corresponding criteria (1/70) stipulated in the local design code (BTS) [23]. Fig. 17 shows the curves of the displacement envelope in the y direction under the excitation of Taiwan1115 record with different return periods, it is noted that under the seismic action with 70 gal peak ground acceleration, the deformation curve is smooth without inflexions, similar to a deforming cantilever, indicating that the structure remains in an elastic stage. When the peak ground acceleration is increased to 390 gal, the curve has an inflexion, which implies that some structural components have deformed plastically. The curves of the displacement envelope in the y direction subjected to the four accelerograms with a 950-year return period are shown in Fig. 18 and the seismic responses of the lateral displacement at the top storey are shown in Fig. 19. Since the spectral compositions of the four earthquake records are different, there are differences in the curves of timehistory responses of the lateral displacements. However, it is observed from Fig. 18 that the curves of the displacement envelope have a similar shape, showing that the mega-frame has a shear deformation mode. Fig. 20 plots the dynamic amplification factors for the case of earthquake events with a 50-year return period and a 950-year
A detailed study on the dynamic characteristics and seismic responses of Taipei 101, the world’s tallest building, was presented in this paper. The constitutive relationships for rectangular CFT columns were established based on the unified theory, and then were verified through comparison between the shaking table test data and numerical analysis results. A 3-D finite element model of Taipei 101 structure was established based on the verified constitutive relationships for the rectangular CFT columns and selected finite element types for the structural members. The seismic analysis results of the super-tall building indicated that the structural system, with belt trusses at every eighth or tenth story, provides equal stiffness along the height of the building, which can decrease the lateral deformation efficiently. Meanwhile, for such a mega-frame structural system with a central braced core connected to perimeter columns on each building face, the total dead and live loads at every floor are transferred to the sloping exterior columns, thereby the structural capacity to withstand lateral loading is enhanced. The results of this research also showed that Taipei 101 has relatively high earthquake resistance and could guarantee the structural safety under a seismic action with a moderate seismic fortification, as stipulated in the local seismic design code. However, it was revealed that there are abrupt changes in the shear force in the columns near the floors with outrigger belts. This issue should be adequately dealt with in the earthquake-resistant design of high-rise structures of this type.
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Fig. 19. Time-history responses of the top storey lateral displacement for the four records with a 950-year return period.
(a) The results with a 50-year return period.
(b) The results with a 950-year return period.
Fig. 20. Dynamic magnification factors.
Acknowledgement The work described in this paper was fully supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project No: CityU 116906). Appendix The tested model with 0.58 m storey height is a 1/5 scale model of a 5-storey steel frame. The cross-sections of the columns are 60 mm × 60 mm × 2 mm, and the cross-sections of I-shape steel beams are 40 mm × 60 mm × 2 mm × 2 mm. The steel tube columns were filled with concrete, and the concrete compressive strength is 40 MPa. Braces were used in the perimeter bays parallel with the direction of earthquake excitation. The model, instrumented with an accelerometer and a displacement-meter at the basement and each floor, was bolted to the shaking table (shown in Fig. 6). References [1] Bungale ST. Structural analysis and design of tall building. New York: The William Byrd Press; 1988.
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