Seismic analysis of a super high-rise steel structure with horizontal strengthened storeys

Abstract

Horizontal strengthened storeys are widely used in super high-rise steel structures to improve the lateral structural rigidity. This use has great effects on the seismic properties of the entire structure. The seismic properties of the Wuhan International Securities Building (a 68-storey super high-rise steel structure with three horizontal strengthened storeys) were evaluated in this study. Two approaches, i.e., mode-superposition response spectrum analysis and time-history analysis, were employed to calculate the seismic response of the structure. The response spectrum analysis indicated that transition parts near the three strengthened storeys were weak zones of the structure because of the abrupt change in rigidity. In the response spectrum analysis approach, the Square Root of Sum of Square (SRSS) method was recommended when the vertical seismic effects could be ignored. However, the complete quadratic combination (CQC) method was superior to SRSS method when the vertical seismic effects should be considered. With the aid of time-history analysis, the seismic responses of the structure were obtained. The whiplash effect that spectrum analysis cannot reveal was observed through time-history analysis. This study provides references for the seismic design of super high-rise steel structures with horizontal strengthened storeys.

Keywords

seismic analysis / steel structure / super high-rise / horizontal strengthened storey / response spectrum analysis / time-history analysis

Introduction

The lateral load, such as earthquake and wind loading, becomes the main control factor for high-rise and super high-rise structures. Compared with concrete structure, the lateral rigidity of steel structure is smaller. An effective approach to reduce lateral deformations is to install several horizontal strengthened stories along the height of the building [1]. In China, the horizontal strengthened storey was initially applied in steel structures and then popularized in outer steel frame, inner concrete tube structures. In recent years, the number of high-rise and super high-rise structures with steel-braced frames or steel frame-tubes has increased. The domestic high-rise structures with horizontal strengthened stories are listed in Table 1.

In general, if the horizontal strengthened storey is set appropriately in the super high-rise structure, Thedisplacement of the top floor can be reduced by 10%—32% compared with that of the structure without a horizontal strengthened storey. Thus, the horizontal strengthened storey can remarkably improve the structural rigidity. Until now, research on horizontal strengthened storeys has been limited to concrete structures [2], and little work has been conducted for steel structures whose inner tubes are composed of steel-braced frames. Investigations were limited in the optimization of horizontal strengthened storey quantity and locations [3,4]. Further studies (such as transition paths for internal forces and stress distributions near the weak floors) are still necessary for braced frame structures with horizontal strengthened storeys. In a previous study [5], a three-dimensional model of a 32-storey building was used to analyze location effects of horizontal strengthened stories on the seismic response of the structure. The results indicated that three strengthened stories were sufficient for ordinary high-rise buildings, while special attention should further be paid to the weak floors in the vicinity of the strengthened storeys.

The seismic properties of the Wuhan International Securities Building, which is a 68-storey super high-rise steel structure with three horizontal strengthened storeys, were investigated in this study using finite element software ANSYS [6]. The technical problems of the modeling in the response spectrum analysis and time-history analysis of super high-rise steel structures are discussed and solved. This paper provides valuable references for seismic analyses and design of super high-rise steel structures with horizontal strengthened storeys.

Seismic analysis of Wuhan International Securities Building

Introduction of the project

The Wuhan International Securities Building is the tallest building in the central area of China. The height of the major structure is 281.3 m. A 50 meters communication antenna is set in the top of the building, and thus, the maximum building height reaches 331.3 m. The building includes three storeys underground and 68 storeys over ground. The typical plane view and elevation view are shown in Fig. 1.

The project was originally a 48-storey reinforced concrete building with a tube-in-tube structure system. When the concrete structure reached the 7th floor, the building was adjusted to a 68-storey braced-frame steel structure according to the owner’s request. To satisfy the design index of the high-rise steel structure, transition stories from the 8th to the 12th floor were installed on the constructed concrete structure. Above the 12th storey, a steel structure was adopted. Concrete-filled steel tubular columns were applied in out-frames, while steel columns with a centrically and eccentrically brace system were used in the core tube. Composite slabs with profiled steel sheets were also used in this building.

The 1st to the 40th storey of the building consists of offices (the height of a standard storey is 3.6 m), while the 41st to the 67th storey consists of hotel rooms (the height of a standard storey is 3.2 m); the top floor is for sightseeing. Three horizontal strengthened storeys are installed in the 25th, the 43rd and the 65th storey. These strengthened storeys, together with two transition sections (i.e., one from the concrete to steel structure and the other from the offices to the hotel), are weak zones of the structure because the structural rigidity changes suddenly in these areas. Commissioned by the design institute, the authors have analyzed the seismic properties of the structure and observed the responses of the horizontal strengthened storeys under earthquake loadings. The design parameters are referred to the Chinese standard GB 50011–2010 [7].

The geometrical sizes of cross sections of representative columns and beams are listed in Table 2. The locations of box columns Z-1, Z-2 and Z-3 in Table 2 are indicated in Fig. 1(a). The load parameters can refer to the Ref. [8,9]. The seismic fortification intensity is seven, and the seismic fortification criterion of the building is at level C. The security of the building is at the highest level, and construction site is class II. The basic assumptions are listed as follows:

1) The structure material is simplified as a perfect elastoplastic model.

2) Because the basement floor is a box foundation with high stiffness and the restraint effects of ground are very strong, the boundary conditions between the ground and basement floor can be regarded as fixed.

3) The concrete-filled steel tubular components and the steel-reinforced concrete components are simplified according the equivalent principle of rigidity.

4) The concrete slabs above and below the strengthened stories are considered elastic and calculated as thin shells, and the other slabs are assumed to be infinitely rigid in the plane.

Calculation and analysis

The finite element software ANSYS [6] was used in the analysis of the dynamic properties and seismic responses of the structure. According to the structural characteristics and the simulation precision, three element types (i.e., Beam188 for columns and beams, Shell181 for slab and Mass21 for loads) were selected in the model. The 3D model includes a total of 71294 elements and 48741 nodes. The dynamic properties were obtained by modal analysis, while the seismic responses were evaluated by response spectrum analysis and time-history analysis. The Lanczos algorithm was selected to solve the eigenvalue problem, and the Newmark iteration method was chosen in the time-history analysis. The structural damping was simplified as Rayleigh damping [10,11]. The results obtained by finite element analysis are shown as follows.

Modal analysis

As the effect of non-structural components, the natural vibration period of the structure must be modified. Referring to Refs. [12,13], the modification coefficient was determined to be 0.70. The natural vibration periods of the first 30 modes are listed in Table 3. The basic natural vibration period is 8.0767 s, which indicates that the structure rigidity is low. The vibration period ratio of torsion mode to translation mode is 0.46 (i.e., 3.7515/8.0767), which reflects the fact that the torsion effect is not very significant.

Deformation analysis

The storey drifts obtained by the mode-superposition response spectrum analysis of the structure under unidirectional seismic loading are shown in Fig. 2. In the locations of the strengthened storeys, lateral deformations and storey drifts obviously decrease.

The El-Centro wave, Taft wave and two artificial waves were selected for time-history analysis in the X and Y directions, respectively. The storey lateral displacement envelops under four seismic loadings are shown in Fig. 3. The results are similar in terms of structure characteristics with those obtained by response spectrum analysis. In addition, they can reflect seismic properties that response spectrum analysis cannot reveal, such as abnormal displacement in local places and the whiplash effect on the top floor.

The magnification factor curves of storey lateral acceleration under four seismic loadings are shown in Fig. 4. The structure acceleration is largest in the location from the 8th to the 15th floor, where the building changes from a concrete structure to a steel structure. Figure 4 indicates that this location is the weak zone of the structure, and special construction measures should be used in these parts.

Internal force analysis

The minimum allowed ratio of shearing force to self weight is 1.2% according to the China Seismic Code [7]. The base shearing force is 20641 kN calculated by the software SATWE, which agrees quite well with the result obtained by ANSYS. The results obtained by response spectrum analysis under unidirectional, bidirectional and three-directional seismic loadings are listed in Table 4. As shown in Table 4, the base shearing force under the X directional earthquake is comparable to that under the Y directional earthquake, which reflects that the rigidities in the two main axes of the structure are approximate and that the seismic responses for the X and Y directional earthquakes are comparable. The base shearing force coefficients are in a reasonable range, indicating a good design case. There is an approximate 42% difference in the base shearing force between the unidirectional and bidirectional earthquake, while only 9% difference between the bidirectional and three-directional earthquake, which indicates that the vertical earthquake effect is quite small.

From the previous analysis of a 32-storey model [5], floors near the strengthened storeys are considered to be the weak zones of the structure. Therefore, more attention was paid to the three strengthened storeys and the adjacent storeys of the Wuhan International Securities Building. Three groups of columns (i.e., columns Z-1, Z-2 and Z-3 shown in Fig. 1), which are located in outer-frame and inner-frame, were selected for discussion. The variations of the internal forces with the column height are shown in Fig. 5.

For column Z-1, the maximal axial force occurs on the base floor, while the maximum shearing force occurs on the 25th floor, where the strengthened storey is located. As for column Z-2, the maximum axial force occurs on the 24th storey, the maximum shearing force occurs on the 25th storey and the maximum moment occurs on the 26th storey. From the above results, conclusions can be drawn that the column of the outer-frame in the location of the 25th storey is the critical component. Stresses concentrate in this part due to the sudden change in structural rigidity. For column Z-3, the maximum axial force is on the 15th storey, and the maximum shearing force and moment simultaneously occur on the 43rd storey, where the strengthened storey is located. The variation of internal forces with the column locations shows that the effect of the strengthened storey on the seismic properties of the structure differs to some extent due to its different mechanisms.

The seismic fortification intensity in Wuhan is six degrees, which is relatively lower than average level in China. By the structure calculation, it is known that the wind load plays a controlling role in the structure design [9]. The strengthened storeys are designed to be used in the concept of “limited rigidity” because they can reduce the lateral deformation effectively and the earthquake is not strong, either. Though the stresses on the strengthened storeys are large, they are still within the allowed range.

Technical points in the response spectrum analysis of a super high-rise steel structure

The damage mechanism of earthquakes to high-rise structures is complicated, making seismic analysis very important for the design of super high-rise buildings. In the early 1940s, Biot of USA first put forward the concept of the earthquake response spectrum. At the beginning of 1950s, Housner set up the method of response spectrum analysis. This method can reflect the seismic properties precisely and clearly. The dynamic properties of the structure itself and the earthquake ground motion are considered together. Some practical parameters are offered by observing the records of the strong earthquakes. Because of these advantages, the response spectrum analysis method has won international recognition and plays a leading role in the field of seismic design [14].

There are still some key factors, such as the analysis model, reasonable vibration modes and suitable summation method, which should be considered in the response spectrum analysis of super high-rise steel structures. The factors mentioned above will be discussed according to the seismic analysis of the Wuhan International Securities Building.

Analysis model

According to the importance and complexity of the structure and the requirements of result precision, a 3D model was selected for analysis. Using modal analysis, some representative deformations of the vibration modes are shown in Fig. 6. The basic modes are primarily in horizontal deformation and torsion deformation, while the high modes reflect vertical deformation, but not as much. The structural modes show that a complicated super high-rise structure like the Wuhan International Securities Building should be simulated by a 3D model because a simple model, such as the equivalent mass model or plane model, cannot reflect the true dynamic properties.

Number of reasonable vibration modes

Calculation method

According to the China Seismic Code [7], the number of reasonable vibration modes included in the response spectrum analysis is on the order of 2~3. For a building whose basic natural vibration period is greater than 1.5 s or whose height to width ratio is greater than 5.0, the number of modes included in the analysis should be increased accordingly. This topic has been discussed in China by several experts [15]. The ETABS program in the USA proposed an effective modal mass approach, which is a quantitative method and is employed extensively all over the world. The definition of effective modal mass is shown in Eq. (1).

(1)M_{ej}={\gamma }_j^2\cdot {\left\{\Phi \right\}}_j^{\text{T}}\left[M\right]{\left\{\Phi \right\}}_j,

where γ_j is the jth modal participation factor, {Φ}_j is the eigenvector for vibration mode j and [M] is the mass matrix of the system.

Modal participation factor

The modal participation factors of the structure in the X, Y and Z directions for each mode are shown in Fig. 7. In the X and Y directions, the lower vibration modes play the predominant role. In the Z direction, the 12th and 16th vibration modes play the controlling role.

Excitation effect on structural effective mass

The expression for the jth modal participation factor is shown in Eq. (2). Here, the cumulative effective mass fraction of the structure for different earthquake excitations is calculated according to Eq. (3), where M_ej can be calculated by Eq. (1) and M_T is the total actual mass of the structure. The cumulative effective mass fractions of the first 40 modes are shown in Fig. 8.

(2){\gamma }_j=\frac{{\left\{\Phi \right\}}_j^{\text{T}}\left[M\right]}{{\left\{\Phi \right\}}_j^{\text{T}}\left[M\right]{\left\{\Phi \right\}}_j},

(3)\alpha ={\displaystyle \sum _1^j{M}_{ej}}/M_{\text{T}}.

According to the seismic code [7], the number of vibration modes included in the analysis should be increased accordingly for tall buildings whose rigidities are small. Perhaps the number should be sufficient so that the total effective modal mass of the structure is at least 90% of the actual mass. The smallest values for the summation modes satisfying the above rule for different earthquake excitations are shown in Fig. 8 and listed in Table 5.

For structures whose layout is uniform such as the Wuhan International Securities Building (torsion effect is relatively small), few summation modes are required to satisfy the seismic code. When using the first 20 modes for calculation, the cumulative effective mass fraction is already greater than 90%, and thus, seismic analysis can be performed accurately with only the first 20 modes. However, compared with the general high-rise steel structure with a smaller size and lower complexity, the super high-rise steel structure still requires more summation modes.

Results of different summation mode numbers

Here, a three-directional earthquake excitation is taken as an example to discuss the effect of the number of summation modes on the calculation results. In a three-directional earthquake, the cumulative mass fraction of five summation modes (including the 1st, 2nd, 4th, 5th and 16th modes) is already over 88.9%. The response of the structure in the three-directional earthquake is computed, and the base shearing force is listed in Table 6. The structure displacements in the X and Z directions as shown in Fig. 9 visually reflect the effect of the number of summation modes on the analysis results.

Figure 9(a) indicates that the number of summation modes has little effect on the structural lateral displacements because the first ten modes already contain the principle mode of lateral vibration. As shown in Fig. 9(b), the principle mode of vertical vibration occurs after the 10th mode, so the number of summation modes has a great effect on the vertical displacements. The difference between the 20 summation modes and 30 summation modes is quite small, which agrees well with the result obtained by effective modal mass analysis, as shown in Fig. 8. Figure 9(b) also shows that the number of summation modes has a lower effect on the vertical displacement of the bottom structure compared to that of the top structure.

Modal summation method

When the mode-superposition response spectrum analysis method is used, the response of each vibration mode is combined according to a certain summation method to form the response of the entire structure. For the buildings such as the super high-rise steel structure with remarkable space effect, different modal summation methods may lead to different calculation results.

Introduction of the summation method

The main process of the response spectrum method is described as follows. First, simplify the structure into a discrete multi-degree of freedom system; then, dissociate this system to the summation of several single degree of freedom systems; third, calculate the maximum seismic response of each single degree of freedom system by the design response spectrum method; finally, combine the maximum response to form the response of the entire structure.

The result obtained by following these steps is the maximum seismic response of a certain vibration mode. Because the maximum response of each mode does not occur simultaneously, the real seismic response is not the simple superposition of the maximum response of each mode but the superposition of seismic responses of some principle modes by certain summation methods. Presently, there are two modal summation methods, i.e., the square root of sum of square (SRSS) method and complete quadratic combination (CQC) method, which are widely used in China [7].

The SRSS method is based on the simplification that the seismic wave is a stable random process and the response of each mode is independent of each other. Thus, this method is only suitable for the situation in which the vibration period of each mode is far apart. The relationship between the entire structure response and the response of each mode can be expressed as follows [12]:

(4)S=\sqrt{{\displaystyle \sum _{j=\mathrm{1}}^m{S}_j^{\mathrm{2}}}},

where S and S_j are the effects of the characteristic value in horizontal earthquake action corresponding to the entire structure and the jth vibration mode.

For the combined vibration of translation and torsion for asymmetric high-rise buildings, some vibration periods may be very close due to the intervention of the torsion effect. Therefore, another modal summation method, called CQC, for asymmetric structures is superior to the SRSS method for considering the correlations between each vibration mode. The torsion effect of the structure in a unidirectional earthquake can be evaluated by Eqs. (5) and (6) [12].

(5)S=\sqrt{{\displaystyle \sum _{j=\mathrm{1}}^m{\displaystyle \sum _{k=\mathrm{1}}^m{\rho }_{jk}{S}_j{S}_k}}},

(6){\rho }_{jk}=\frac{8{\xi }_j{\xi }_k\left(1+{\lambda }_{\text{T}}\right){\lambda }_T^{1.5}}{{\left(1-{\lambda }_T^2\right)}^2+4{\xi }_j{\xi }_k{\left(1+{\lambda }_{\text{T}}\right)}^2{\lambda }_{\text{T}}},

where S_j and S_k are the effects of the characteristic value of mode j and k in an earthquake, ρ_jk is the coupling coefficient of modes j and k, ξ_j and ξ_k are the damping ratio of modes j and k, and λ_T is the ratio of the vibration period of mode k to that of mode j.

Calculation of vibration period

The effects of two summation methods on the analysis results are discussed according to the calculation of the Wuhan International Securities Building. Indicated by Eq. (6), the modal couple coefficient of the structure decreases quickly with increasing distance between two vibration periods. Generally, the correlation between two vibration modes can be ignored when the coupling coefficient is smaller than 0.70. Modal analysis also indicates that the space effect of super high-rise steel structures is relatively larger than that of an ordinary high-rise structure. The space effects on the dynamic properties are expressed as follows: 1) In the lower modes, the distribution of the vibration period is similar to that ordinary high-rise structures, and the period is discrete. 2) In the higher modes, the vibration period of each mode is compressed because of the vertical vibration, and the interaction of the near modes becomes stronger.

To quantitatively evaluate the correlation between the modes, coupling coefficients of modes 1, 10, 20 and 30 shown in Fig. 10 are calculated according to Eq. (6). Coupling coefficients between modes 1, 10 and other modes are both less than 0.4, while coupling coefficients between modes 20, 30 and other modes contain five and four modes larger than 0.4, respectively. These results further indicate that more summation modes are required for calculation with the CQC method than with the SRSS method to receive comparative accuracy.

Results of the different modal summation methods

The SRSS method and the CQC method were both used to analyze the seismic response of the Wuhan International Securities Building to discuss the influence of different summation methods on the results. The design response spectrum provided by the Institute of Earthquake Science of China Earthquake Administration (IESCEA) was selected according to a seismic fortification intensity of 7 and site class II. The damping ratio of the structure was ζ = 0.02.

The three-dimensional model was applied in the structure analysis, and the horizontal and vertical seismic responses were calculated. The first 20 modes were combined to analyze the seismic response under a bidirectional horizontal earthquake, and the results are shown in Table 7. More summation modes were included in the vertical earthquake response analysis. Twenty or thiety modes were used for the CQC method, and only twenty modes were used for the SRSS method. The primary results of the response spectrum analysis under a vertical earthquake are shown in Table 8.

As shown in Tables 7 and 8, the results obtained by the SRSS and CQC methods are quite similar. This trend was mainly because the effects of the higher vibration modes themselves are slight, though their interaction effects are strong. The number of summation modes (20 modes or 30 modes) has little influence on the CQC method analysis results, which agrees quite well with the conclusion of the effective mass method. Comparatively, the SRSS method is suitable and conservative for seismic analysis when vertical earthquake effects and correlations between each mode are relatively small. In contract, the CQC method is applicable when the vertical earthquake effect cannot be ignored.

Conclusions

Through the analysis of the Wuhan International Securities Building, the following conclusions can be drawn:

1) There are great effects of multi-directional earthquakes on the base shearing forces and lateral deformations indicated by response spectrum analysis. It is necessary to consider seismic responses in multi-directions for the Wuhan International Securities Building. The two transition sections, including the floors from the concrete structure to the steel structure and the floors from the office to the hotel as well as three horizontal strengthened storeys, are the weak zones of the structure. Stresses concentrate in these areas, in which the rigidities change suddenly.

2) Time-history analysis can be used to observe the entire process of structure motion under seismic loading, which is an effective supplement to response spectrum analysis. Time-history analysis can reveal some local deformations that response spectrum analysis cannot reflect, such as the whiplash effect. Structural responses under various seismic waves have great differences; thus, the selection of the seismic wave requires special attention.

3) The number of summation modes required differs according to the earthquake excitations. This number is determined by the shape of the vibration mode and the distribution of vibration periods. The effective mass method can help to control the calculation precision.

4) When analyzing a super high-rise steel structure with the response spectrum analysis method, the space effect cannot be ignored, and the structure is affected by vertical vibration modes in the high frequency region. The SRSS method is recommended when the vertical earthquake effect can be neglected. In contrast, the CQC method is superior when the vertical earthquake effect should be considered.

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