Introduction
The Shanghai Qizhong Tennis Center covering about 85000 m2 consists of main and minor indoor stadiums, and dozens of outdoor courts. The structure of the main stadium is composed of retractable steel roof and concrete stands, as shown in Fig. 1. The circularly arranged columns of the stand support the oblique beams in radial direction and are connected by beams in circumferential direction. Those beams in the two directions are pre-stressed, forming an overall configuration of the stand like inverse-cone type. The inside and outside diameter of the inverse-cone is 70.8 m and 127.9 m, respectively, and the overall height is 24.0 m. As shown in Fig. 2, all columns are arranged in three concentric circumferences, located in 64 radial axes. Except from the middle ring where the columns are arranged in each junction of radial and circumferential axis, the column in inner ring are drawn one in each two, while in out ring, are drawn two in each four. The retractable roof is divided into eight separate but identical leaves, which are irregularly spatial truss structures able to move along their own rails. The strength grade of concrete used in stand is C55, and the members of steel roof are hollow circular sections by the connection of unstiffened welded joint. Unlike the normal frames in which horizontal vibration in one direction is predominant, torsional vibration becomes predominant in this stand frame because of its ring configuration. The other reason for these characteristics is that no adequate shear walls and braces were employed by the limitation of architectural requirements. Furthermore, the opening or closing status of the retractable roof will cause the change of mass distribution and structural stiffness. Therefore, the dynamic characteristics and earthquake response were different. Hence, the seismic performance of the stand affected by such factors should be investigated.
Most research work relating to the seismic performance of prestressed concrete frame system was focused on members, connections, or normally arranged frames [
1-
3]. No report dealing with the seismic performance of closed-ring type frame has been found. Based on the practical demands of the project of Shanghai Qizhong Tennis Center, this study put the emphases on the natural vibration characteristics in the structure supporting moveable roof, elasto-plastic seismic response, and the failure mechanism under rarely occurred earthquake action.
Method
Shaking table model test
A shaking table test on a model with the scale of 1∶25 was conducted. The stand and circular steel truss beam were only composed in the model, and equivalent mass as its distribution in prototype was loaded on the stand model. It is a simulation of the structure in the status of roof opening, because by preanalysis, the stiffness of whole structure is reduced noticeably in opening status and thus brings about severe seismic response. The additional mass was added in order to adjust the gravity distortion caused by the small scale model. The similarity of prestress was implemented by means of tensioning the radial and hoop beams in batch to realize the final pretension status according to the yardstick of designed value.
Earthquake excitations for the test included bi-directional El Centro records, Pasadena records, and unidirectional SHW2 (man-made wave in accordance with the Chinese local code). In total, 36 excitations were input, many of which were horizontal vibration, with the minor, medium, and major level of intensity VII, referring to the Chinese seismic code. More details on the test model can be found in Ref. [
4].
Numerical analysis of FE
In order to explore the vibration characters and all-around seismic behaviors of the structure, nonlinear seismic analyses were conducted for both of roof closing and opening status but excluded from the situation of opening or closing process of the roof, considering the fact that it is a rare probable event that the structure is subject to earthquake action during the infrequent as well as very short moving process.
The key factors of the numerical model are listed in Table 1. The element possessing prestress function was adopted and was assigned the preload equivalent to the products of the effective prestressing and the area of the tendons. The initially tensioned FE elements were bonded with ambient concrete elements by means of constraint equations. For El Centro wave input, X direction denotes the original records of north-south direction while Y direction east-west direction. The excitation for elasto-plastic analyses included 750 data points selected from the strong record with interval of 0.02 second.
Main analysis cases are listed in Table 2, in which S35 signified excitation of SHW2 with peak acceleration of 35 gal and E35 signified excitation of El Centro with peak acceleration value of 35 gal. This rule applied to all other cases. Based on the fact that the stand was mainly affected by the horizontal earthquakes, vertical earthquake inputs were not included in the analyses waged here.
The data picking points which will be used later as given in Table 3, can be grouped into three types, referring to Fig. 2∶ 1) the top ends of columns in outer ring WKZ lie in YZ plane or XZ plane; 2) the intersection of prestressed circular beam PL3 with YZ plane or XZ plane; 3) and the intersection of the outer members of steel circular beam CB with YZ plane or XZ plane. WKZ+Y signifies the top end of outer ring columns along the positive Y direction and lies in the YZ plane. PL3+Y denotes the section of PL3 intersected by YZ plane which points to positive Y direction. CB+Y denotes the intersection of the outer member of circular steel beam to YZ plane that pointes to positive Y direction.
Torsional mode and seismic response analysis
The first six natural periods and corresponding vibration modes of the structure are shown in Table 4 for closing and opening roof, respectively, from which it can be seen that the fundamental vibration mode is torsional for both situations.
In order to explore the actual seismic response of systems that is characterized by its fundamentally torsional vibration mode, a series of elastic and elasto-plastic analyses were conducted for both states of roof closing and opening according to the cases set in Table 2. The configurations of transient vibration under excitations of SHW2 and El Centro for both closing and opening states of roof are outlined in Fig. 3. It can be noticed that the structure responds to the former excitation in the translational X direction while in a 45 degree of angle relative to X axis for the latter excitation.
The time history of displacements on top end of the outer ring columns to the case of E220, i.e., WKZ+Y, WKZ+X, WKZ-Y, WKZ-X, are illustrated in Fig. 4. Figure 4 (a) is the result of numerical analysis, corresponding to the roof opening state, while Fig. 4 (b) is the result of shake table test. It can be proven from the figure by means of both numerical and experimental study that the X directional displacement time history of all the top column ends located at the same height agree with each other, which presents a sound evidence that the overall response is translational rather than rotational. It should be explained again that numerical analysis based on a full-scale prototype, while the shaking table model is a small scale one.
Both vibration mode analysis and time-history transient analyses show that the primary displacement response modes in free vibration and forced vibration are different from each other. It is based on such factors as the deployment of lateral load resistant members and the characteristics of the excitations. For the structure discussed here in closed ring with evenly distributed mass, the seismic response mode is dominantly translational.
In normal frame system, torsional vibration shall induce shear forces unevenly distributed among columns, and corner columns are usually subjected to a larger part of shear than the other columns do. However, a further study reveals that the structure where the columns deployed in concentric rings can guarantee a relatively even distribution of shear force among columns even in case of torsional response mode is dominant. Thus, the structure behaves satisfied seismic performance against torsional vibration.
Effects of roof states on seismic performance
The states of retractable steel roof affect the seismic behavior of the stand structure. Compared with the opening state, the closing roof provides stronger rigidity in its own plane and, thus, enhance the lateral rigidity of stand; on the other hand, the change of mass distribution due to the different roof states changes the natural frequency of stand, too. As shown in Table 4, all the natural periods corresponding to overall torsion, X and Y direction translations which are closely related to the lateral stiffness of the whole structures are larger in roof opening state than that in closing state, which demonstrates the closing state increases the overall integrity.
By comparison of the dynamic response and the occurrence of plastic hinges through elasto-plastic time history analyses, the roof closing state shows more favorable to stand the structure than the opening state. Figure 5 shows the results in case of E220 by displacement at WKZ+Y. It exhibits notable plastic displacement with the lasting of seismic duration in opening state compared to the closing state, although the peak deformation has no remarkable difference. Furthermore, it can also be detected that more plastic hinges occur on top of the columns in the opening state than in the closing state after the excitation of rarely occurred earthquake level in intensity VII, as shown in Fig. 6.
The sequences of the occurrence of plastic hinge on column ends for both closing and opening states in case of E220 are compared in Table 5. It can be seen that in the opening state, the plastic hinges first occur at columns located in middle ring, next in the inner ring, and then the outer ring follows. While for the closing state, the plastic hinges initially occur on the columns in the inner ring and are limited within this region through the whole excitation duration. The distribution of plastic hinges in the opening state predicted by numerical analysis agrees fairly well with that observed in shake table test (see Fig. 7). That is, they show the same sequence of hinge occurrence and distribution on columns.
Effects of prestress on seismic performance
As shown in Fig. 2, the prestress system consists of two rings of hoop beams (PL3, PL4), 64 lines of radial beams (XL2), and all columns in inner, middle, and outer ring, with the number of 32, 64, and 32, respectively. Effects of prestress on structural properties can be analyzed in three phases, which are tensioning phases, during an earthquake, and after an earthquake.
1) Tensioning phase. The tensioning of prestress system results in notable change of axial force in columns. As shown in Table 6, axial compression occurred in columns located at inner and middle ring while axial tensile force at outer ring. It is demonstrated in Table 7 that tensioning also makes change of natural characteristics due to the change of structural stiffness, that is, the periods closely related to lateral stiffness tend to be reduced after prestress tensioning.
2) During earthquake. The axial force variation of tendons in various cases, as shown in Table 8, are less than 0.3 percent of the effective tensioning force which are listed in Table 1. The measurement results of peak strain variation of tendons by shake table test are given in Table 9, based on which the maximum value is merely 17 μϵ. Thus, it can be concluded that the variation of prestressing force during earthquake is relatively small. Figure 8 shows the change of axial forces of tendons during the excitations at minor and middle level in intensity VII degree. Compared with the effective tension strain after pretension, the fluctuation range can be reasonably ignored. Thus, in elasto-plastic dynamic analysis, the initial stiffness of tendon elements is adopted.
3) After earthquake. No cracks remained for most of the prestressed beams through the whole excitation process except for PL4, which did crack transversely in the initial excitation stage, as shown in Fig. 9. However, no further development of cracks was observed during the consequent excitations, and the cracks closed fairly well after the excitations. Along with trivial residual strain within prestressed members, it provided strong evidence that the prestress system possesses such merits as structural integrity, capacity of crack closing, and deformation recovery.
Failure mode analysis
By Chinese Seismic Design Code for Buildings [
7], the plastically relative drift of interstorey should be limited within 0.02 in case of frame systems. For the stand structure, the corresponding maximum displacement on the top end of columns located at outer ring equals 397.4 mm. Table 10 lists the results of numerical analyses response to the input of rarely occurred earthquake level of intensity VII, and Table 11 shows the extrapolated peak displacement of prototype under the same level of input by the shaking table model test. It can be drawn that the overall plastic displacements of the structure are quite small.
It has been revealed by both analysis and test that the seismic behaviors of the structure under the serious earthquake input have three characteristics: 1) the predominant displacement response is translational; 2) the overall displacement is small; 3) with the adequate duration of major excitation the plastic hinge might be fully generated.
By the shaking table test, it was observed that the remained residual lateral displacement of the stand structure was not distinguished, as shown in Fig. 7, though most columns lost their moment capacity after several strong excitation inputs. It is obviously different from normal frame structures.
To explain the reason of this phenomenon, two mechanisms are demonstrated in Fig. 10. By Fig. 10 (a), it is clear that the stand structure discussed here is with a cone configuration, as we know from section 4 that the undamaged prestressed ring beams could keep the integrity of the cone structure after several severe earthquakes. Thus, the structure could avoid push-over failure. However, if the stand is a completely platform structure, a moveable mechanism shall be possible, as shown in Fig. 10 (b), with the generation of enough plastic hinges. In that case, the collapse of the structure is unavoidable. We observed that the former structure is robust enough to sustain gravity load after loss of moment resistant capacity in individual columns.
Conclusion remarks
1) The natural vibration characteristics and its earthquake response of the structure are dependent on the roof states, and roof closing state makes the structure in a favorable situation against horizontal earthquake.
2) The predominant seismic response mode of the structure system is translational rather than rotational, though the fundamentally natural vibration mode of it is characterized by torsional vibration.
3) The stand can maintain integrity even after severe earthquake inputs due to the excellent capability of crack closing and deformation recovery of prestressed circular beams.
4) Owing to its geometrical configuration and adequate overall rigidity of the platform, the structure possesses quite good resistance against side-sway deformation after most columns develop plastic hinges, and thus, more excellent capacity bearing gravity loads to avoid collapse can be expected.
5) The structure can guarantee structural safety in rarely occurred earthquake action of intensity VII by Chinese code.
Higher Education Press and Springer-Verlag Berlin Heidelberg
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