Effect of the CHS on Seismic Responses of the Single-Layer Spherical Reticulated Shell under Vertical Seismic Motions

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Introduction
Te single-layer spherical reticulated shell (SPRS) is a typical structural form of long span spatial structure, which has the characteristic of good economy, stability, and seismic behavior.Te SPRS is commonly used in large-scale public buildings, and is often used as temporary disaster shelters [1].At present, the seismic performance of SPRS has been systematically studied [2].However, the efect of the centerhung scoreboard on the seismic responses of the SPRS has been ignored in most studies.
Te research on seismic response of the SPRS has reached many achievements.Cao and Zhang analysed the seismic response of a SPRS and discussed the infuence of spans, rise-to-span ratios on the seismic responses in the elastic range [3].Lin et al. applied the pseudoexcitation method to analyse the seismic response of reticulated shells in the elastic stage [4].Shen and Zhi concerned the nonlinear response and studied failure mechanisms under severe earthquakes, and classifed failure modes of the single-layer reticulated shells [5].Considering the material nonlinearity and geometric nonlinearity, Ishikawa et al. [6][7][8] systematically investigated the seismic response of the SPRSs.Te overall stability and collapse mechanism of the SPRSs under earthquake action were discussed in detail.Te research results provide a reference for the seismic design of the actual SPRSs.Xue et al. summarized development and progress in seismic design methods and analysis methods for long span spatial structures [9] in the past 30 years.
Investigation on infuence factors of the seismic response of the SPRSs has many achievements.Fan et al. studied the seismic response of the SPRSs with semirigid joints [10,11].
Wang and Shen investigated the stability of reticulated shell structures in practical engineering by using fnite element analysis technology [12].Du et al. considered the infuence of damage accumulation efect on the dynamic stability of the SPRS [13].Zhang et al. conducted incremental dynamic analysis on the SPRS and found that roof quality, rise-span ratio, and span have non-negligible efects on the seismic response [14].Zhong et al. studied nine reticulated shells under 40 far-feld and near-feld ground motions and found that near-feld ground motions caused more serious damage [15].Yu et al. found that the efect of supporting fexibility signifcantly infuences the failure characteristics of the SPRSs subjected to severe earthquakes [16].Zhang et al. [17,18] studied the efects of diferent initial geometric defect modes on the seismic performance of the SPRSs.In terms of structural design for engineering, neglect of spatial variation of ground motions would underestimate seismic response of spatial space truss structures [19].But onedimensional seismic input is often used in academic research for revealing the deep mechanism of seismic response.
In recent years, researchers showed concern on the infuence of roofng system and hanging devices on the seismic response of long span spatial structures.Cao et al. [2] investigated the infuence of metal roof panels on the seismic performance of reticulated shells and confrmed that the seismic failure load decreased after the roof panels were considered.Zhou et al. [20] conducted shaking table tests and found that the skin efect of roofng systems could reduce node acceleration response.Huo et al. [21] analysed infuence of the roofng system on the seismic performance of the SPRSs, and the results show that the roofng system could greatly change the seismic response and failure under strong earthquake conditions.Tere are some common nonstructural components in large-span spatial structures, such as catwalks, air ducts, lamps, large screens, suspended ceilings, and other roof pendants.Cai et al. transformed them into suspended mass pendulums to control the vibration response of large-span spatial structures [22].In the past decade, with the development of professional sports events and other activities, the number of center-hung scoreboard (CHS) applications has increased signifcantly [23].Te CHS is a large display device hanging in the center of the roof structure, and the heaviest CHS is about 55 t [24].Xue et al. conducted a shaking table test on a 94 m suspendome structure with a 30 t CHS and found that the CHS made a great increase on axial forces and node acceleration under seismic motions [25,26].Liu et al. [19] analysed the infuence of the CHS on natural dynamic characteristics of space truss structures, and found that the infuence on the dynamic characteristics cannot be ignored, especially for low-number frequencies and mode shapes.However, research achievements considering the infuence of the CHS on the seismic response of the SPRSs are not enough in current references.
In this paper, the infuence of the CHS on the seismic response of the SPRS under vertical seismic motion is investigated.Te fexibly suspended models and the simplifed models are built in Abaqus software, respectively.Dynamic characteristics are analysed by the Lanczos method [27], and seismic response is analysed by the dynamic explicit method.Te dynamic characteristics and seismic response of the fexibly suspended models and the simplifed models are compared and discussed.Te infuence of the weight and the sling length of the CHS on the dynamic characteristics and seismic response of the SPRS under vertical seismic motion are analysed.

FE Models.
A long span roof structure uses a single-layer spherical K6 reticulated shell with a diameter of 60 m and a rise-span ratio of 1/6, as shown in Figure 1.Te CHS is supported by a support platform, as shown in Figure 2. Te support platform and the shell are connected by vertical rods.Te CHS and the support platform are connected by slings, and the vertical location of the CHS can be controlled by a hoist system.Te slings are made of high vanadium coated cables and other members of the structure are made of steel Q355B.Te strain-stress curve of the Q355B steel is shown in Figure 3. Abaqus software is used to establish the FE model, the B31 beam element is used for lattice shell members and platform members, and the T3D2 truss element is used for slings.Te B31 is a 2-node beam element with linear interpolation formulations in three-dimensional space.Tis element allows for transverse shear deformation [28].Te T3D2 is a two-node, 3-dimensional truss element used in two and three dimensions to model slender, line-like structures that support only axial loading along the element [29].Section specifcations of the structural members are shown in Table 1, Figures 2 and 4.
Similar to the mass pendulum, the sling length and the weight of the CHS are the main parameters that afect dynamic characteristics, diferent sling lengths and diferent weights are designed to study infuence laws.In practice, a safety distance of about 1.0 m is reserved between the CHS and the support platform, and a sling length is selected every 0.5 m among 1.0 m and 9.0 m.Te length of the sling is taken as 0.0 m when the CHS is simplifed as fxed masses on the suspension nodes on the support platform.Since most of the CHSs used in recent years exceed 20 t and the heaviest ones have exceeded 55 t [23,24], the weight is selected every 5 t among 20 t and 60 t.
Te standard value of dead load D includes the standard value of uniformly distributed dead load on the roof, which is 1.0 kN/m 2 and the self-weight of members and nodes.Te standard value of uniformly distributed live load on the roof L is taken as 0.5 kN/m 2 , and the representative value of gravity load is 1.0D + 0.5L.Te boundary conditions are assumed to be three-way fxed hinge supports, see Figure 2. Te representative value of gravity load of the roof without the CHS is about 500 t.Te weight of the CHS among 20 t and 60 t is about 1/15 to 1/9 of the representative value of gravity load of the roof.

Analysis Methods.
Lanczos method is a common method for extracting eigenvalues of space grid structures [27].Te Lanczos method is a very powerful and fast 2 Advances in Civil Engineering convergence tool for extracting some of the extreme eigenvalues of real symmetric matrixes, which usually employs a sequence of Krylov subspaces K 1 , K 2 , . .., K m , and computes Ritz pairs from each other or some of the subspaces [30].In this paper, the Lanczos method is used to analyse the dynamic characteristics.Commonly used seismic response analysis methods for large-span spatial structures include mode shape decomposition response spectrum method, time history analysis method, and simplifed analysis method provided by the regulations [31].Te time history analysis method is a direct dynamic analysis method, which can analyse both the linear elastic dynamic response and the elastic-plastic dynamic response [32].In the fexibly suspended model, the CHS is hanging by only-tension slings, the model is a mechanism, and the overall stifness matrix is singular.Terefore, the dynamic explicit analysis in the Abaqus software is used for calculating the seismic response.For the explicit algorithm, the convergence of the analysis is no problem.Te numerical method above has been verifed by shaking table test on a suspen-dome structure with a CHS [25,26], and the accuracy of the numerical results was acceptable if the structure was meshed according to the grid size.Table 1: Section specifcations of structural members.

Section number Section specifcations Materials
High vanadium coated cable Te section specifcation of circular pipe φ245 × 10 means the outer diameter is 245 mm and the thickness is 10 mm.Te section specifcation of spiral strand φ12 means the nominal diameter is 12 mm.
Advances in Civil Engineering 3

Seismic Motions.
In general, the seismic waves available for structural time history analysis include actual seismic records of the proposed site, typical past seismic records, and artifcial seismic waves.According to provisions of the regulation [32], when the time history analysis method is used, the actual strong earthquake records and the artifcially simulated acceleration time history curve should be selected according to the type of construction site and the design earthquake group, and the number of actual strong earthquake records should not be less than 2/3 of the total number, the average seismic infuence coefcient curve of multiple sets of time history curves should be consistent with the seismic infuence coefcient curve used by the mode shape decomposition response spectrum method in a statistical sense.When three sets of acceleration time history curves are input, the calculation result should take the envelope value of the time history method.
Considering the spectral characteristics of ground motion, the predominant period of the selected seismic wave is as consistent as possible with the design characteristic period, and the epicentral distance of the selected seismic wave is as consistent as possible with that of the proposed site.Te design conditions of site Class II, the design earthquake group is the second group, the seismic fortifcation intensity is 8 degrees, and the design basic acceleration is 0.3 g are taken as an example.Natural seismic waves El-centro, Taft, and artifcial RH4TG040 are selected.Figure 5 shows that the seismic wave response spectrum curves after amplitude modulation is in agreement with the design response spectrum curve and the average seismic wave response spectrum curve.If the seismic fortifcation intensity is 8 degrees, for the spatial grid structure such as the single-layer reticulated shell structure, the vertical seismic efects should be checked [31,32].
Te traveling wave efect of seismic waves could act on the large-span spatial roof structures [33].However, it is difcult to defne the minimum span that needs to consider the traveling wave efect.GB 50011-2010 [32] stipulates that large-span spatial structures with specifc plane projection size, including structures with span greater than 120 m, length greater than 300 m, or cantilever greater than 40 m, shall consider the traveling wave efect.According to the seismic response analysis of single-layer cylindrical latticed shells and square pyramid latticed frames by scholars, the traveling wave efect should be considered when the structure length exceeds 200 m [34].Since the span of the models in this paper is 60 meters, the traveling wave efect is not very signifcant.Te consistent input method is adopted for seismic wave input.Te infuence of other input methods on the seismic response of the single-layer reticulated shells will be studied separately in the future.

Results and Discussion
3.1.Infuence on Dynamic Characteristics.Te frst 80 natural vibration frequencies and modes of both the simplifed models and the fexibly suspended models.In the simplifed models, the CHS is simplifed as fxed masses on the suspension nodes, as shown in Figure 2. Te vibration modes and frequencies of the simplifed models and the fexibly suspended models are compared, and the infuence of the sling length and the CHS weight on the vibration modes and frequencies is discussed.

Infuence on the Vibration Modes.
Table 2 shows that the frst two modes of the simplifed model are antisymmetric with vertical deformation, and the third mode is symmetric with main vertical deformation.Te reason is that the vertical stifness is far less than the horizontal stifness.It is displayed that when the CHS is fexibly suspended, the frst three modes are the horizontal swing of the CHS, and the fourth to sixth modes are the torsion of the CHS itself, and the seventh mode is the vertical mode with the coupling of the CHS and the reticulated shell.Te only tension feature of the slings leads to the horizontal swing or torsion of the CHS.Te frequency of the frst three modes of the fexibly suspended model is signifcantly lower than that of the simplifed model.Tis is because the horizontal constraint of the CHS is low, so the frst three vibration modes are mainly the rigid body displacement of the CHS.It is shown that the fexibly suspended model is divided into two parts due to the use of slings.Te modes of the overall structure show the motion of the CHS itself and the mode of coupling efect.

Infuence on Vibration
Frequencies. Figure 6 shows that when the CHS is simplifed to fxed masses, the frst three natural frequencies decrease along with the increase in the weight w of the CHS.Tis is because increasing the weight of the CHS is equivalent to increasing the mass of the overall structure, but there is no obvious change in the structural stifness.From the fourth mode onwards, the weight has little efect on the natural frequency of the simplifed model.For the fexibly suspended models, the frequencies under diferent sling lengths l show similar laws.Figure 7 shows that the weight has little infuence on the frst three numbers of natural frequencies, but has a signifcant infuence on the fourth to the thirteenth numbers, and has little infuence on higher numbers.Te 4 th to 13 th modes show the interaction between the CHS and reticulated shell, which is greatly afected by the CHS.Te 14 th and higher numbers are mainly the vibration mode of the lattice shell itself, which is less afected by the concentrated mass of the CHS.Te results show that the infuence of the CHS weight on the natural frequency of the fexibly suspended models is obvious.In general, the weight has little infuence on the high numbers natural frequency, but has a greater infuence on the low numbers natural frequency.
Figures 8 and 9 show the curves of the natural frequencies varying with the sling length under w � 20 t and w � 40 t, respectively.It is shown that the infuence rules of the sling length are the same under diferent CHS weights.Furthermore, the natural frequencies of the simplifed model (l � 0 m) are higher than those of the fexibly suspended model (l > 0 m) from the top 80 numbers of frequencies.Compared with the simplifed model, the frst three natural frequencies of the fexibly suspended model are signifcantly reduced, but the frst three natural frequencies are basically the same under diferent sling lengths.Tis is because the frst three vibration modes are mainly the free swing of the CHS.Te results show that the infuence of the sling length on the natural frequencies is obvious, especially, it has a greater impact on some low numbers natural frequencies, while it has no efect on the high numbers natural frequencies.

Infuence on Seismic
Responses.Te infuence on the axial forces of the reticulated shell members is mainly concerned since underestimation on the internal forces could afect the structural safety.Te axial forces and nodal acceleration of the fexibly suspended cases and the simplifed cases are compared.Te introduction of the acceleration reveals the relationship between motion and force to a certain extent, and it is convenient to evaluate the seismic response of local part of the shell.Te degree that the axial forces are afected, the position of the most afected members are analysed.Te deep mechanisms by which the CHS afects seismic responses under vertical seismic motions are discussed based on both axial forces and nodal acceleration.Te infuence laws of the sling length and the scoreboard weight on the seismic responses are also discussed.

Infuence on Axial Forces.
Te envelope peak values of the time history of axial forces under three sets of seismic waves are taken as the peak axial force of a structural member.Te symbol F w,l j,max is set as the peak axial force of Advances in Civil Engineering the j th member when the weight of the CHS is w and the sling length is l, where j is a positive integer.Ten, the change rate c w,l j,max of the j th member can be obtained by equation ( 1), where the symbol F w,0 j,max represents the peak axial force when the weight of the CHS is w and the CHS is simplifed as fxed masses on the support platform.Te maximum change rate c w,l max of the axial forces of the single-layer reticulated shell members is calculated, by equation ( 2), for analysing the degree of the infuence of w and l on the axial forces of all members, where p is the total number of members in the shell.
Figure 10 shows that the c w,l max values of shell members are between 46.9% and 130.4%. Figure 11 shows that the c w,l max values of support platform members are between 6% and 532.0%.It shows that the axial forces of some members on the shell and on the platform in the fexibly suspended model could increase by up to 1.3 times and 5.32 times of those in the simplifed model.It indicates that the amplifcation efect of the CHS on the axial forces of the shell members and the platform members are signifcant, and axial forces could be underestimated if a simplifed model is used for analysis subjected to vertical seismic motions.Te deep reason is that the diferent dynamic characteristics between the CHS and the structure result in the vertical impact efect on the structure under vertical seismic motions.
Figure 10 shows that the maximum c w,l max value appears when the weight is 30 t and the sling length is 1 m.When the weight is less than 40 t, the c w,l max value of the reticulated shell  Advances in Civil Engineering 7 members gradually decreases with the increase in the sling length.When the sling length is greater than 2.5 m, the c w,l max value of the shell members is not signifcantly afected by the weight.Figure 11 shows that the maximum c w,l max value appears when the weight is 20 t and the sling length is 1 m.When the sling length is less than 3 m, the c w,l max value of the platform members decreases frst and then increases with the increase of the weight.Te c w,l max value of the platform members reaches a local peak when the weight is 50 t and the sling length is 1 m.When the sling length is greater than 3 m, the c w,l max value of the platform members is less afected by the weight and the sling length, and the c w,l max value is maintained below 70%.Generally, the infuence laws of the weight and the sling length on axial forces under vertical seismic motions are complicated.Te deep reason is that the dynamic characteristics of the integrated structure are signifcantly infuenced by the weight and the sling length.However, Figures 10 and 11 show only overall degree of the infuence of the CHS on the axial forces of the shell members with the variation of the weight and the length.Te position of the most afected members needs to be displayed and discussed.

Te Position of the Most Afected Members.
Contours and peak axial force change rates c w,l max of structural members are displayed in Table 3, so as to illustrate the position of the most afected members.It shows that when the sling length is 1 m and 2 m, the c w,l j,max value of the central part of the reticulated shell is greater than that near the boundary.As the length of the sling increases, the c w,l j,max value of the reticulated shell member gradually decreases as a whole.It is displayed that the hoop members have signifcantly larger c w,l j,max value than other shell members.For the platform members, it is shown that the c w,l j,max value of all is quite high in some cases.It indicates that hoop members and members in the center region of the reticulated shell are the most afected members, and all members of the platform are the most afected members.
Te platform members are directly connected with the CHS, and the vertical impact efect afects the platform members frst.Te same theory can explain the most afected region of the center part of the shell.But there are two reasons why the hoop members are most afected members.Te frst and main reason is that the acceleration of the relevant nodes increases signifcantly, which is illustrated in Section 3.3, under vertical seismic motions.Te second is that the cross-sections of hoop members are controlled by slenderness, and the axial forces of hoop members are very low.Ten, the c w,l j,max values of hoop members are sensitive to the increase of axial forces.
It is also displayed that the parameters w and l signifcantly afect the distribution of change rate c w,l j,max of the reticulated shell members and the platform members, but the infuence laws are complicated.Tere are many types of single-layer reticulated shells, and the shells of one type are usually unique with diferent parameters in practice.It is hard to fnd a general rule for all single-layer reticulated shells.It is suggested that axial forces of the structural  Advances in Civil Engineering γ w,l =-5.5% Te legend of the contours is shown as < -80% -60% -40% -20% 0% 20% 40% 60% 80% 100% 120% .
Advances in Civil Engineering members, considering all possible lengths under seismic motions should be used for design of cross-sections of the members.

Infuence on Nodal Acceleration.
With the same theory for numbering the change rate of axial forces, the acceleration of the i th node on the reticulated shells is set as a w,l i, max , and the peak acceleration change rate of the i-th node is ρ w,l i,max (equation ( 3)).Te ρ w,l max values (equation ( 4)) of the reticulated shells and the platform members are displayed in Table 4. Te parameter n is the number of the members.Te legend of the contours is shown as.
< -80% -60% -40% -20% 0% 20% 40% 60% 80% 100% 120%   Figure 12 shows that the ρ w,l max values of the reticulated shell nodes are between 84% and 564%.Figure 13 displays that the ρ w,l max values of the support platform nodes are between 11% and 250%.It indicates that the CHS signifcantly infuences the acceleration of the structure under the action of vertical seismic, and the acceleration could be underestimated if the CHS is simplifed as fxed masses on the suspension nodes.Te maximum ρ w,l max values of both the shell nodes and the platform nodes occur when the weight is 20 t and the sling length is 1 m.With the increase of the weight and the sling length, the ρ w,l max value of the shell nodes decreases suddenly.When the weight is more than 30 t and the sling length is more 2.5 m, the weight and the sling length have little infuence on the ρ w,l max value.With the increase of the weight and the sling length, the ρ w,l max value of the platform nodes also shows a complicated decrease trend.However, Figures 12 and 13 show only the overall degree of the infuence of the CHS on the acceleration of the nodes with the variation of the weight and the length.Te position of the most afected nodes needs to be displayed and discussed.
Contours and peak acceleration change rates ρ w,l max of nodes are displayed in Table 4, so as to show the position of the most afected nodes.It is shown that the most afected nodes are located at two regions, one is the central part of the shell and another is the third hoops of the shell.It is also shown that all the platform nodes are signifcantly afected.Te distribution of the most afected node acceleration is consistent with the position of the most afected axial forces.Tis is due to the vertical impact efect of the CHS under the vertical seismic motions.

Conclusions
In this paper, the seismic responses on the SPRS with a CHS under vertical seismic motions are investigated.Te dynamic characteristics and seismic responses of the fexibly suspended models and the simplifed models are analysed and compared in the Abaqus software.Te infuence of the weight and the sling length of the CHS on the seismic response of the SPRS is also discussed.
(1) Under diferent CHS weight and the sling length, the frst three vibration modes are all free swing of the CHS, and the CHS weight and the sling length have a signifcant impact on the fourth and subsequent vibration modes.Compared with the simplifed model, the frst three natural frequencies of the fexibly suspended model are signifcantly reduced.Te length of the sling only has a large impact on some low numbers natural frequencies, but has little impact on the high numbers natural frequencies.(2) Compared with the simplifed model, the axial forces of some structural members and some nodal acceleration in the fexibly suspended under vertical seismic motions would increase by as high as 523% and 564%, respectively.It turns out that the seismic responses of the SPRS would be underestimated if a simplifed model is used for analysis.
(3) Te parameters including the weight of the CHS and the sling length signifcantly afect the distribution of peak axial force change rate of the reticulated shell members and the distribution of peak acceleration change rate, but the infuence laws are complicated.(4) Te region in the central of the SPRS, the hoop members of the SPRS, and the support platform are the most afected regions in terms of both axial force and nodal acceleration.Tis is due to the vertical impact efect of the CHS.(5) Te envelope results of the fexibly suspended cases taking diferent CHS weights and sling lengths into account are recommended for structural design.But if a simplifed model is used, the most afected regions must be concerned and strengthened.(6) Te weight of the CHS is usually determined by the owner of the gym.If the weight is determined, it is suggested that the vertical location of the CHS when it is not being used is quite important.Because the sling length has a signifcant efect on the seismic response.Since every gym is diferent, the best location should be studied case by case.

Figure 1 :Figure 2 :Figure 3 :
Figure 1: Front view of the integrated model.

Figure 4 :
Figure 4: Te layout of the shell members.

Figure 10 :Figure 11 :Figure 12 :
Figure 10: Te c w,l max values of shell members.

Figure 13 :
Figure 13: Te ρ w,l max values of support platform nodes.

Table 2 :
Modes and frequencies of simplifed models and fexibly suspended models.

Table 3 :
Contours and peak axial force change rates c w,l max of structural members.

Table 4 :
Contours and peak acceleration change rates ρ w,l max of the nodes.