Analysis and Design of the Accessory Structure under the Large Deformation of a Flexible Roof

An increasing number of large-span space structures use exible roofs to achieve a light and splendid architectural eect. When a exible structural system such as a cable net is applied in a large-span stadium, the roof will deform signicantly under the selfweight and wind load. Accessory structures of the roof such as catwalks, radial drainage pipes, circular drainage channels, and radial cable trenches need to cooperate with the large deformation of the roof to prevent damage caused by the large deformation of the roof. To thoroughly unveil the analysis and design method of an accessory structure, this paper rst carried out the wind tunnel test to determine the wind load of the structure. en, the gust response factors of each roof area and the dynamic amplication coecient of the accessory structure considering the roof vibration were determined. Next, circular and radial catwalks were designed based on static analysis. A sliding joint was set in the accessory structure to adapt to the large deformation of the roof. Finally, a time history analysis of the catwalk was carried out to obtain the maximum deformation value of the sliding joint for the safety of the structure. e results demonstrate that the maximum deformation is less than the value given by the design and meets the specication.


Introduction
e large-span roof structures are widely used in sports buildings because of their capacity of providing huge space without inner columns [1,2]. e roof is usually divided into the rigid roof, exible roof, and rigid-exible hybrid roof. In recent years, a growing number of stadiums adopt exible roof structures to achieve soft and polished architectural appearance [3]. e large-span roof structure has the characteristics of lightweight, large exibility, and small damping. e wind load is generally one of the control loads in the structural design. e damping of the exible roof is smaller, and the natural frequency is closer to the predominant frequency of the wind, and it is more sensitive to the wind load, especially the uctuating wind compared with other structures [4].
Consequently, the wind-induced response is ordinarily signi cant, and the vertical displacement usually far exceeds the limit value of the speci cation [5]. e roof of sports buildings will be equipped with accessory structures such as radial catwalks, circular catwalks, radial drainage pipes, circular drainage channels, and radial cable trenches. e large deformation of the roof will cause damage to the accessory structure. erefore, the stress state of the accessory structure under the large deformation of the roof needs to be further analyzed and specially designed. evolved from the stress mechanism of the bicycle spoke with the advantages of a simple structural system and clear force transmission path, as shown in Figure 1. e radial cable is connected to the outer compression ring and the inner tension ring. e cable's tension and the outer ring's pressure are balanced. e overall structure belongs to a self-balancing system. is structure is usually covered with membrane materials, which have lightweight and beautiful appearance characteristics and are mainly applied in large stadiums. e single-layer cable-net structure generally has the ups and downs of the outer compression ring, forming a saddle-shaped spatial curved surface shape. e saddle-shaped roof can provide a sizeable vertical stiffness for the structure.
Suzhou Olympic Sports Center Stadium is the largest spoke single-layer cable-net structure in China [7,8], as shown in Figure 2. Figure 3 illustrates the main geometric dimensions of the structure. e roof is constructed as a single-layer cable-net structure with a membrane covering and axial diameters of 260 m and 230 m. e height difference of the saddle shape is 25 m, creating the outer contour of the bowl. e maximum height of the structure is 40m, and the radial cable length is between 51 m and 54 m. e main structure of the stadium is a new supporting system with reasonable stress and high structural efficiency. e structure is made of modern and novel materials to realize a lightweight super large-span space structure. Highstrength materials are applied to save natural resources in the structure for environmental protection and the economy. It comprises the outward inclined V-shaped column, saddle-shaped outer compression ring, inner tension ring, and cable net, as shown in Figure 4. e membrane roof is arranged above the cable net structure formed by radial and inner ring cables. e concave radial cable mainly bears the gravity load, and the convex radial cable mainly bears the upward wind suction load, as shown in Figure 5. e radial cable is prestressed and anchored on the outer ring beam. e outer ring beam mainly bears the axial pressure through a reasonable cable net form-finding analysis. It forms a spatial structure with excellent stiffness with a V-shaped column, which can resist horizontal wind loads and earthquakes. All radial cables and circular cables are Galfancoated full-locked coil ropes [9]. e wind-induced response analysis of the stadium shows that the maximum vertical displacement of the inner ring cable can reach 2.8 m [7], which is far beyond the deformation limit of the code. erefore, it is necessary to analyze the accessory structure of the roof to avoid damage to the accessory structure caused by the large deformation.
Since there are few such projects in the world and the relevant research is basically blank, Suzhou Olympic Center Stadium was taken as an example to introduce the analysis and design methods to the accessory structure under large deformation of the roof in detail. In Section 3 of this paper, the wind tunnel test was conducted to determine the wind load of the structure, and the dynamic amplification factor of the accessory structure was calculated. In Section 4, the design method of the catwalk was introduced in detail to adapt to the large deformation of the roof. In Section 5, the     wind-induced response analysis of the catwalk was carried out, and the results were compared with the results obtained by the design method to ensure the safety of the structure. Finally, Section 6 summarizes the whole study.

Wind Tunnel Test.
Pressure measurement experiments of rigid models have always been the main wind resistance research method for long-span roof structures [10]. e pressure measurement test of the rigid model ignores the coupling between the structure and the wind [11]. It measures the average and fluctuating wind load acting on the structure. To determine the wind load acting on the structure, a detailed wind pressure measurement was carried out through the rigid model wind tunnel test. e model is made of plexiglass and ABS (acrylonitrile butadiene styrene) plates with sufficient strength and stiffness, as shown in Figure 6. ere is no deformation or noticeable vibration under the test wind speed to ensure pressure measurement accuracy.
e geometric scale ratio of the model is 1 : 250, and the shape of the model and the real object is geometrically similar [12]. e wind direction angle in the test was simulated by rotating the turntable. Figure 7 illustrates the definition of the wind direction angle. When the wind blows from the north of the building to the model, it is defined as a 0°wind direction angle. e wind direction angle increases clockwise, and the interval of the wind direction angle is 15° [ 13]. e structure is divided into four areas A, B, C, and D. e red part is the cantilever end of the roof, and the blue part is the wall and external ring beam. Four hundred fortythree measuring points were arranged on the rigid model. Considering the wind on the inner and outer sides of the wall of the long-span roof structure, a pair of measuring points were arranged at each position, and the pressure of the measuring points is the difference between outer and inner surface pressures [14]. e landform type is class B; the basic wind pressures of the 50-year return period and 100-year return period are 0.45 kPa and 0.50 kPa, respectively, and the corresponding gradient wind speeds are 45.7 m/s and 48.2 m/s. e wind load F ji on the structure is determined by the following formula: where ρ is the air density, U i is the wind speed at the height of the ith pressure measuring point, A i is the auxiliary area of the measuring point, and C P,ji is the wind pressure coefficient at the ith measuring point on the surface of the rigid model. e wind load acting on the actual structure can be obtained by proportional conversion of the wind load in the wind tunnel test.

Gust Response
Factors of the Structure. e wind load time histories obtained from the rigid model wind tunnel test were used to act directly on the corresponding nodes of the roof. e wind-induced vibration response of the structure was analyzed in the time domain through the ANSYS transient analysis method, taking into account the nonlinear geometric effect caused by the large deformation of the structure. e first two vibration modes of the roof are shown in Figure 8. e ring beam and the V-shaped column were simulated by Beam 188, and the cable was simulated by Link 10. e dynamic balance equation of the structure is given by where [M], [C], and [K] are the mass matrix, damping matrix, and stiffness matrix of the structure, respectively. {U (t)} is the node displacement and {P (t)} is the column vector of the structural node load under the action of fluctuating wind. e wind pressure sequence is applied to the finite element model by the Newmark-β method. e wind-induced response of the structure can be obtained by step-bystep integration. e damping of the structure adopts the proportional damping model [15]. Fourteen specific wind directions in the wind tunnel test were selected for calculation. e wind loads acting on structures under stationary winds have traditionally been treated by the gust response factor method [16,17] in most major codes and standards worldwide. Based on the theory of stationary random vibration [18,19], the peak displacement response at a ref- where Y(H) is the mean displacement response, g is the peak factor, taken as 2.5 in the Chinese specification, and σ Y represents the root mean square (RMS) of the displacement. e gust response factor, which is defined as the ratio of the peak load effect on the structure to the mean load effect corresponding to the design wind speed, can be expressed as follows: Advances in Civil Engineering (4) Figure 9 presents the time histories of the vertical displacement of the cantilever end in areas A, B, C, and D under the control wind direction. e gust response factor β of each area can be calculated by Formula (4), which is listed in Table 1. From the vertical displacement time histories of the roof cantilever end in each area, it can be observed that the vertical displacement of the cantilever end in area C is the largest, which is the most unfavorable working condition. erefore, the gust loading factor of the whole roof can be subject to the result of area C, taken as 1.90. rough the gust loading factor, the equivalent static wind load of the structure can be determined to carry out the subsequent static analysis.

Dynamic Amplification Factor of the Accessory Structure.
e roof vibrates under the wind load, and the dynamic amplification caused by the roof vibration should be considered for the accessory structure. e accessory structure's dynamic amplification factor (DAF) was calculated by analyzing the vertical acceleration of the cantilever end. Figure 10 shows the cantilever's vertical acceleration time histories in area D and the acceleration power spectral density (PSD). e maximum variance of the acceleration root reaches 2.575 m/s 2 . e dynamic amplification factor can be obtained by Formula (4): where g 1 is gravity acceleration, taking a value of 9.8 m/s 2 , and σ a is the RMS of acceleration. erefore, the dynamic amplification factor of area D can be given by DCF � (9.8 + 2.5×2.575)/9.8 � 1.66. e dynamic amplification factor and the control wind direction angle of each area are shown in Table 2. Considering this dynamic amplification factor, the static analysis of the accessory structure is carried out to consider the dynamic influence of roof vibration under wind-induced action.

Deformation Analysis of the Catwalk.
e design idea of an accessory structure is actively adapted to the roof's large deformation, so many sliding connection nodes and rotating connection nodes are set in the accessory structure. is can reduce the adverse effects of the large roof deformation and the stiffness of the accessory structures [20], avoiding unnecessary secondary stress and making the main structure analysis closer to reality. Unstructured connecting elements are set in the overall model to simulate and analyze the deformation of the accessory structure, as shown in Figure 11. e element is equivalent to a linear spring, so the deformation of the connecting element is the release demand of the sliding node of the accessory structure. e total length of the circular catwalk is 435 m, which is simulated by 40 unstructured connection elements (Nos. 0-39). Figure 12 illustrates each element's deformation under different working conditions. It can be seen that the expansion amount of each element is −5.0 mm to 11.3 mm (tension is defined by a positive sign and pressure is defined by a negative sign). e radial catwalk is simulated by six unstructured elements (Nos. 1 to 6), and four radial catwalks (axes 1 to 4) are selected for analysis. Figure 13 displays the envelope values of tension (max) and compression (min) of the four radial catwalks under different working conditions. It can be seen that the deformation of each component of the radial catwalk is large.

Design of the Circular Catwalk.
For the purpose of meeting the architectural requirements for the roof, the traditional "hanging" form was abandoned for the catwalk. All catwalks are turned up and arranged above the cable head at the intersection of the inner ring cable and the radial cable to ensure that spectators cannot see catwalk components. e circular catwalk is divided into two cases, catwalk with lamps and catwalk without lamps, considering lighting layout requirements. e three-dimensional diagram of the circular upturned catwalk is shown in Figure 14.
Two transverse beams connect the supporting columns on the adjacent ring cable clamps on the circular catwalk. e maximum span of the beam is nearly 12 m. Two railing columns are set at the three equally divided positions on the cross beam of each span to reduce the span of the upper transverse member, as shown in Figure 15. e main stressbearing component of the circular catwalk is transverse beams, and both sides are hinged with the ring cable clamp through the support column. Under structural deformation, the transverse beam is always parallel to each ring cable. erefore, the deformation to be set for the beam sliding joint is the envelope value of the ring cable's deformation under various working conditions. Based on the deformation of the circular connecting element, a deformation of     Advances in Civil Engineering 5 20 mm is reserved for the sliding joint. One end adopts a fixed hinge, and the other adopts a hinge plus axial sliding. e detailed structure is shown in Figure 16.

Catwalk without Lamps.
e railing column is set on the transverse beam. e railing column is rigidly connected to the beam. When there is a significant vertical deformation difference between adjacent cable clamps, the railing will turn a certain angle with the deformation of the beam, resulting in a large amount of deformation of the side span. If the conventional three-span railing is adopted, the nodes of the side spans on both sides need to reserve a large amount of deformation, increasing the node length. Meanwhile, the large number of nodes will increase the load of the inner ring cable, resulting in difficulties in processing and a poor overall effect on the building. After studying the    Advances in Civil Engineering deformation mechanism of the single-span catwalk, it can be found that when the vertical relative deformation δ occurs on the catwalk, the total length of the three sections of railings between adjacent cable clamps remains unchanged, as shown in Figure 17. erefore, we abandoned the traditional railing and changed it to a single-span cable, as shown in Figure 18. e cable is only anchored to the supporting column on the cable clamp, a circular hole is set on the railing column, and the cable passes through the circular hole. e inner wall of the hole is provided with an EPDM (ethylene propylene diene monomer) protective sleeve to avoid scratching the cable, as shown in Figure 19. e cable makes the overall appearance effect of the catwalk concise and eliminates the cumbersome railing release node. e cable material is stainless steel, which has the advantages of a low elastic modulus and a strong ability to adapt to deformation.

Catwalk with Lamps.
When lamps are installed on the catwalk, the load of the lamps is significant, and a reciprocating dynamic load will be generated under the windinduced vibration response. In this case, if a 12 m beam is used to support the lamp, the span is too large, resulting in a large deflection, which is unfavorable to the structure. erefore, the lamp beam was segmented to reduce its calculated length, and support nodes were set up, as shown in Figure 20. e lamp beam has significant torque because the lamp load is large and has an inclination angle. e support node adopts the form of a double ear plate so that the lamp beam can rotate and expand axially along the pin shaft and effectively transmit the lamp's torque.
Due to the rigid connection between the railing column and the transverse beam, when the beam is deformed, the length of the middle span lamp beam remains unchanged, which only causes the sliding of the side span lamp beam. erefore, fixed hinge joints are adopted on both sides of the middle span lamp beam, and the side span lamp beam adopts joints with one end hinged and one end sliding axially. e maximum deformation of the lamp beam can reach 50 mm to ensure its safety. To reduce the node size, each side of the node releases ±25 mm deformation.

Design of the Radial Catwalk.
e radial catwalk is canceled due to the large deformation of the radial cable. e upper part of the radial cable is covered with a protective film, which forms a natural berm by using the load-bearing characteristics of the membrane structure. Staff can reach the inner circular berm when walking on the protective film. e cable is placed on the radial berm. It can be seen from Figure 21 that the deformation of each component needs to be effectively released.
erefore, slotted holes of +90/ −90 mm are adopted for each section of cable route components. Figure 21 shows the detailed structure of the radial components. rough the catwalk set by the protective film and the slotted holes set between the components, the radial secondary components have enough flexibility to adapt to the large deformation of the structure.

Wind-Induced Response Analysis of the Circular Catwalk
To avoid incomplete consideration of the static analysis, the wind-induced response dynamic analysis of the catwalk was carried out, and the analysis method is shown in Figure 22. First, the gymnasium's wind-induced response time history analysis was carried out. en, the three-dimensional displacement time history of the inner ring cable head is extracted and applied to the support of the catwalk as an input load to obtain the time history curve of node deformation. e maximum deformation of the joint is compared with the static analysis results. e local finite element model of the circular berm is shown in Figure 23. e nodes are divided into two types: the first type is the transverse beam node and the second type is the lamp beam node. e top ten maximum deformations of the beam node at the bottom of the circular catwalk are shown in Table 3, and the maximum deformation reaches 10 mm. e top ten maximum deformations of the lamp beam nodes are shown in Table 4, up to 33 mm. e demand value of node expansion is less than the reserved number of nodes in the design (transverse beam: 20 mm and lamp beam: 50 mm).

Conclusion
In this paper, the analysis process and design countermeasures of the accessory structure under the large deformation of the inner ring are given in detail based on the engineering background of Suzhou Olympic Sports Center Stadium. e relevant conclusions are as follows:     (1) rough the wind tunnel test of the rigid model of the stadium, the wind load time history is obtained, and the time history analysis of the stadium is carried out. e vertical displacement of the cantilever end in area C is the largest, and the gust response factor of the whole roof can be subject to area C, which is taken as 1.90. (2) e roof vibrates under the wind load, and the dynamic amplification caused by the roof vibration should be considered for the accessory structure. e dynamic amplification factor is obtained from the acceleration response, which is 1.66. (3) e accessory structure needs to set sliding joints to adapt to the large deformation of the roof. e innovation of the accessory structure in this paper lies in that a double ear plate with a slotted hole is set on the lamp beam, which can effectively transmit torque while adapting to telescopic deformation. e cable is used to replace the traditional railing to avoid complex joints. (4) e deformation of the sliding joint is determined by static analysis. e time history analysis of the accessory structure shows that the maximum deformation of the sliding joint does not exceed the design value.
Data Availability e data used in this paper are derived from independent analysis of participants during the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.