Parametric Design and Static Performance Comparison of Six Typical Spherical Semi-open Reticulated Shell

By using ANSYS Parametric Design Language (APDL), this paper compiled the parametric design macro program for six types of single-layer semi-open spherical reticulated shell structure. Parametric design of the six types of reticulated shell structure was realized with given parameters of span S, the vector high F, radial node cycles number Nx, ring to symmetric regional copies number Kn, and the upper removed laps Ns. The stress performance of the six types of reticulated shell structure was compared and analyzed. Modelling examples demonstrated that the parametric design macro program is simple, practical and improves the efficiency of the selection of the reticulated shell design and the stress analysis under different geometric parameters. Through the comparative analysis of stress performance, some conclusions with engineering significance were obtained.


Introduction
The semi-open reticulated shell structure is a new curved space mesh structure, which has beautiful shape, reasonable stress, and both the characteristics of the rod system and thin shell structure.It has a broad development space and application prospect [1].    The changes of the parameters such as the type, span, vector high, ring to symmetric regional copies number and radial node cycles will result in the redistribution of the internal forces and change of the structure cost [1]. In the stress analysis and optimization of the reticulated shell, the workload of re-modeling is large [2][3]. Therefore, studying the distribution law of rods and nodes of different types of reticulated shell, and applying the parametric design method of the semi-open reticulated shell based on the typical reticulated shell [4], can improve the efficiency of structural design, stress analysis and the selection and optimization design of various semi-open spherical reticulated shell [5][6][7].

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Geometric description of structure The main geometric parameters are: span S, the vector high F, radial node cycles number Nx, ring to symmetric regional copies number Kn and upper removed laps Ns.

Parametric design of semi-open reticulated shell
The parametric design method of the Kiewit semi-open reticulated shell is introduced in this section. According to the sports building design code [8], the structure size is selected as: S=120m, F=45m, Kn=6, Nx=6, Ns=3.
(1) Determining input control parameters: span S, the vector high F, radial node cycles number Nx, ring to symmetric regional copies number Kn and upper removed laps Ns.
(2) Naming nodes and determining their coordinates. Establishing a spherical coordinate system, the nodes of each circle are named from the Ns-th circle. Then numbering the nodes at the i-th (Ns≤i≤Nx) circle and j-th (1≤j≤Kn) symmetrical region as 1+kn×(i-1)×i/2+j. The node coordinates of these nodes are (R, (j-1)×360/ (kn×i), 90-i×Dpha).
(3) Loop rod connection. The loop rods are connected by the node 1+kn×(i-1)×i/2+j and the node 1+kn×(i-1)×i/2+j+1 at the i-th (Ns≤i≤Nx) circle and j-th (1≤j≤kn×i-1) area. The loop rods at the last symmetrical region is connected by the first node and the last node of each circle.
(5) Applying node load and boundary constraint. Since the number of nodes increases successively from the inner circle to the outer circle, it can be obtained the first node at last circle is numbered as 1+kn×(nx-1)×nx/2+1. Applying load when node number is smaller than the first node number. Otherwise, the movable hinge or fixed hinge shall be applied as required by the project.

Parametric design examples of six typical spherical semi-open reticulated shell
Because the Geodesic type and the Three Way Grid type Kn can only be 6 or 5, so the Kiewit type Kn=6, others Kn=26 is selected to make the number of joints and rod comparable. The modeling examples of six semi-open spherical reticulated shell are shown in figure 5 when S=120m, F=45m, Nx=6, Ns=3. The total number of structural nodes, circular members, radial members and structural members corresponding to these six kinds of reticulated shell and different Kn are shown in table 1.  The following conclusions can be obtained from figure 5 and table 1.
(1) When the reticulated shell covers the same space (that is, S, F, Nx, Ns are the same), the changes of reticulated shell type and Kn will cause the total number of structural nodes and members to change greatly. Moreover, the internal force of the structure will be redistributed, and the stress performance of the reticulated shell structure will also change greatly.
(2) With the increase in the number of nodes and bars, the dead weight and cost of the structure will increase accordingly. The increase and the change of the stress performance of net shell are the factors which should be paid attention to in the selection and optimization design of the reticulated shell.

Stress behavior of semi-open spherical reticulated shell structure
The structure size is selected as: S=120m, F=45m, Nx=6, Ns=3. The form of structural support is fixed hinge support. Uniform load is applied here and the value is q=2.35kN/m 2 [9]. The dead weight of the structure is considered. The structural members are Q235 steel tubes with a section area of 46.62cm 2 (d=219mm, t=7mm). The maximum allowable displacement of the structure is 1/400 of the span [10]. The allowable stress is 215Mpa. Figure 6 and figure 7 show the displacement and stress cloud diagrams. Table 2 and table 3 show the calculated and analyzed data.   The following conclusions can be obtained from figure 6, figure 7, table 2 and table 3. (1) Under uniform distribution load, fixed hinge support is used around the structure, the maximum displacements and stresses of these six semi-open types of spherical reticulated shell structures occur in the topmost ring of the structure. The maximum displacement and stress of the Geodesic type appear at the main rib joint of the top ring. The maximum displacement and stress of the Kiewit type and the Three Way Grid type occur at the middle joint of the main rib adjacent to the top ring of the structure. The maximum displacement and stress of the other three types is evenly distributed in the top ring of the structure. The stress characteristics of the above mentioned structures should be paid attention to in engineering application.
(2) Among the six types of semi-open reticulated shells, the Kiewit type and the Lamella type have the best stress performance. Their maximum displacement and stress are less than the other four reticulated shell structures, and far less than the allowable value of the structure. Under the same conditions, the Kiewit type and the Lamella type have larger strength and stiffness reserves. The mechanical properties of the Ribbed type and the Schwedler type are close to each other. Their maximum displacement and stress of both are not beyond the allowable value of the structure. The maximum stress and displacement of the Geodesic type and the Three Way Grid type are both beyond the allowable value of the structure. They are not suitable for large-span reticulated shell structures.

Conclusion
This paper draws the following conclusions.
(1) In this paper, six methods are proposed for the joint generation and rod connection of semi-open spherical reticulated shell by using ANSYS software parametric design language APDL. These design methods are simple and efficient, which can improve the efficiency of structural design, stress analysis and the selection and optimization design of six kinds of reticulated shell in practical engineering.
(2) The modeling example shows that when the semiopen spherical reticulated shell covers the same space, the changes of the Kn and the reticulated shell type will cause the total number of nodes and bars to change greatly, which will lead to the great changes of the dead weight and the cost of the structure. Moreover, the internal force of the structure will be redistributed, and the mechanical performance of the reticulated shell will also change greatly.
(3) With the increase of the number of Kn, the number of nodes and bars of reticulated shell will increase. At the same time, the dead weight and construction cost will also increase. Such change and the consequent change in the stress performance are the key cost performance factors to be analyzed in the selection and optimization design of reticulated shell.
(4) The maximum stress and displacement of the six semi-open spherical reticulated shells appear in the top ring of the structure. The maximum stress and displacement of the Geodesic type and the Three Way Grid type both exceed the allowable value of the structure. So they only apply to the small-span structure (S≤80m). While the maximum stress and displacement of the Kiewit type and the Lamella type are far less than the allowable value of the structure. They have good mechanical performance, which can be applied to the large-span semi-open structure.