Stability of cable-supported spherical reticulated shell with tension members

Abstract

The suspendome has been widely employed in large-span space structures in recent years, and it has stronger structural stiffness and higher load-carrying capacity than single-layer spherical reticulated shell. In general, it is negligible for enhancement of load-carrying capacity to integrate cables and struts into the inner ring of reticulated shell. Based on the suspendome structure, a new hybrid space structure system, namely, cable-supported reticulated shell with tension member, is proposed in this study. To elucidate and verify its feasibility, the buckling mode and buckling form are obtained by the eigenvalue buckling analysis and nonlinear buckling analysis using ANSYS package, respectively. Furthermore, to determine the optimal structural form, this article investigates the effect of the main ribbed strut length, the initial geometric imperfection, asymmetric load, pretension in cables, and the material nonlinearity on its stability. The result shows that the proposed new structural system is of high load-carrying capacity. Tension member integrated to cable-supported reticulated shell can effectively improve the overall stiffness and greatly reduce the deformation of spherical reticulated shell. The plastic failure shape occurs with the similar pattern. The instable region mainly occurs on the main ribs with tension members, and each main rib only has one local failure dimple. The load-carrying capacity is remarkably affected by the asymmetric load, the initial geometric imperfection, and material nonlinearity. Based on the parametric analyses, Type C is the optimal choice, that is, appending cables and struts to the outermost ring of single-layer spherical reticulated shell, and arranging out-of-plane tension members under the four main ribs.

Keywords

arrangement of tension members cable-supported spherical reticulated shell with tension members initial geometric imperfection load-carrying capacity stability

Introduction

Kawaguchi et al.^1–3 from Hosei University in Japan first put forward the concept of suspendome structure, which is composed of the rigid structure, tension-only cables, and compression-only struts. Experimental research on the structural static and dynamic behavior of the suspendome was well investigated through a full-size model, in which the lower tensegrity system could not only greatly decrease the boundary reaction but also observably enhance load-carrying capacity and overall stiffness compared with the single-layer reticulated dome. In Kawaguchi et al.,⁴ experiment test and finite element method (FEM) analysis on a small-scale suspendome with a diameter of 3.0 m and a rise of 0.45 m were carried out, and finite element analysis result accorded well with the one obtained from experiments.

The suspendome has become popular in recent years due to its attractive structural performance, which is exemplified by the sports buildings built worldwide in recent decades. The suspendome has also been developed rapidly in China, such as the Kiewitt suspendome with a span of 34.5 m and a rise of 4.6 m constructed in Tianjin, in 2001,^5–7 and the badminton gymnasium for the 29th Beijing Olympic Games held in 2008.⁸

The stability of reticulated shell has always been a hot topic.^9,10 Many attentions have been also paid on stability of suspendome structure, in which structural characteristics such as the buckling strength and the structural stiffness are proved to be increased greatly.¹¹ Many researchers have made great achievements not only on the stability of single-layer reticulated shell,^7,12 but also the influence of cable arrangement on the stability of single-layer reticulated shell.^9,12

The suspendome has draw much attention from engineers and researchers throughout the world since it was put forward, and it possesses vigorous vitality.⁵ The upper rigid member can resist both the axial force and the bending moments to increase the rigidity of the structure on one hand, and on the other hand, the internal stress flow in suspendome could be formed in a closed loop, thus making the structure to be a self-equilibrated system, and a weak boundary bearing system becomes possible.^6,7 The suspendome performs like a double-layer reticulated shell when prestressing is introduced to the lower tensegrity system, therefore enhancing the stiffness of the reticulated shell and improving its load-carrying capacity. While studies^14,15 have shown that the main rib joints of the two outmost rings are prone to lose stability. It is a direct and effective method to improve the stability of the structure by integrating lower flexible tensegrity (cable–strut) system with one or the two outmost rings of the upper single-layer latticed dome, and introducing reasonable magnitude of prestressing. Therefore, it is of no great practical significance to arrange cables and struts under the inner ring of latticed dome. Zhang et al.^16–18 proposed the tension members strengthening two-way grid single-layer cylindrical shell by arranging in-plane diagonal tension members and struts connected to out-plane prestressed tension members based on the two-way grid single-layer cylindrical shell, which could greatly improve the load-carrying capacity of the structure.

Based on aforementioned weaknesses of suspendome, that is the inner ring cable–strut systems make little contribution to load-carrying capacity of the rigid primary structure, this article introduces the concept of out-of-plane tension member into the suspendome structure, that is, the inner rings of lower tensegrity system are substituted for out-of-plane prestressed tension member to form a new type of hybrid space structure, namely, cable-supported spherical reticulated shell with tension member.

To investigate stability of cable-supported spherical reticulated shell with tension member in this study, large-scale parametric numerical analyses are performed for a reticulated dome under various tension member arrangements, such as rise-to-span ratio, cable prestress, initial geometric imperfection, and asymmetrical load. The limit load, buckling mode, and plastic development level of cable-supported spherical reticulated shells with different layouts of tension members are examined by numerical study, considering both geometrical and material nonlinearities based on ANSYS package and self-compiled programs. Finally, the optimal arrangement of cable–strut in reticulated dome is determined.

Finite element model

A Kiewitt-8 (K8) cable-supported tension members reticulated shell with a span of 50 m is chosen as numerical object, shown in Figure 1, and three kinds of rise-to-span ratio 1/3, 1/5, and 1/7 are considered, respectively. Steel pipe with diameter 114 mm and wall thickness 6 mm is used for all members in the upper reticulated shell, and steel rod with diameter 20 mm for all other members. The length of the vertical struts under main rib is 1/3 of the rise of reticulated shell, and the length of cable-supported struts is 1/2 of the rise. The Young’s moduli of the steel Q235 and the cable are 2.1 × 10⁵ and 1.9 × 10⁵ MPa, respectively. Poisson’s ratio of the steel is 0.3. The density of the steel and the cable is 7.85 × 10³ and 7.80 × 10³ kg/m³, respectively. Perfect elasto-plastic model is adopted here for steel Q235 with yielding strength 235 MPa. The dead load and the live load are full-span and half-span uniformly distributed, respectively. All joints are rigid-connected. The joint 1 shown in Figure 1 is hinged, and the other peripheral joints of reticulated shell are supported in tangential and vertical directions. The slenderness ratio λ of K8 reticulated shell is shown in Table 1.

Figure 1.

Formation of K8 cable-supported reticulated shell with tension members.

Table 1.

Range of slenderness ratio λ of Kiewitt-8 reticulated shell.

Type of member	f/L	1/3	1/5	1/7
Ribbed bar	Member length	4.545	3.939	3.762
	λ	59.562	51.621	49.301
Ring bar	Member length	[2.804, 3.484]	[2.804, 3.051]	[2.804, 2.924]
	λ	[36.746, 45.658]	[36.746, 39.983]	[36.746, 38.319]
Diagonal rod	Member length	[4.567, 5.397]	[3.963, 4.751]	[3.787, 4.581]
	λ	[59.850, 70.728]	[51.935, 62.262]	[49.629, 60.034]

The numerical analysis is carried out using general finite element software ANSYS. As shown in Figure 1, the members in upper reticulated shell and vertical members under main rib are simulated by Beam188 element, which is suitable for analyzing slender to moderately stubby/thick beam structures. Shear deformation effects are also included. This element is based on the Timoshenko beam theory. Link8 is used to model the vertical struts, diagonal struts of cable-supported structure, and mid-span tension members under main rib, which is a uniaxial tension–compression element with 3 degrees of freedom at each node: translations in the nodal x-, y-, and z-directions. The element is not capable of carrying bending loads. The stress is assumed to be uniform over the entire element. Link10 is an element having the unique feature of a bilinear stiffness matrix resulting in a uniaxial tension-only (or compression-only) element. With the tension-only option, the stiffness is removed if the element goes into compression. Link10 (tension-only element) is used to simulate the prestress hoop cables and other tension members under main rib.

There are six types of cable–strut arrangements, as shown in Figure 2, involving three kinds of tension member arrangement and two kinds of suspendome structure. One ring and two rings, shown in Figure 2, represent for appending cables and struts to the outermost ring and two outermost rings of single-layer spherical reticulated shell, respectively. One main rib, two main ribs, and four main ribs, shown in Figure 2, denote that arrange out-of-plane tension members under one main rib, two main ribs, and four main ribs, respectively.

Figure 2.

K8 cable-supported spherical reticulated shells with tension members. (a) Type A: one ring + one main rib. (b) Type B: one ring + two main ribs. (c) Type C: one ring + four main ribs. (d) Type D: two rings + one main rib. (e) Type E: two rings + two main ribs. (f) Type F: two rings + four main ribs.

Comparison of load-carrying capacity

To validate the rationality of cable-supported spherical reticulated shell with tension member, its load-carrying capacity is compared with the single-layer spherical reticulated shell and suspendome structure based on the same span, load, and boundary conditions, as shown in Figure 3. Through nonlinear analysis, their load-carrying capacities are obtained, shown in Table 2, from which we can see that load-carrying capacity of the structure can be greatly enhanced when tension members are introduced into suspendome structure.

Figure 3.

Reticulated shell and its improved patterns: (a) single-layer reticulated shell, (b) suspendome, and (c) cable-supported reticulated shell with tension members.

Table 2.

The load-carrying capacity of different reticulated shells.

Type	Limit load (kN/m²)	Increasing rates to Figure 3(a) (%)
Figure 3(a)	3.87	–
Figure 3(b)	4.19	8.3
Figure 3(c)	4.83	27.8

The load–displacement curves of the three structures are depicted in Figure 4, from which we can conclude that the elastic phases of three structures are roughly coincident. The single-layer spherical reticulated shell with lower limit load is the first one to enter the plastic phase, and cable-supported reticulated shell with tension member is the last one. It illustrates that cable can evidently improve the elastic load-carrying capacity of the single-layer spherical reticulated shell. Furthermore, cable-supported structure with tension member has better plastic deformation capability than the other two structures, so tension members can be employed to improve the plastic property of the single-layer spherical reticulated shell. It makes the structure tend to be more “soft,” and takes full advantage of the material property. Ultimately, the load-carrying capacity of the structure is enhanced.

Figure 4.

The load–displacement curves of different structures.

Linear and nonlinear buckling analysis

Linear buckling analysis

Linear buckling mode accounts for the displacement trend of the structure at the critical point and can predict the possible instability shapes. The weak positions of the structure can be determined by buckling modal analysis. Figure 5 shows the first eigenvalue buckling mode of K8 single-layer spherical reticulated shell with rise-to-span ratio 1/7. Buckling modes of six structures are similar, that is, one side at the apex of structure is convex and the other side is concave. The stiffness of mid-span is weak.

Figure 5.

First buckling modes of K8 cable-supported spherical reticulated shells with different layouts of tension members: (a) Type A, (b) Type B, (c) Type C, (d) Type D, (e) Type E, and (f) Type F.

Figure 6 displays the first buckling mode of different structures (shown in Figure 3). The stiffness of the outermost two rings is improved, when the cables and struts are integrated into the outermost two rings of upper reticulated shell, as shown in Figure 6(b). But cable-supported reticulated shell with tension members (Type A) can effectively improve the stiffness of the outermost five rings, and greatly reduce the deformation of the region. The buckling modes of the three reticulated shells are antisymmetric in the vertex region.

Figure 6.

The first buckling mode of reticulated shells: (a) K8 single-layer spherical reticulated shell, (b) reticulated shell with two rings cable-supported members, and (c) Type A.

Nonlinear buckling analysis

Nonlinear buckling analysis is implemented in this section to investigate the buckling behavior of six reticulated shells, as shown in Figure 2. The buckling modes of the six ideal cable-supported spherical reticulated shells with tension members under uniformly distributed load are obtained, as shown in Figure 7. It can be seen that the buckling mode of each reticulated shell is similar. The point instability results from a larger range of collapse, consequently resulting in overall failure of reticulated shell. It is the common phenomenon for plastic failure shape of cable-supported reticular shell with tension members. The instable region mainly occurs on the main ribs with tension members, and each main rib only has one local failure dimple. With the number of main ribs with tension members increasing, the instability nodes of the structure become more and more.

Figure 7.

Plastic failure modes of cable-supported spherical reticulated shells with tension members: (a) Type A, (b) Type B, (c) Type C, (d) Type D, (e) Type E, and (f) Type F.

Parametric analysis

A K8 cable-supported reticulated shell with tension members is chosen as a numerical example with a span of 50 m. Both geometric and material nonlinear analyses are considered in this section. A series of parametrical analysis is carried out in order to determine the optimal type of cable-supported spherical reticulated shells with tension members.

Height of main ribbed struts

To elucidate the relationship of main ribbed struts to load-carrying capacity of ideal structure, six reticulated shells, shown in Figure 2, are researched with rise-to-span ratio 1/7. The uniformly distributed service load is applied to structures. Height of main ribbed strut is chosen as 1/3, 1/4, 1/5, 1/6, and 1/7 of rise of the reticulated shell, respectively. Height of the vertical struts of cable-supported structure keeps the same value as half of the rise. The numerical results are listed in Table 3, from which we can see that as the main ribbed struts get shorter, the load-carrying capacity of the structure becomes weaker, and the maximum reduction rate is 8.3%. Comparing Type A with D, B with E, and C with F, respectively, it can be concluded that more applying more cables and struts has little significance for improving the load-carrying capacity of the reticulated shell. Comparing Type A with C, it is a better approach for enhancing the load-carrying capacity to arrange more main ribbed struts. And the same conclusion can be obtained by comparing Type D with F.

Table 3.

The load-carrying capacity of reticulated shells with different heights of main ribbed struts (kN/m²).

Types	Height of main ribbed struts					Reduction rates to maximum (%)
	f/3	f/4	f/5	f/6	f/7
A	4.83	4.69	4.64	4.55	4.49	7.0
B	4.90	4.80	4.74	4.64	4.55	7.1
C	5.18	5.12	5.00	4.89	4.75	8.3
D	4.95	4.93	4.89	4.84	4.77	3.6
E	4.96	4.94	4.91	4.86	4.83	2.6
F	5.28	5.22	5.17	5.11	5.08	3.8

In addition, it can be seen from Table 3 that the load-carrying capacity of Type F is the largest one, only 1.9% higher than that of Type C with f/3 and f/4. Meanwhile, the steel consumption of Type F is more than the one of Type C. Hence, Type C is the optimal solution through overall consideration and comparison.

Rise-to-span ratio

To investigate the effect of rise-to-span ratio on the load-carrying capacity, the height of main ribbed struts is taken as h/3 constantly, and the height of vertical struts of cable-supported structure is taken as h/2 constantly, h = 50 m/7 = 7.143 m. The rise-to-span ratio f/S is taken as 1/7, 1/5, and 1/3, respectively, and the service load is uniformly distributed without consideration of initial geometric imperfection. Table 4 lists the limit loads of each reticulated shell.

Table 4.

The load-carrying capacity of reticulated shells with different rise-to-span ratios (kN/m²).

Rise-to-span ratio	Type
Rise-to-span ratio	A	B	C	D	E	F
1/3	12.2	12.3	12.3	12.6	12.8	13.0
1/5	6.88	6.97	7.23	7.00	7.06	7.45
1/7	4.83	4.90	5.18	4.95	4.96	5.28

As it shown in Table 4, the load-carrying capacity of reticulated shell gradually increases with the increment of the rise-to-span ratio, and the larger the rise-to-span ratio is, the more quickly the load-carrying capacity increases. It indicates that tension members have a more significant effect on improving the load-carrying capacity of reticulated shell with large rise-to-span ratio. Cable-supported reticulated shell with tension members can satisfy the structural requirements of reticulated shell with large rise-to-span ratio. Therefore, cable-supported reticulated shell with tension members will have a significant impact on large rise-to-span ratio spherical reticulated shell.

The limit loads of reticulated shell with different spans are listed in Table 5, from which we can conclude that the larger the span is, the greater the limit load is, and the load-carrying capacity of all six reticulated shells, shown in Figure 2, is all increased by 19%. For reticulated shells with the same span, the load-carrying capacities of Types C and F are relatively larger than other types, and close to each other. Therefore, Types C and F are both optimal.

Table 5.

The limit load of reticulated shell with different spans (kN/m²).

Span (m)	Type
	1	2	3	4	5	6
50	4.83	4.90	5.18	4.95	4.96	5.28
60	5.82	5.90	6.19	5.95	6.01	6.39
70	6.88	6.97	7.23	7.00	7.06	7.45

Load distribution type

The load-carrying capacity of reticulated shell is closely related to the load distribution type. The distribution of live load on the reticulated shell roof is usually non-uniform, and the load-carrying capacity of reticulated shell is decreased significantly under asymmetric distributed load.¹⁹ Therefore, it is particularly important to research the effect of asymmetric load distribution on the load-carrying capacity of the reticulated shell.

Table 6 enumerates the limit load of reticulated shell under different load distributions, where the symbols p and g denote the half-span and full-span uniformly distributed load, respectively. As mentioned before, the ratio of p/g varies from 0 to 1 considering four separated cases in this section. The load-carrying capacity of the ideal reticulated shell with rise-to-span ratio 1/7 is obtained.

Table 6.

The load-carrying capacity of reticulated shells with different load distributions (kN/m²).

p/g	Type
	A	B	C	D	E	F
0	4.80	4.85	5.13	4.90	4.91	5.21
1/4	4.29	4.33	4.50	4.41	4.41	4.65
1/2	3.95	3.98	4.20	4.08	4.08	4.33
1	3.53	3.58	3.80	3.66	3.66	3.96

It can be seen that with the increase in ratio of p/g, the load-carrying capacity of reticulated shell declines gradually. When the ratio of p/g increases to 1, load-carrying capacity decreases by 26%.

Figure 8 shows deformation configuration of different cable-supported spherical reticulated shells with tension members at limit state with the ratio of p/g = 1/2. The nodal displacement distributed live load becomes large significantly, and the largest deformation region occurs on the main ribs. Type C has minimum red region, which also proves that structural stiffness of the Type C is the largest one and is the optimal structural pattern.

Figure 8.

Deformation configuration of cable-supported reticulated shells with tension members at limit state (p/g = 1/2): (a) Type A, (b) Type B, (c) Type C, (d) Type D, (e) Type E, and (f) Type F.

Initial geometric imperfection

Reticulated shell is a kind of typical imperfection-sensitive structures.²⁰ The aim of this section is to study the influence of initial geometric imperfection on the stability of reticulated shell. Types A, C, and D are selected as the research objects. The consistent imperfection mode method is employed to determine the distribution of initial geometric imperfection. The maximal nodal deviation of initial geometric imperfection provided in Chinese technical specification²¹ is taken as S/300. The nonlinear analysis of reticulated shell is carried out to obtain load-carrying capacity, and the numerical results are all listed in Table 6.

From Table 7, it can be concluded that the limit load of the reticulated shell is declined significantly when initial geometric imperfection is taken into account, which indicates that load-carrying of cable-supported spherical reticulated shell is sensitive to the initial geometric imperfection. Furthermore, limit load reduction of Type D is the largest one, as shown in Table 7, and the one of Type C is the smallest. The load-carrying capacity of Type C is always larger than other types. From the above-mentioned analysis, it can be concluded that Type C is the optimal structural pattern of cable-supported spherical reticulated shell with tension members as before.

Table 7.

The load-carrying capacity of reticulated shells with initial geometric imperfection (kN/m²).

Type	Without imperfection	With imperfection	Reduction to maximum (%)
A	4.80	3.91	18.5
C	5.13	4.34	15.4
D	4.90	3.84	21.6

Figure 9 shows deformation configuration of different reticulated shells at limit state. It can be seen that maximal deformation of the three reticulated shells all occurs on the main ribbed nodes in the second ring, which can be attributed to initial geometric imperfection introduced. It also further indicates that the impact of the initial geometric imperfection on the structural deformation is evident. Furthermore, Type C has the smallest nodal deformation at limit state, which proves definitively the superiority of Type C from the viewpoint of deformation.

Figure 9.

Deformation configuration of different reticulated shells at limit state: (a) Type A, (b) Type C, and (c) Type D.

Pretension in cables

As is well-known, the buckling behavior of cable-supported reticulated shell is obviously affected by the pretension in cables. Previous research works²² have elucidated that pretension in cables can improve the overall structural stiffness and load-carrying capacity of the reticulated shell. The pretension level can generally be determined by the experiment and structural design.²³ In this section, three pretension levels varied from 30 to 90 kN are considered to analyze the effect of different pretension levels on load-carrying capacity of cable-supported spherical reticulated shell with tension members. Initial strain method is adopted to apply the aforementioned pretension in cables. The relationship between the pretension $N_{1}$ and the initial strain $ε$ satisfies the following relationship

N_{1} = E ε A

(1)

where E denotes the Young’s modulus of cable, and A denotes the cross-sectional area of cable.

Table 8 shows the load-carrying capacities of six reticulated shells under different pretension levels. The result indicates that the larger the pretension level considered here is, the higher the load-carrying capacity of cable-supported spherical reticulated shell with tension members is. Furthermore, with the pretension level growing, increasing rates of load-carrying capacity become larger and larger.

Table 8.

Load-carrying capacities of reticulated shells with different pretension levels (kN/m²).

Type	Pretension level (kN)			Increasing rates to minimum (%)
	30	60	90	Increasing rates to minimum (%)
A	4.89	4.96	5.03	2.9
B	4.96	5.03	5.12	3.2
C	5.21	5.27	5.32	2.1
D	5.04	5.11	5.20	3.2
E	5.03	5.14	5.24	4.2
F	5.37	5.46	5.61	4.5

From Table 8, it can be seen that Type C has the least increasing rate. The maximum increasing rate is only 2.1%, which indicates that load-carrying capacity of the Type C is less affected by pretension. Consequently, the effect of relaxation of prestressing on the load-carrying capacity of the Type C is negligible. Considering this factor, Type C is the best choice.

Figure 10 shows the axial force distribution of reticulated shells with the pretension level of 60 kN at limit state. Observably, the internal forces of Type C are higher than other structures. Hence, Type C is the preferable pattern, because increment of internal force after pretension introduced can strengthen surface stiffness, consequently resulting in enhancing load-carrying capacity.

Figure 10.

Axial force distribution of structures at limit state: (a) Type A, (b) Type B, (c) Type C, (d) Type D, (e) Type E, and (f) Type F.

Material nonlinearity

The influence of material nonlinearity on structural stability can be expressed by plastic reduction coefficient c_p, which is the ratio of elastic–plastic limit load to elastic limit load. In the elastic–plastic analysis, the maximal nodal deviation of initial geometric imperfection is also taken as S/300.²¹ The plastic reduction coefficients c_p of the three reticulated shells are given in Table 9. Considering the 95% guarantee rate, the suggested plastic reduction coefficients obtained according to formula (2) are as follows: c_ps = 0.36 of Type A; c_ps = 0.11 of Type C; and c_ps = 0.39 of Type D

c_{p s} = {\bar{c}}_{p} - 1.645 δ_{c p}

(2)

where ${\bar{c}}_{p}$ denotes the mean value of c_p, and $δ_{c p}$ denotes the variance of c_p.

Table 9.

Plastic reduced coefficients c_p of cable-supported spherical reticulated shell with tension members.

Type	Rise-to-span ratio
	1/3	1/5	1/7
A	0.42	0.62	0.81
C	0.30	0.43	0.56
D	0.42	0.63	0.74

Similar to other spherical reticulated shells,²⁴ material nonlinearity is of great significance on the load-carrying capacity of cable-supported spherical reticulated shell with tension members. The greater the rise-to-span ratio is, the greater the influence is. Material nonlinearity must be considered in practical engineering.

Conclusion

A new spatial structure system, namely, cable-supported spherical reticulated shell with tension members, is put forward in this study. The load-carrying capacities of six types of cable-supported spherical reticulated shell with tension members are extensively studied, and the effect of height of main ribbed struts, initial geometric imperfection, asymmetric load, pretension in cables, and material nonlinearity on load-carrying capacity of the structure is well-elaborated. The main conclusions obtained in this study are highlighted as follows:

Integrating tension members with suspendome structure can remarkably increase the load-carrying capacity of reticulated shell. Tension members in cable-supported spherical reticulated shell can effectively improve structural stiffness of the outermost five rings, and observably reduce deformation of the region. The plastic failure states of six cable-supported spherical reticulated shells with tension members exhibit the similar pattern. The instable region mainly concentrated on the main ribs with tension members, and each main rib only has one local failure dimple.

Load-carrying capacity of cable-supported spherical reticulated shell gradually decreases with the decrease in main ribbed struts, and can be effectively improved by increasing the number of main ribbed struts.

Load-carrying capacity of reticulated shell with larger rise-to-span ratio is higher than the one with smaller rise-to-span ratio. With the increase in the ratio of p/g, the load-carrying capacity of reticulated shell declines gradually. When the ratio of p/g increases to 1, load-carrying capacity decreases by 26%.

Load-carrying capacity of reticulated shell significantly decreases if initial geometric imperfection is taken into account. The larger the pretension level is, the higher the load-carrying capacity of cable-supported spherical reticulated shell with tension members is. Material nonlinearity is of great significance on the structural stability of cable-supported spherical reticulated shell with tension members. Furthermore, the greater the rise-to-span ratio is, the greater the influence is.

It is proved that Type C is the optimal structural pattern, that is, appending cables and struts to the outermost ring of single-layer spherical reticulated shell, and arranging out-of-plane tension members under the four main ribs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (grant no. 51408490).

ORCID iDs

Zhen Lu

Hui-jun Li

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