Analytical Study of Structural Conformation and Prestressing State of Drum-Shaped Honeycomb Quad-Strut Cable Dome Structure with Different Calculation Methods

Abstract

Building upon the analytical study of the structural configuration and prestress state of the drum-shaped quad-strut cable dome, we conducted further investigation into its structural configuration. By employing the nodal equilibrium equations to solve the prestress state analysis of the cable dome, we compared the effects of two different cable laying methods on the prestress state of the cable dome structure. These methods include equal length of the radial horizontal projection of the upper chord ridge cables and equal radial chord length of the upper chord ridge cables. The analysis results show that the radial length of the top chord and its corresponding radial horizontal projection length of the cable dome structure can effectively reflect the trend of the prestress state of the structural configuration. Furthermore, by using a rise-to-span ratio of 0.11 as a threshold, the cable dome configuration is categorized into the flat spheroidal structural configuration and the small hemispheroidal structural configuration. When the structure is analyzed using a small rise-to-span ratio, the difference in prestress calculations between the two structural configurations is found to be less than 10%. Additionally, the structure exhibits a more uniform distribution of prestress, with the prestress gradually increasing from the inner circle to the outer circle. However, when the rise-to-span ratio increases, the difference between the prestress calculation results of the two configurations also increases, emphasizing the need to deploy upper chord ridge cables based on equal radial chord lengths (arc lengths). The research presented in this paper provides a novel insight into the structural topological form and prestress state calculation of cable domes with this configuration.

1. Introduction

The innovation and application of large-span space structural systems serve as a measure of a country’s building modernization and comprehensive national strength [1]. At present, extensive research efforts in the field of large-span space structures are being conducted all over the world. These research studies primarily focus on various types of prestressed space structures, which include cable dome structures, chord-bearing dome structures, tensile truss structures, and cable network structures. The cable dome structure originated from R. B. Fuller’s idea of tension integral structure. Subsequently, scholars have inherited and further developed Fuller’s concept, leading to the practical application of cable domes in engineering projects [2,3,4]. At present, several representative cable dome structure projects have been successfully completed worldwide, as presented in Table 1.

Zhang Ailin et al. [5,6,7,8,9,10] have carried out a comprehensive series of studies focusing on the static performance and structural optimization of different cable dome configurations, including T-shaped three-strut-rod cable domes, star-shaped tetrahedral cable domes, and single-double-strut staggered cable domes. Their research findings emphasize the significant influence of the structural topology of cable dome configurations on the mechanical properties of these structures. Moreover, their research indicates that the force transmission mechanism within the cable dome structure is complex, and there is a lack of theoretical calculation formulas that accurately capture the behavior of these structures. In response, scholars have proposed innovative honeycomb configuration cable domes [10,11,12,13,14] and sunflower configuration cable domes [15,16,17] based on the design concept of traditional cable domes. They have discussed general pre-stress state analysis methods for the new configurations of cable domes. Inheriting the advantages of honeycomb configuration cable domes, researchers have conducted pioneering studies on drum-shaped honeycomb configuration cable domes [18,19,20,21,22]. In this paper, further innovative research is carried out on the structural configuration of the original drum-shaped honeycomb quad-strut cable dome structure. The study highlights the importance of the topological configuration of the cable dome of the small hemispherical surface, specifically in relation to the topological configuration of the radial chord length (arc length) of the upper chord ridge cable. The significance of this paper lies in enriching the configuration and prestress calculation method of the drum-shaped honeycomb quad-strut cable dome structure. Additionally, the researchers have redefined the cable dome upper chord structure schemes and analyzed the difference in prestress among different design schemes. By addressing these aspects, this paper provides new ideas and schemes for practical engineering applications.

Table 1. Representative cable dome structural projects completed worldwide.

Name	Year Built	Structural Form	Plane Size
Gymnastics Pavilion for the Asian Games in Seoul, Korea [3]	1986	ribbed ring	Diameter 119.8 m
Suncoast Dome, Florida, USA [23]	1990	ribbed ring	Diameter 210 m
Georgette Dome, Olympic Games, Atlanta, USA [24]	1992	sunflower-shaped	Oval 240 m × 192 m
La Plata Stadium, Argentina [25]	2011	sunflower-shaped	Intersection of two circles 219 m × 171 m
Tianjin Polytechnic University [26]	2016	Sunflower + Rib Ring Type	Oval 102 m × 83 m
Ya’an Tianquan County Gymnasium [4,27]	2017	Sunflower + Rib Ring Type	Diameter 77.3 m
Chengdu Fenghuangshan Sports Center	2021	Sunflower + inner ring truss type	279 m × 234 m

Table 1. Representative cable dome structural projects completed worldwide.

Name	Year Built	Structural Form	Plane Size
Gymnastics Pavilion for the Asian Games in Seoul, Korea [3]	1986	ribbed ring	Diameter 119.8 m
Suncoast Dome, Florida, USA [23]	1990	ribbed ring	Diameter 210 m
Georgette Dome, Olympic Games, Atlanta, USA [24]	1992	sunflower-shaped	Oval 240 m × 192 m
La Plata Stadium, Argentina [25]	2011	sunflower-shaped	Intersection of two circles 219 m × 171 m
Tianjin Polytechnic University [26]	2016	Sunflower + Rib Ring Type	Oval 102 m × 83 m
Ya’an Tianquan County Gymnasium [4,27]	2017	Sunflower + Rib Ring Type	Diameter 77.3 m
Chengdu Fenghuangshan Sports Center	2021	Sunflower + inner ring truss type	279 m × 234 m

2. Structural Configuration and Characteristics

The drum-shaped honeycomb quad-strut cable dome represents a further innovation following the drum-shaped multi-braced rod system [10,14,15]. Unlike the traditional cable dome concept of an “ocean of tension ropes and island of compression rods”, this new structure combines the concept of the multi-braced rod concept. Depending on the number of ring ropes used, these structures are classified into three types of open-drum shaped honeycomb quad-strut type III cable domes: a large one, referred to as cable dome ${}_{24}\overline{H}_{14\mathrm{III}}$ (see Figure 1), an intermediate one, referred to as cable dome ${}_{24}\overline{H}_{24\mathrm{III}}$ (see Figure 2), and a small one, referred to as cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ (see Figure 3). The arrow symbols in the figures indicate the direction of the cogging force.

The large-opening drum-shaped honeycomb quad-strut cable dome is characterized by having one drum-shaped honeycomb mesh per substructure in plan projection. It consists of top chord inner ridge cables, bottom chord ring cables, rigid outer ring beams, two diagonal cables, and four struts intersecting at the bottom chord node. In the small-opening drum-shaped honeycomb quad-strut cable dome configuration, two additional ring cables are added on top of the original one. Moreover, this configuration introduces three drum-shaped honeycomb meshes for each substructure in plan projection. Each substructure exhibits three honeycomb grids per bay. Under the influence of axisymmetric prestress, the nodal equilibrium equations can be established to solve the various openings of honeycomb quad-strut cable domes. For the double-axis-symmetric nodes, one vertical nodal equilibrium equation is established. For the single-axis-symmetric nodes, one vertical equilibrium equation and one radial-horizontal equilibrium equation are established, resulting in a total of two equations. For the nodes without an axis of symmetry, three nodal equilibrium equations are established to ensure equilibrium in the vertical, radial, and horizontal directions. The number of nodes, internal forces in rods, and nodal equilibrium equations for the three configurations are shown in

Table 2. Number of nodes, cable and rod internal forces and nodal balance equations of drum-shaped honeycomb quad-strut type III cable dome with large, intermediate, and small openings.

Form	${}_{24}\overline{\mathit{H}}_{14\mathrm{III}}$	${}_{24}\overline{\mathit{H}}_{24\mathrm{III}}$	${}_{24}\overline{\mathit{H}}_{34\mathrm{III}}$
Node with axis of symmetry	$1^{\prime }$ Two in total	$1^{\prime }$$2^{\prime }$ Two in total	$1^{\prime }$$2^{\prime }$$3^{\prime }$ Two in total
Node without axis of symmetry	$1_a$$1_b$ Two in total	$1_a$$1_b$$2_a$$2_b$ Four in total	$1_a$$1_b$$2_a$$2_b$$3_a$$3_b$ Four in total.
Nodal equilibrium equations	(2 × 1) + (3 × 2) = 8	(2 × 2) + (3 × 4) = 16	(3 × 2) + (3 × 6) = 24
Internal force on the cable stem	$N_1$$N_{1a}$$N_{1b}$$T_{1a}$$T_{1b}$$V_{1a}$ $V_{1b}$$B_1$$H_1$ 9 in total.	$N_1$$N_{1a}$$N_{1b}$$T_{1a}$$T_{1b}$$V_{1a}$ $V_{1b}$$B_1$$H_1$$N_{2a}$$N_{2b}$$T_{2a}$ $T_{2b}$$V_{2a}$$V_{2b}$$B_2$$H_2$ 17 in total.	$N_1$$N_{1a}$$N_{1b}$$T_{1a}$$T_{1b}$$V_{1a}$ $V_{1b}$$B_1$$H_1$$N_{2a}$$N_{2b}$$T_{2a}$ $T_{2b}$$V_{2a}$$V_{2b}$$B_2$$H_2$$N_{3a}$ $N_{3b}$$T_{3a}$$T_{3b}$$V_{3a}$$V_{3b}$$B_3$ $H_3$ 25 in total.

. Compared to other cable domes, the drum-shaped honeycomb quad-strut type III cable dome exhibits the following characteristics:

(1)

The drum-shaped honeycomb grid design of the cable dome includes only one ring cable and two diagonal cables connected to four struts. This design reduces the number of rings and diagonal cables compared to a Levy cable dome.

(2)

The drum-shaped honeycomb quad-strut type Ⅲ cable dome structure has a simple topology. For example, a small opening drum-shaped honeycomb quad-strut cable dome (${}_{24}\overline{H}_{34\mathrm{III}}$) consists of only 320 rod and cable units. The ratio of the total number of rods and cables in the structure is 3:7, which is higher compared to the commonly used Levy cable dome (1:5). Since the material cost of cables is often higher than that of the rods, the drum-shaped honeycomb quad-strut cable dome offers a cost advantage in terms of engineering and holds promising prospects for popularization and application.

3. Calculation of Internal Forces in Structural Prestressed State Rods

Since the cable dome structure requires prestressing to provide stiffness, the prestressing state serves as the fundamental basis for studying cable domes. In this context, the internal forces of the rods in the structural prestressing state are now being calculated using the drum-shaped honeycomb quad-strut type III cable dome as a case study.

3.1. Succinct Analysis of Pre-Stressed States

In the cable dome ${}_{24}\overline{H}_{14\mathrm{III}}$, there are nine unknown internal forces of the cable and rods, and the number of nodal equilibrium equations is eight. In the cable dome ${}_{24}\overline{H}_{24\mathrm{III}}$, there are 17 unknown internal forces of the cable and rods, and the number of nodal equilibrium equations is 16. Finally, in the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$, there are 25 unknown internal forces of the cable and rods, and the number of nodal equilibrium equations is 24. These configurations indicate that these cable dome structures belong to the statically indeterminate structures with one-degree indeterminacy. Therefore, these structures cannot be solved solely by the nodal equilibrium equations. To analyze these structures, the following unification of the equations is required:

$$At=H_{m}c$$

(1)

Solving for this gives:

$$t=H_m{A}^{-1}c$$

(2)

In Equations (1) and (2), $A\in R^{(M-1)\times (M-1)}$ is a dimensionless (M − 1)-order matrix (M is the total number of sets of cords), with the matrix order expressed by the (M − 1)-order cords’ azimuthal trigonometric function. $A^{-1}$ is the inverse matrix of A. $t$ is the vector representing the internal forces in the cords excluding the internal force in the outermost loop of the ring cords. $H_m$ and $c$ are the vectors representing the dimensionless coefficients of the free term expressed by the $H_m$-ring cords’ azimuthal trigonometric function.

3.2. Succinct Analysis of Pre-Stressed States

This paper takes the small-opening drum-shaped honeycomb quad-strut cable dome (${}_{24}\overline{H}_{34\mathrm{III}}$) as an analyzed case, and the structural plan and section of the cable dome can be seen in Figure 3. Additionally, the plan view and section view of the analysis model can be referred seen in Figure 4 and Figure 5.

From the inside out, the equilibrium equations can be established for each node as follows.

For nodes 1a, there are

$$\begin{array}{l}{T}_{1a}\cos {\alpha }_{1a}\cos {\gamma }_{1a}-N_{1a}\sin \frac{\pi }{\mathrm{n}}=\\ N_1\sin \frac{2\pi }{\mathrm{n}}-V_{1a}\cos {\phi }_{1a}\cos ({\delta }_{1a}+\frac{\pi }{\mathrm{n}})\end{array}$$

(3)

$$\begin{array}{l}{T}_{1a}\cos {\alpha }_{1a}\sin {\gamma }_{1a}+N_1\cos \frac{2\pi }{\mathrm{n}}=\\ N_{1a}\cos \frac{\pi }{\mathrm{n}}+V_{1a}\cos {\phi }_{1a}\sin ({\delta }_{1a}+\frac{\pi }{\mathrm{n}})\end{array}$$

(4)

$$T_{1a}\sin {\beta }_{1a}+V_{1a}\sin {\phi }_{1a}=0$$

(5)

For nodes $i_a$, $2\le i$, there are

$$\begin{array}{l}{T}_{ia}\cos {\alpha }_{ia}\cos {\gamma }_{ia}+V_{ia}\cos {\phi }_{ia}\cos ({\delta }_{ia}+\frac{\pi }{n})\\ -N_{ia}\sin \frac{\pi }{n}=B_{(i-1)}\cos {\beta }_{(i-1)}\cos ({\gamma }_{(i-1)\beta }-\frac{\pi }{n})\\ +T_{(i-1)b}\cos {\alpha }_{(i-1)b}\cos ({\gamma }_{(i-1)b}-\frac{\pi }{n})\end{array}$$

(6)

$$\begin{array}{l}{T}_{ia}\cos {\alpha }_{ia}\sin {\gamma }_{ia}-V_{ia}\cos {\phi }_{ia}\sin ({\delta }_{ia}+\frac{\pi }{n})\\ -N_{ia}\cos \frac{\pi }{n}=B_{(i-1)}\cos {\beta }_{(i-1)}\sin ({\gamma }_{(i-1)\beta }-\frac{\pi }{n})\\ -T_{(i-1)b}\cos {\alpha }_{(i-1)b}\sin ({\gamma }_{(i-1)b}-\frac{\pi }{n})\end{array}$$

(7)

$$\begin{array}{l}{T}_{ia}\sin {\alpha }_{ia}+V_{ia}\sin {\phi }_{ia}+B_{(i-1)}\cos {\beta }_{(i-1)}=\\ T_{(i-1)b}\sin {\alpha }_{(i-1)b}\end{array}$$

(8)

For nodes $i_b$, $1\le i$, there are

$$\begin{array}{l}{T}_{ib}\cos {\alpha }_{ib}\cos {\gamma }_{ib}=T_{ia}\cos {\alpha }_{ia}\cos ({\gamma }_{ia}-\frac{\pi }{n})\\ +V_{ib}\cos {\phi }_{ib}\cos ({\delta }_{ib}-\frac{2\pi }{n})+N_{ib}\sin \frac{\pi }{n}\end{array}$$

(9)

$$\begin{array}{l}{T}_{ib}\cos {\alpha }_{ib}\sin {\gamma }_{ib}+T_{ia}\cos {\alpha }_{ia}\sin ({\gamma }_{ia}-\frac{\pi }{n})=\\ -V_{ib}\cos {\phi }_{ib}\sin ({\delta }_{ib}-\frac{2\pi }{n})+N_{ib}\cos \frac{\pi }{n}\end{array}$$

(10)

$$T_{1b}\sin {\alpha }_{1b}-T_{1a}\sin {\alpha }_{1a}+V_{1b}\sin {\phi }_{1b}=0$$

(11)

For nodes $i^{\prime }$, $1\le i$, there are

$$\begin{array}{l}2B_i\cos {\beta }_i\cos {\gamma }_{\iota \beta }+2V_{ib}\cos {\phi }_{ib}\cos {\delta }_{ib}=\\ 2H_i\sin \frac{3\pi }{n}+2V_{ia}\cos {\phi }_{ia}\cos {\delta }_{ia}\end{array}$$

(12)

$$2B_i\cos {\beta }_i+2V_{ib}\sin {\phi }_{ib}+2V_{ia}\sin {\phi }_{ia}=0$$

(13)

where n is the number of sides of the polygon in the structure. ${\alpha }_{ia}$, ${\alpha }_{ib}$, ${\phi }_{ia}$, ${\phi }_{ib}$, and ${\beta }_i$ indicate the angles between the top chord ridge cable, strut, and diagonal cable, respectively, and the horizontal plane. ${\gamma }_{ia}$, ${\gamma }_{ib}$, ${\delta }_{ia}$, and ${\delta }_{ib}$ are the angles between the horizontal projections of the top chord ridge cable, strut, and diagonal cable, respectively, and the radial line of the structure. The formulas for the geometric dimensions of the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ are shown in

Table A1. Formulas for calculating the geometric dimensions of the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ ($i=$ 1, 2, 3).

Lengths

Angles

$\begin{array}{l}R=\raisebox{1ex}{$L^2$}\!\left/ \!\raisebox{-1ex}{$8f$}\right.+\frac{f}{2}\\ r_{ia}=R\sin {\theta }_{ia}\\ r_{ib}=R\sin {\theta }_{ib}\\ r_{i^{\prime }}=r_i=R\sin {\theta }_i\\ s_{ia}=\sqrt{{(r_{ib}\cos \frac{\pi }{n}-r_{ia})}^2+{(r_{ib}\sin \frac{\pi }{n})}^2}\\ s_{ib}=\sqrt{{(r_{(i+1)a}\cos \frac{\pi }{n}-r_{ib})}^2+{(r_{(i+1)a}\sin \frac{\pi }{n})}^2}\\ s_{ia}^{\prime }=\sqrt{{(r_i-r_{ia}\cos \frac{\pi }{n})}^2+{(r_{ia}\sin \frac{\pi }{n})}^2}\\ s_{ib}^{\prime }=\sqrt{{(r_{ib}\cos \frac{2\pi }{n}-r_i)}^2+{(r_{ib}\sin \frac{2\pi }{n})}^2}\\ s_i^{\prime }=\sqrt{{(r_{(i+1)a}\cos \frac{\pi }{n}-r_i)}^2+{(r_{(i+1)a}\sin \frac{\pi }{n})}^2}\\ h_{ia}=R(\cos {\theta }_{ia}-\cos {\theta }_{ib})\\ h_{ib}=R(\cos {\theta }_{ib}-\cos {\theta }_{(i+1)a})\\ h_{ia}^{\prime }=h_i+R(\cos {\theta }_{ia}-\cos {\theta }_i)\\ h_{ib}^{\prime }=h_i+R(\cos {\theta }_{ib}-\cos {\theta }_i)\\ h_i^{\prime }=h_i-R(\cos {\theta }_i-\cos {\theta }_{(i+1)a})\\ \end{array}$

$\begin{array}{l}{\alpha }_{ia}={\tan }^{-1}(\frac{h_{ia}}{s_{ia}})\\ {\alpha }_{ib}={\tan }^{-1}(\frac{h_{ib}}{s_{ib}})\\ {\phi }_{ia}={\tan }^{-1}(\frac{h_{ia}^{\prime }}{s_{ia}^{\prime }})\\ {\phi }_{ib}={\tan }^{-1}(\frac{h_{ib}^{\prime }}{s_{ib}^{\prime }})\\ {\beta }_i={\tan }^{-1}(\frac{h_i^{\prime }}{s_i^{\prime }})\\ {\gamma }_{ia}={\tan }^{-1}(\frac{r_{ib}\sin \frac{\mathsf{\pi }}{\mathrm{n}}}{r_{ib}\cos \frac{\mathsf{\pi }}{\mathrm{n}}-r_{ia}})\\ {\gamma }_{ib}={\tan }^{-1}(\frac{r_{(i+1)a}\sin \frac{\mathsf{\pi }}{\mathrm{n}}}{r_{(i+1)a}\cos \frac{\mathsf{\pi }}{\mathrm{n}}-r_{ib}})\\ {\delta }_{ia}={\tan }^{-1}(\frac{r_{ia}\sin \frac{\mathsf{\pi }}{\mathrm{n}}}{r_i-r_{ia}\cos \frac{\mathsf{\pi }}{\mathrm{n}}})\\ {\delta }_{ib}={\tan }^{-1}(\frac{r_{ib}\sin \frac{2\mathsf{\pi }}{\mathrm{n}}}{r_{ib}\cos \frac{2\mathsf{\pi }}{\mathrm{n}}-r_i})\\ {\gamma }_{i\beta }={\tan }^{-1}(\frac{r_{(i+1)a}\sin \frac{\mathsf{\pi }}{\mathrm{n}}}{r_{(i+1)a}\cos \frac{\mathsf{\pi }}{\mathrm{n}}-r_i})\end{array}$

.

3.3. Main Parameters of the Cable Dome Structure

When the radial horizontal projected length of the top chord ridge cable is taken as the basis for calculation (see Figure 6), the main parameters of the structure are as follows:

(1)

The span of the cable dome structure is L.

(2)

The ratio of the aperture to the span is L₁/L.

(3)

The ratio of the radial horizontal projection lengths of the front and back sections of the top chord ridge cable is

$${\eta }_{\mathrm{i}}=\frac{{\Delta }_{ia}}{{\Delta }_{ib}}$$

(14)

(4)

The structural thickness-to-span ratio is given by $h_i/L$, where $h_i$ represents the thickness and is calculated by

$$h_i=h(1+\frac{r_i}{r}\xi )$$

(15)

and in the case of equal thickness, $\xi =0$.

(5)

The nodal location for node $i^{\prime }$ is denoted as ${\zeta }_i$ and is described by

$${\zeta }_i=2(\frac{r_i-r_{ia}}{r_{ib}-r_{ia}})+1\in \left[1,3\right]$$

(16)

(a)

The mapping point $i$ of $i^{\prime }$ on the sphere coincides with the point $i_a$ on the sphere, as in programmatic 1.

(b)

The mapping point $i$ of $i^{\prime }$ on the sphere is located at the midpoint of $i_a$ and $i_b$, with x = 2, as in scheme 2.

(c)

The mapping point $i$ of $i^{\prime }$ on the sphere coincides with $i_b$, ${\zeta }_i=3$, as in method 2.

When taking the radial chord length of the top chord ridge cable as the basis for calculation (see Figure 7), the expressions of the main parameters of the structure change as follows:

$${\eta }_{\mathrm{i}}={\overline{\Delta }}_{ia}/{\overline{\Delta }}_{ib}$$

(17)

$$h_i=h(1+\frac{{\theta }_i}{\theta }\xi )$$

(18)

$${\zeta }_i=2(\frac{{\theta }_i-{\theta }_{ia}}{{\theta }_{ib}-{\theta }_{ia}})+1\in \left[1,3\right]$$

(19)

The corresponding formulas for the geometric dimensions of the cable dome in Table A1 are changed to the following (as $r_i\ne r_{i^{\prime }}$):

$$\begin{array}{l}{r}_{i^{\prime }}=(R-h_i)\sin {\theta }_i\\ h_{ia}^{\prime }=h_i\cos {\theta }_{ia}+R(\cos {\theta }_{ia}-\cos {\theta }_i)\\ h_{ib}^{\prime }=h_i\cos {\theta }_{ib}+R(\cos {\theta }_{ib}-\cos {\theta }_i)\\ h_i^{\prime }=h_i\cos {\theta }_i-R(\cos {\theta }_i-\cos {\theta }_{(i+1)a})\\ s_{ia}^{\prime }=\sqrt{{(r_{i^{\prime }}-r_{ia}\cos \frac{\pi }{n})}^2+{(r_{ia}\sin \frac{\pi }{n})}^2}\\ s_{ib}^{\prime }=\sqrt{{(r_{ib}\cos \frac{2\pi }{n}-r_{i^{\prime }})}^2+{(r_{ib}\sin \frac{2\pi }{n})}^2}\\ s_i^{\prime }=\sqrt{{(r_{(i+1)a}\cos \frac{\pi }{n}-r_{i^{\prime }})}^2+{(r_{(i+1)a}\sin \frac{\pi }{n})}^2}\end{array}$$

4. Example Calculation of a Cable Dome with Rise-to-Span Ratio ≤ 0.11

4.1. When the Radial Lengths of the upper Chord Ridge Cords Are Equal

Given L₁/L = 1/7, ${\eta }_{\mathrm{i}}=1$, $\xi =0$, $\zeta =1$,${\Delta }_{1a}={\Delta }_{1b}={\Delta }_{2a}={\Delta }_{2b}={\Delta }_{3a}={\Delta }_{3b}=(L-L_1)/12$, the internal forces of the cables in the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ were calculated using Matlab (version MatlabR2022a), a computational software, for different rise-to-span ratios ($f/L$ = 0.06, 0.07, 0.08, 0.09, 0.10, and 0.11). The calculation results are shown in

Table 3. Calculation results of radial horizontal prestressing for cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ with rise-to-span ratio ≤ 0.11.

$f/L$	0.06				0.07				0.08
$h/L$	0.06	0.07	0.08	0.09	0.06	0.07	0.08	0.09	0.06	0.07	0.08	0.09
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.246	0.352	0.462	0.575	0.160	0.245	0.335	0.429	0.101	0.170	0.244	0.322
$N_{1a}$	0.267	0.381	0.500	0.623	0.174	0.266	0.363	0.465	0.110	0.184	0.265	0.350
$N_{1b}$	0.054	0.077	0.102	0.128	0.035	0.053	0.074	0.095	0.022	0.037	0.053	0.071
$T_{1a}$	0.102	0.146	0.191	0.238	0.067	0.102	0.139	0.178	0.042	0.070	0.101	0.134
$T_{1b}$	0.111	0.160	0.212	0.264	0.072	0.111	0.153	0.196	0.045	0.076	0.111	0.147
$V_{1a}$	−0.005	−0.007	−0.010	−0.012	−0.004	−0.006	−0.008	−0.010	−0.003	−0.005	−0.007	−0.009
$V_{1b}$	−0.007	−0.009	−0.011	−0.013	−0.005	−0.007	−0.009	−0.011	−0.003	−0.005	−0.007	−0.009
$B_1$	0.027	0.033	0.039	0.044	0.021	0.027	0.033	0.038	0.015	0.022	0.028	0.033
$H_1$	0.054	0.064	0.072	0.079	0.042	0.053	0.062	0.070	0.031	0.043	0.053	0.061
$N_{2a}$	0.115	0.160	0.207	0.254	0.077	0.115	0.153	0.193	0.050	0.081	0.114	0.148
$N_{2b}$	0.165	0.234	0.305	0.378	0.109	0.164	0.223	0.283	0.070	0.115	0.163	0.214
$T_{2a}$	0.169	0.235	0.303	0.372	0.113	0.168	0.225	0.283	0.074	0.120	0.168	0.217
$T_{2b}$	0.202	0.285	0.371	0.459	0.134	0.202	0.273	0.346	0.086	0.142	0.201	0.263
$V_{2a}$	−0.020	−0.027	−0.035	−0.042	−0.016	−0.023	−0.030	−0.037	−0.012	−0.018	−0.025	−0.032
$V_{2b}$	−0.018	−0.023	−0.027	−0.031	−0.014	−0.018	−0.023	−0.027	−0.010	−0.015	−0.019	−0.023
$B_2$	0.100	0.116	0.129	0.140	0.084	0.101	0.115	0.127	0.067	0.086	0.102	0.115
$H_2$	0.215	0.241	0.261	0.275	0.182	0.214	0.237	0.254	0.147	0.185	0.212	0.232
$N_{3a}$	0.390	0.517	0.643	0.769	0.280	0.388	0.497	0.604	0.196	0.291	0.386	0.481
$N_{3b}$	0.540	0.729	0.921	1.115	0.377	0.536	0.698	0.862	0.258	0.394	0.532	0.673
$T_{3a}$	0.422	0.556	0.689	0.821	0.305	0.421	0.536	0.651	0.216	0.319	0.421	0.522
$T_{3b}$	0.533	0.716	0.901	1.088	0.377	0.532	0.690	0.848	0.261	0.395	0.532	0.670
$V_{3a}$	−0.081	−0.102	−0.121	−0.141	−0.068	−0.090	−0.110	−0.130	−0.055	−0.077	−0.098	−0.118
$V_{3b}$	−0.075	−0.089	−0.102	−0.113	−0.061	−0.077	−0.090	−0.102	−0.048	−0.064	−0.079	−0.091
$B_3$	0.464	0.476	0.488	0.502	0.455	0.467	0.479	0.492	0.447	0.459	0.470	0.483
$f/L$	0.09				0.10				0.11
$h/L$	0.09	0.010	0.11	0.12	0.09	0.010	0.11	0.12	0.09	0.010	0.11	0.12
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.243	0.312	0.384	0.457	0.182	0.242	0.304	0.367	0.135	0.186	0.240	0.296
$N_{1a}$	0.263	0.338	0.416	0.495	0.198	0.262	0.329	0.398	0.147	0.202	0.260	0.321
$N_{1b}$	0.053	0.069	0.085	0.101	0.039	0.053	0.067	0.081	0.029	0.040	0.052	0.065
$T_{1a}$	0.101	0.130	0.159	0.190	0.076	0.100	0.126	0.153	0.056	0.077	0.100	0.123
$T_{1b}$	0.110	0.143	0.176	0.210	0.082	0.110	0.139	0.169	0.061	0.085	0.110	0.136
$V_{1a}$	−0.007	−0.010	−0.012	−0.014	−0.006	−0.008	−0.010	−0.012	−0.005	−0.007	−0.009	−0.011
$V_{1b}$	−0.008	−0.009	−0.011	−0.013	−0.006	−0.008	−0.010	−0.012	−0.005	−0.007	−0.008	−0.010
$B_1$	0.028	0.033	0.038	0.042	0.024	0.029	0.033	0.038	0.019	0.024	0.029	0.033
$H_1$	0.052	0.059	0.065	0.070	0.044	0.052	0.058	0.063	0.036	0.044	0.051	0.057
$N_{2a}$	0.113	0.143	0.173	0.204	0.087	0.113	0.139	0.166	0.065	0.088	0.112	0.136
$N_{2b}$	0.162	0.207	0.254	0.301	0.122	0.161	0.201	0.243	0.091	0.125	0.160	0.196
$T_{2a}$	0.167	0.211	0.256	0.302	0.128	0.167	0.206	0.246	0.097	0.131	0.166	0.202
$T_{2b}$	0.201	0.256	0.312	0.370	0.153	0.200	0.250	0.300	0.115	0.156	0.200	0.245
$V_{2a}$	−0.028	−0.035	−0.042	−0.049	−0.024	−0.030	−0.037	−0.044	−0.019	−0.026	−0.033	−0.040
$V_{2b}$	−0.020	−0.024	−0.028	−0.032	−0.017	−0.021	−0.025	−0.029	−0.014	−0.018	−0.022	−0.026
$B_2$	0.102	0.115	0.126	0.137	0.090	0.103	0.115	0.126	0.078	0.092	0.104	0.116
$H_2$	0.210	0.229	0.243	0.255	0.187	0.208	0.225	0.238	0.163	0.187	0.206	0.221
$N_{3a}$	0.384	0.467	0.551	0.634	0.306	0.381	0.456	0.531	0.242	0.310	0.378	0.446
$N_{3b}$	0.528	0.652	0.778	0.905	0.413	0.523	0.634	0.746	0.320	0.418	0.517	0.618
$T_{3a}$	0.420	0.510	0.600	0.689	0.338	0.420	0.501	0.581	0.269	0.344	0.419	0.493
$T_{3b}$	0.531	0.654	0.778	0.902	0.421	0.530	0.641	0.753	0.330	0.430	0.530	0.631
$V_{3a}$	−0.107	−0.127	−0.147	−0.167	−0.095	−0.116	−0.136	−0.156	−0.083	−0.104	−0.124	−0.145
$V_{3b}$	−0.081	−0.093	−0.105	−0.116	−0.070	−0.083	−0.095	−0.107	−0.060	−0.073	−0.086	−0.098
$B_3$	0.474	0.487	0.502	0.517	0.466	0.479	0.492	0.507	0.459	0.470	0.483	0.498

. Based on the calculation results in the table, the following characteristics of the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ can be concluded.

(1)

The rise-to-span ratio and thickness-to-span ratio are important parameters that impact the overall prestress of a cable dome structure. When the rise-to-span ratio increases, there is a decreasing trend in the overall prestress of the structure. Conversely, as the thickness-to-span ratio increases, the prestress of the structure tends to increase.

(2)

When the rise-to-span ratio and thickness-to-span ratio of the structure are the same, the prestress values of the same type of cables in the structure are similar. However, as the gap between the two ratios becomes larger, the magnitude of prestress between the same type of cables also increases.

(3)

The internal forces in the rods of the inner ring of the cable dome are relatively small. However, as the number of cables in the outer ring increases, the internal forces in the rods of the inner ring also increase.

4.2. When the Radial Chord Lengths (Arc Lengths) of the upper Chord Ridge Cords Are Equal

Given L₁/L = 1/7, ${\eta }_{\mathrm{i}}=1$, $\xi =0$, $\zeta =1$, $\theta =\arcsin \left(\frac{4Lf}{L^2+4f^2}\right)$, $\Delta \theta =\frac{\theta }{7}$, the internal forces of the cables in the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ were calculated for different rise-to-span ratios ($f/L$ = 0.06, 0.07, 0.08, 0.09, 0.10, and 0.11), and the results can be found in

Table A2. Calculation results for the upper chord ridge laying based on radial chord length (arc length) in the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ with rise-to-span ratio ≤ 0.11.

$f/L$	0.06				0.07				0.08
$h/L$	0.06	0.07	0.08	0.09	0.06	0.07	0.08	0.09	0.06	0.07	0.08	0.09
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.240	0.345	0.453	0.564	0.155	0.238	0.326	0.417	0.096	0.162	0.234	0.311
$N_{1a}$	0.261	0.374	0.491	0.612	0.168	0.258	0.353	0.452	0.104	0.176	0.254	0.337
$N_{1b}$	0.052	0.076	0.100	0.126	0.033	0.052	0.072	0.092	0.021	0.035	0.051	0.068
$T_{1a}$	0.100	0.143	0.188	0.234	0.064	0.099	0.135	0.173	0.040	0.067	0.097	0.129
$T_{1b}$	0.109	0.157	0.208	0.260	0.069	0.108	0.148	0.191	0.043	0.073	0.106	0.142
$V_{1a}$	−0.005	−0.007	−0.010	−0.012	−0.004	−0.006	−0.008	−0.010	−0.003	−0.005	−0.007	−0.009
$V_{1b}$	−0.006	−0.008	−0.010	−0.012	−0.005	−0.007	−0.009	−0.011	−0.003	−0.005	−0.007	−0.009
$B_1$	0.027	0.033	0.039	0.044	0.021	0.027	0.033	0.038	0.015	0.022	0.027	0.033
$H_1$	0.054	0.064	0.072	0.079	0.042	0.053	0.062	0.070	0.031	0.043	0.052	0.061
$N_{2a}$	0.113	0.158	0.204	0.251	0.075	0.112	0.150	0.189	0.048	0.079	0.111	0.144
$N_{2b}$	0.163	0.231	0.302	0.374	0.106	0.161	0.219	0.279	0.067	0.111	0.159	0.210
$T_{2a}$	0.166	0.231	0.299	0.367	0.110	0.164	0.220	0.277	0.071	0.115	0.162	0.211
$T_{2b}$	0.198	0.281	0.366	0.453	0.130	0.197	0.267	0.339	0.083	0.137	0.195	0.256
$V_{2a}$	−0.020	−0.027	−0.034	−0.042	−0.015	−0.022	−0.030	−0.037	−0.011	−0.018	−0.025	−0.032
$V_{2b}$	−0.018	−0.022	−0.027	−0.031	−0.013	−0.018	−0.023	−0.027	−0.010	−0.014	−0.019	−0.023
$B_2$	0.100	0.115	0.128	0.140	0.083	0.100	0.115	0.127	0.066	0.085	0.101	0.114
$H_2$	0.214	0.241	0.260	0.275	0.181	0.213	0.236	0.254	0.145	0.183	0.211	0.232
$N_{3a}$	0.390	0.517	0.643	0.769	0.279	0.388	0.496	0.604	0.194	0.290	0.385	0.480
$N_{3b}$	0.542	0.733	0.927	1.123	0.378	0.539	0.703	0.868	0.258	0.395	0.536	0.679
$T_{3a}$	0.419	0.552	0.685	0.817	0.301	0.417	0.532	0.646	0.212	0.314	0.415	0.516
$T_{3b}$	0.532	0.716	0.902	1.088	0.375	0.531	0.689	0.848	0.258	0.393	0.530	0.669
$V_{3a}$	−0.081	−0.102	−0.121	−0.140	−0.068	−0.089	−0.110	−0.129	−0.055	−0.077	−0.098	−0.118
$V_{3b}$	−0.075	−0.090	−0.103	−0.114	−0.062	−0.077	−0.091	−0.103	−0.048	−0.065	−0.079	−0.092
$B_3$	0.464	0.476	0.489	0.503	0.456	0.468	0.480	0.493	0.449	0.460	0.472	0.484
$f/L$	0.09				0.10				0.11
$h/L$	0.09	0.010	0.11	0.12	0.09	0.010	0.11	0.12	0.09	0.010	0.11	0.12
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.231	0.298	0.367	0.437	0.170	0.227	0.286	0.347	0.123	0.171	0.222	0.275
$N_{1a}$	0.250	0.323	0.398	0.474	0.184	0.246	0.310	0.377	0.133	0.186	0.241	0.299
$N_{1b}$	0.050	0.065	0.081	0.097	0.037	0.050	0.063	0.077	0.027	0.037	0.049	0.061
$T_{1a}$	0.096	0.124	0.152	0.182	0.071	0.094	0.119	0.144	0.051	0.071	0.092	0.115
$T_{1b}$	0.105	0.136	0.168	0.201	0.077	0.103	0.131	0.159	0.055	0.078	0.101	0.126
$V_{1a}$	−0.007	−0.009	−0.011	−0.014	−0.006	−0.008	−0.010	−0.012	−0.005	−0.007	−0.008	−0.010
$V_{1b}$	−0.007	−0.009	−0.011	−0.013	−0.006	−0.008	−0.010	−0.011	−0.005	−0.006	−0.008	−0.010
$B_1$	0.028	0.033	0.037	0.042	0.023	0.028	0.033	0.037	0.019	0.023	0.028	0.033
$H_1$	0.052	0.059	0.065	0.070	0.043	0.051	0.058	0.063	0.035	0.043	0.050	0.056
$N_{2a}$	0.109	0.139	0.168	0.198	0.083	0.108	0.134	0.160	0.061	0.083	0.106	0.130
$N_{2b}$	0.157	0.202	0.247	0.294	0.117	0.155	0.195	0.235	0.086	0.119	0.153	0.188
$T_{2a}$	0.160	0.203	0.247	0.291	0.121	0.158	0.197	0.236	0.090	0.122	0.156	0.190
$T_{2b}$	0.193	0.247	0.302	0.359	0.145	0.191	0.239	0.288	0.107	0.147	0.189	0.232
$V_{2a}$	−0.027	−0.034	−0.041	−0.048	−0.023	−0.029	−0.036	−0.043	−0.019	−0.025	−0.032	−0.038
$V_{2b}$	−0.020	−0.024	−0.028	−0.032	−0.016	−0.020	−0.024	−0.028	−0.013	−0.017	−0.021	−0.025
$B_2$	0.101	0.114	0.125	0.136	0.089	0.102	0.114	0.125	0.076	0.090	0.102	0.114
$H_2$	0.209	0.228	0.242	0.255	0.185	0.206	0.224	0.237	0.161	0.185	0.204	0.220
$N_{3a}$	0.383	0.467	0.551	0.634	0.304	0.380	0.455	0.531	0.239	0.308	0.376	0.445
$N_{3b}$	0.532	0.659	0.788	0.917	0.415	0.528	0.642	0.758	0.321	0.421	0.524	0.627
$T_{3a}$	0.414	0.503	0.592	0.680	0.330	0.411	0.492	0.572	0.261	0.335	0.409	0.482
$T_{3b}$	0.529	0.652	0.777	0.902	0.417	0.528	0.639	0.752	0.325	0.425	0.526	0.628
$V_{3a}$	−0.106	−0.126	−0.146	−0.166	−0.094	−0.114	−0.134	−0.154	−0.081	−0.102	−0.123	−0.143
$V_{3b}$	−0.082	−0.094	−0.106	−0.118	−0.071	−0.084	−0.097	−0.108	−0.061	−0.074	−0.087	−0.099
$B_3$	0.476	0.489	0.503	0.518	0.468	0.480	0.494	0.508	0.461	0.472	0.485	0.499

. Based on the calculation results, the following characteristics of the cable dome ${}_{24}\overline{H}_{34\mathrm{III}}$ can be concluded.

(1)

The internal forces in the inner ring of the structure with an equal chord length arrangement are small. However, their magnitudes increase gradually from the inner ring to the outer ring.

(2)

When the rise-to-span ratio and thickness-to-span ratio are kept equal, the differences in prestress magnitudes within the structure are not significant. However, there is a decreasing trend in prestress magnitudes as both the rise-to-span ratio and thickness-to-span ratio increase. This suggests that the rise-to-span ratio has a greater influence on the structure compared to the thickness-to-span ratio.

(3)

The calculation results show that the internal forces of the braces are relatively small. In this structure, the primary load-bearing mechanism is based on the tension of the cables, with the braces providing additional support through compression.

5. Example Calculation of a Cable Dome with Rise-to-Span Ratio ≥ 0.12

5.1. Comparison of Radial Horizontal and Radial Chord Length Projected Lengths of the upper Chord Ridge Cord of the Drum-Shaped Honeycomb Quad-Strut Dome

When the radial horizontal projection length of the upper chord is equal, the overall shape of the structure is flatter in the vertical plane, and the structural curvature is not significant. However, when the radial arc length is equal, the structural curvature increases, and the appearance tends to resemble a hemisphere. This leads to a noticeable difference in the overall appearance of the structure. This difference can be reflected by the ratio of the radial chord length of the upper chord (${\overline{\Delta }}_{ia}$, ${\overline{\Delta }}_{ib}$) to the radial horizontal projection length of the upper chord (${\Delta }_{ia}$, ${\Delta }_{ib}$), providing insight into the vertical shape of the structure and enabling a comprehensive analysis of the overall form. When the thickness-to-span ratio of the structure remains unchanged at 0.06, the overall shape of the structure in the vertical plane exhibits an increasing trend as the rise-to-span ratio increases. This consistent trend indicates that the shape of the structure changes uniformly regardless of the rise-to-span ratio. When the structure is laid with equal lengths of the radial horizontal projection of the upper chord and equal length of the chord, the ratio of radial chord length to radial horizontal projection increases as the number of structural ring ropes, which indicates that as the number of ring ropes increases, the structure tends to become more hemispherical in shape. Additionally, when the structure is laid with a rise-to-span ratio of 0.11, the maximum vertical inclination of the structure is 24.81°, and the corresponding ratio of the outermost ring is 1.086. Under the two laying methods, at this rise-to-span ratio, the shape of the structure is flatter compared to the overall shape. The shape difference between the two layouts is insignificant for the structure with a rise-to-span ratio of less than 0.11, which is referred to as a flat surface structure. However, as the rise-to-span ratio increases, the difference in structure between the two layouts gradually becomes more significant. Consequently, a structure with a rise-to-span ratio larger than 0.11 is referred to as a small hemispherical structure.

Table 4. Comparison of radial length and horizontal projection length between the two laying methods.

Equal distribution of radial horizontal projections	$f/L$	$\theta$	${\overline{\Delta }}_{1a}/{\Delta }_{1a}$	${\overline{\Delta }}_{1b}/{\Delta }_{1b}$	${\overline{\Delta }}_{2a}/{\Delta }_{2a}$	${\overline{\Delta }}_{2b}/{\Delta }_{2b}$	${\overline{\Delta }}_{3a}/{\Delta }_{3a}$	${\overline{\Delta }}_{3b}/{\Delta }_{3b}$
	0.09	20.41°	1.002	1.008	1.012	1.026	1.031	1.057
	0.10	22.62°	1.003	1.008	1.019	1.026	1.049	1.057
	0.11	24.81°	1.003	1.011	1.019	1.039	1.049	1.086
	0.12	26.99°	1.005	1.013	1.027	1.046	1.070	1.103
	0.13	29.15°	1.005	1.016	1.031	1.053	1.083	1.121
	0.14	31.28°	1.006	1.018	1.036	1.061	1.095	1.142
	0.15	33.40°	1.007	1.020	1.040	1.069	1.109	1.164
	0.16	35.49°	1.008	1.022	1.045	1.078	1.124	1.188
	0.17	37.56°	1.009	1.025	1.050	1.087	1.140	1.214
	0.18	39.60°	1.009	1.027	1.055	1.097	1.156	1.242
Radial chords are laid out with equal length	$f/L$	$\theta$	${\overline{\Delta }}_{1a}/{\Delta }_{1a}$	${\overline{\Delta }}_{1b}/{\Delta }_{1b}$	${\overline{\Delta }}_{2a}/{\Delta }_{2a}$	${\overline{\Delta }}_{2b}/{\Delta }_{2b}$	${\overline{\Delta }}_{3a}/{\Delta }_{3a}$	${\overline{\Delta }}_{3b}/{\Delta }_{3b}$
	0.09	20.41°	1.003	1.008	1.016	1.027	1.040	1.057
	0.10	22.62°	1.004	1.010	1.020	1.033	1.050	1.071
	0.11	24.81°	1.004	1.012	1.024	1.040	1.061	1.087
	0.12	26.99°	1.005	1.014	1.028	1.048	1.073	1.104
	0.13	29.15°	1.006	1.017	1.033	1.056	1.086	1.123
	0.14	31.28°	1.007	1.019	1.038	1.065	1.100	1.144
	0.15	33.40°	1.008	1.022	1.044	1.075	1.115	1.167
	0.16	35.49°	1.009	1.025	1.050	1.085	1.131	1.192
	0.17	37.56°	1.010	1.028	1.056	1.096	1.149	1.219
	0.18	39.60°	1.011	1.031	1.063	1.108	1.168	1.248

below compares the radial length and horizontal projection length under the two layout methods.

5.2. Calculation and Analysis of Large Rise-to-Span Ratio and Thickness-to-Span Ratio

In the previous section, in the calculation of the prestressing state of the flat spherical structure, it was observed that the calculation error of the structure is minimal under both laying methods. However, in the calculation of the prestressing state of the small hemispherical structure, it has been found that the calculation error gradually increases as the rise-to-span ratio of the structure increases. When the structure is constructed with the top chord ridge cables according to the radial horizontal projection length, the internal forces of the cables of the same horizontal projection length decrease linearly as the rise-to-span ratio increases. The difference in internal forces among the rods in the inner ring of the structure is relatively small, but it gradually increases with the number of ring ropes. The rods in the outermost ring experience the highest internal forces, and increasing the prestress of the third ring rope has a larger impact on the structure’s prestress compared to increasing the prestress of the second ring rope. When the upper chord of the structure is laid based on the equal radial chord length (arc length), the structural internal force exhibits a linear increasing trend with a change in the structural thickness-to-span ratio. The internal forces of the rods in the inner ring of the structure remain relatively small. As the number of channels as a division of the step increases, the structural prestress gradually increases from the inside to the outside of the ring, and this trend becomes more pronounced with the choice of a higher rise-to-span ratio. The calculation results are shown in

Table A3. Calculation results of structural prestress for equal radial horizontal projection of the upper chord.

$f/L$	0.12				0.14				0.16
$h/L$	0.12	0.14	0.16	0.18	0.14	0.16	0.18	0.20	0.14	0.16	0.18	0.20
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.239	0.342	0.450	0.560	0.235	0.323	0.413	0.506	0.231	0.307	0.385	0.464
$N_{1a}$	0.259	0.371	0.487	0.607	0.255	0.350	0.448	0.548	0.251	0.332	0.417	0.503
$N_{1b}$	0.052	0.075	0.100	0.125	0.051	0.071	0.091	0.113	0.051	0.067	0.085	0.103
$T_{1a}$	0.099	0.142	0.187	0.233	0.098	0.134	0.172	0.211	0.096	0.128	0.160	0.194
$T_{1b}$	0.109	0.157	0.208	0.260	0.108	0.149	0.192	0.236	0.107	0.142	0.179	0.217
$V_{1a}$	−0.010	−0.014	−0.018	−0.022	−0.011	−0.015	−0.019	−0.023	−0.012	−0.016	−0.020	−0.024
$V_{1b}$	−0.009	−0.012	−0.016	−0.020	−0.010	−0.013	−0.017	−0.020	−0.010	−0.014	−0.017	−0.020
$B_1$	0.029	0.038	0.046	0.054	0.030	0.038	0.046	0.054	0.031	0.038	0.046	0.053
$H_1$	0.050	0.060	0.068	0.074	0.048	0.057	0.064	0.069	0.047	0.054	0.060	0.065
$N_{2a}$	0.111	0.155	0.201	0.246	0.109	0.147	0.185	0.224	0.107	0.139	0.172	0.206
$N_{2b}$	0.159	0.226	0.295	0.366	0.156	0.212	0.271	0.331	0.153	0.201	0.252	0.303
$T_{2a}$	0.165	0.231	0.299	0.367	0.164	0.220	0.277	0.336	0.162	0.211	0.260	0.311
$T_{2b}$	0.199	0.282	0.368	0.456	0.198	0.269	0.342	0.417	0.196	0.258	0.322	0.387
$V_{2a}$	−0.035	−0.049	−0.063	−0.077	−0.040	−0.053	−0.067	−0.081	−0.044	−0.057	−0.070	−0.084
$V_{2b}$	−0.022	−0.030	−0.038	−0.046	−0.024	−0.032	−0.040	−0.048	−0.026	−0.034	−0.041	−0.049
$B_2$	0.105	0.127	0.147	0.168	0.107	0.128	0.148	0.168	0.108	0.128	0.148	0.168
$H_2$	0.203	0.229	0.248	0.263	0.198	0.220	0.238	0.251	0.191	0.211	0.227	0.240
$N_{3a}$	0.375	0.499	0.622	0.745	0.368	0.473	0.578	0.682	0.360	0.451	0.541	0.632
$N_{3b}$	0.511	0.695	0.882	1.070	0.497	0.653	0.810	0.969	0.482	0.616	0.751	0.887
$T_{3a}$	0.418	0.554	0.688	0.822	0.416	0.533	0.649	0.765	0.414	0.517	0.619	0.721
$T_{3b}$	0.529	0.715	0.903	1.093	0.527	0.688	0.851	1.014	0.525	0.667	0.811	0.955
$V_{3a}$	−0.133	−0.173	−0.213	−0.252	−0.150	−0.190	−0.229	−0.268	−0.166	−0.206	−0.245	−0.284
$V_{3b}$	−0.089	−0.112	−0.134	−0.157	−0.095	−0.118	−0.141	−0.164	−0.101	−0.125	−0.148	−0.172
$B_3$	0.488	0.519	0.555	0.594	0.499	0.532	0.570	0.611	0.511	0.546	0.585	0.628

and

Table A4. Distribution of prestressed states with equal chord lengths (arc lengths).

$f/L$	0.12				0.14				0.16
$h/L$	0.12	0.14	0.16	0.18	0.14	0.16	0.18	0.20	0.14	0.16	0.18	0.20
$H_3$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$N_1$	0.218	0.315	0.416	0.521	0.207	0.288	0.372	0.458	0.196	0.264	0.334	0.406
$N_{1a}$	0.236	0.342	0.452	0.565	0.225	0.312	0.403	0.496	0.213	0.286	0.362	0.440
$N_{1b}$	0.048	0.070	0.093	0.117	0.046	0.064	0.083	0.102	0.043	0.058	0.074	0.091
$T_{1a}$	0.091	0.131	0.173	0.217	0.086	0.120	0.155	0.191	0.082	0.110	0.139	0.170
$T_{1b}$	0.099	0.145	0.193	0.242	0.095	0.133	0.172	0.213	0.090	0.122	0.156	0.190
$V_{1a}$	−0.009	−0.013	−0.017	−0.022	−0.010	−0.014	−0.018	−0.022	−0.011	−0.014	−0.018	−0.022
$V_{1b}$	−0.008	−0.012	−0.015	−0.019	−0.009	−0.012	−0.016	−0.019	−0.009	−0.012	−0.016	−0.019
$B_1$	0.028	0.037	0.045	0.053	0.029	0.036	0.044	0.052	0.029	0.036	0.043	0.050
$H_1$	0.049	0.059	0.067	0.074	0.047	0.056	0.063	0.069	0.045	0.053	0.059	0.064
$N_{2a}$	0.104	0.147	0.191	0.235	0.100	0.136	0.172	0.209	0.095	0.126	0.157	0.188
$N_{2b}$	0.150	0.216	0.283	0.353	0.145	0.199	0.256	0.314	0.138	0.185	0.233	0.282
$T_{2a}$	0.154	0.217	0.281	0.347	0.148	0.201	0.255	0.310	0.142	0.187	0.233	0.280
$T_{2b}$	0.186	0.266	0.349	0.433	0.180	0.248	0.317	0.388	0.174	0.231	0.291	0.352
$V_{2a}$	−0.034	−0.047	−0.061	−0.075	−0.037	−0.050	−0.064	−0.077	−0.040	−0.053	−0.066	−0.079
$V_{2b}$	−0.022	−0.029	−0.037	−0.045	−0.023	−0.030	−0.038	−0.045	−0.024	−0.031	−0.038	−0.046
$B_2$	0.103	0.124	0.145	0.165	0.103	0.124	0.144	0.163	0.103	0.123	0.142	0.162
$H_2$	0.201	0.228	0.247	0.262	0.195	0.218	0.236	0.250	0.188	0.209	0.225	0.238
$N_{3a}$	0.373	0.498	0.622	0.746	0.365	0.472	0.577	0.683	0.356	0.449	0.541	0.632
$N_{3b}$	0.519	0.709	0.903	1.099	0.508	0.671	0.836	1.003	0.495	0.637	0.781	0.927
$T_{3a}$	0.406	0.541	0.674	0.806	0.401	0.516	0.630	0.744	0.394	0.495	0.595	0.695
$T_{3b}$	0.524	0.713	0.904	1.096	0.521	0.684	0.849	1.016	0.516	0.661	0.807	0.955
$V_{3a}$	−0.131	−0.170	−0.209	−0.248	−0.146	−0.185	−0.224	−0.262	−0.160	−0.199	−0.237	−0.275
$V_{3b}$	−0.090	−0.113	−0.136	−0.159	−0.096	−0.119	−0.142	−0.166	−0.102	−0.126	−0.149	−0.173
$B_3$	0.490	0.520	0.555	0.594	0.501	0.533	0.569	0.609	0.512	0.545	0.583	0.625

.

6. Conclusions

The following conclusions can be obtained through the exploratory research conducted in this paper:

(1)

According to the geometric shape of the structure, the cable dome can be categorized into two types: flat spherical cable dome and small hemispherical cable dome. The dividing line between these two types can be determined by setting the spherical rise-to-span ratio equal to 0.11.

(2)

The drum-shaped honeycomb quad-strut cable dome can be constructed using two different laying methods based on the upper chord ridge cable. The first method involves laying the upper chord ridge cable according to the radial horizontal projection length, which is equivalent to the thickness-to-span ratio measured vertically on the spherical surface. The second method involves laying the upper chord ridge cable according to the radial chord length (arc length), which is equivalent to the thickness-to-span ratio measured along the normal direction of the spherical surface. Both laying methods can be used to construct the topology and shape of the cable dome.

(3)

There are two different topologies for the drum-shaped honeycomb quad-strut cable domes. However, when analyzing the nodal equilibrium equations and the prestressing state of the structure, the formulas remain the same. The only difference lies in the geometry of the members.

(4)

When the structure is a flat spherical cable dome, the calculation results of the two layouts have an error of less than 10%. Therefore, it is more convenient to use the layout with the top chord ridge cable laid according to the radial horizontal projection length for calculation purposes.

(5)

When the vertical configuration of the structure is a small hemisphere, the two laying methods result in a significant calculation error. Therefore, it is more reasonable to use the laying method where the upper chord ridge cable is laid based on the radial chord length (arc length) being equal.