Design Optimisation of a Cable–Strut Tensile Structure According to the Importance of Elements

Abstract

In this study, the design of a flexible cable–strut tensile structure was optimised according to the importance of elements to achieve high structural robustness. First, the importance coefficients of elements were determined by comparing their structural prefailure and postfailure strain energy. Moreover, the effects of the external load, the initial prestress, and the cross-sectional areas of elements on the importance coefficients were analysed. Second, a genetic algorithm was used to optimise element section design and minimise the maximum importance coefficient. Third, an optimised cable arrangement scheme was developed by adding an alternative load transfer path to the outer hoop cable with the highest importance coefficient. In this scheme, outer elements have a Levy-type arrangement rather than a Geiger-type arrangement so that a Geiger–Levy composite cable dome is formed. Finally, the cable arrangement and element section design for the aforementioned scheme were comprehensively optimised to reduce the maximum importance coefficient. The results of this study indicated that different elements had different importance coefficients, which exhibited different trends with changes in the external load, the initial prestress, and the cross-sectional areas of elements. Element section optimisation, cable arrangement optimisation, and the comprehensive optimisation reduced the maximum importance coefficient by 20.5%, 11.6%, and 27.7%, respectively, which indicated that these optimisation processes can effectively improve the robustness of cable–strut tensile structures.

1. Introduction

Cable–strut tensile structures are flexible tensile structures composed of cables and struts. These structures contain high-strength cables and have a controllable prestress. They are characterised by features such as a high-span capability, competitive cost, and low weight and have been widely used in sports venues and other large-scale public buildings worldwide [1,2]. With social progress and economic development, large-scale structures will continue to trend towards larger spans, more complex forms, and more novel materials. Moreover, structures are increasingly likely to be negatively affected by factors such as explosion impacts, extreme weather, and accidental damage. If these high strain-bearing cables are damaged, the entire structure might exhibit continuous collapse [3,4,5]. Therefore, exploring the structural performance of this kind of structure is crucial in structural engineering for preventing continuous structural collapse. In particular, the disproportionate or overall collapse of a structure due to the partial damage to its cables should be prevented by ensuring that the structure has high robustness.

The concept of structural robustness originated from an analysis of the continuous collapse of the Ronan Point apartment block in London in 1968 [6]. The collapse of the World Trade Center in 2001 because of the 9/11 terrorist attack resulted in structural robustness becoming a hot research topic worldwide [7,8]. The 2008 Wenchuan earthquake led to the collapses of numerous buildings, which resulted in Chinese researchers focusing considerable attention on the continuous collapse of structures and on structural robustness [9]. Researchers have proposed that the enhancement of the weak areas of a structure and the setting of multiple lines of defence in the design specification would strengthen the structural resistance to continuous collapse and improve structural robustness. The enhancement of the loading capacity of weak areas, especially key elements, and the reduction of the overall structural collapse caused by partial element damage are crucial structural strengthening methods that have been widely adopted on a global scale [10,11,12]. Thus, the importance of elements has been suggested as a crucial indicator of structural robustness. The higher the importance of an element, the stronger is the effect of its failure on the entire structure. An existing approach for assessing the importance of elements involves obtaining the postfailure structural state by removing elements and comparing the prefailure and postfailure structural performance, including the bearing capacity, displacement, and strain capacity. Frangopol and Curley proposed structural robustness indices such as the structural reserve redundancy coefficient, residual redundancy coefficient, strength redundancy coefficient, and damage strength ratio to determine the structural robustness under changes in the loading capacity caused by structural damage [13]. Li et al. proposed an index to quantify the robustness of steel frames by comparing their bearing capacities in the intact and damaged states under the considerations of dynamic effects and plastic internal force redistribution [14]. Biondini et al. proposed a robustness index based on the displacement ratio after structural damage by using the Euclidean norm [15]. Smith proposed a method for identifying the most critical members, and their method involved finding the sequence of damaged members that would require the least energy to collapse [16]. Beeby defined a coefficient related to material performance and the form of structural failure and expressed the energy dissipation capacity of a structure as the product of the defined coefficient and the structural volume. He also analysed the minimum energy required for structural collapse, and measured the structural robustness of the structure [17]. He and Gao investigated the weak link and key components of truss structures by considering the H2 norm of the system transfer function matrices as a structural performance index and the relative strain energy as an indicator of the initial disturbance [18]. Gharaibeh et al. proposed an evaluation method based on system reliability theory and defined the influence of an element on the overall structural reliability as an evaluation index for the element’s importance [19]. Yang proposed an index for evaluating element importance in which the degree to which the structural reliability worsens after the removal of an element is considered [20].

Current design optimisation methods for cable–strut tensile structures primarily focus on prestress optimisation [21,22,23,24], element section optimisation [24,25,26], structural shape optimisation [27,28,29], and structural topology optimisation [29,30,31,32,33]. Most studies on prestress optimisation have focused on minimizing factors such as the prestress level and minimum support reaction forces. Most studies on element section optimisation and structural shape optimisation have focused on minimising the material weight. However, the quantity of steel required in construction is mostly less than 30 kg/m², and steel is lightweight. Therefore, further research on the optimisation of material weight might not add any value to the literature. Studies on structural topology optimisation have mainly focused on spatial grid structures and other rigid spatial structure systems. Such optimisation has not been fully achieved for cable–strut tensile structures. Therefore, developing an optimisation model for the topology of the aforementioned structures is crucial.

In this study, a structural design optimisation model was developed on the basis of structural robustness and element importance. This model was used to determine the optimal distributions of element sections and cables for achieving high structural robustness. First, the importance coefficients of elements were defined by comparing their prefailure and postfailure strain energy, and the effects of some design parameters on the importance coefficients were analysed. Second, a genetic algorithm was used to optimise element section design and minimise the maximum importance coefficient. Third, an optimised cable arrangement scheme was developed by adding an alternative load transfer path to the outer hoop cable with the highest importance coefficient. In this scheme, outer elements have a Levy-type arrangement rather than a Geiger-type arrangement so that a Geiger–Levy composite cable dome can be formed. Finally, element section optimisation was performed for the aforementioned scheme to achieve the optimal element section distribution and to reduce the maximum importance coefficient. The results of this study indicated that different elements had different importance coefficients and that the optimised design effectively improved the structural robustness. The proposed structural optimisation method can be used in theoretical research and engineering applications.

2. Element Importance Analysis

2.1. Definition of the Importance of Elements

Research has indicated that structural robustness can be improved and can be resisted by strengthening the weak areas and important elements of a structure. Thus, the importance of a structural element should be appropriately defined, and this parameter is considered a crucial indicator for reflecting structural robustness. The stronger the effect generated by the failure of an element on a structure, the more critical is the element to the structure, the more severe is the damage produced by the element in the structure, and the lower is the structural robustness. The dynamic responses of the structure, such as the displacement, internal force, and energy response, were studied after the removal of each kind of element. The results indicated that the removals of different elements produced different dynamic responses, which meant that different elements bore different importance. In this study, the importance of elements was determined by examining the postfailure structural state by removing elements and then comparing the prefailure and postfailure structural strain energy. The relevant formula is expressed as follows:

$$k_{\mathrm{i}}=\frac{{\mathrm{U}}_0-{\mathrm{U}}_{\mathrm{i}}}{{\mathrm{U}}_0}$$

(1)

where ${\mathrm{U}}_0$ and ${\mathrm{U}}_{\mathrm{i}}$ represent the strain energy of the structure before and after the removal of element i, respectively, and $k_{\mathrm{i}}$ represents the importance coefficient of element i. The higher the value of $k_{\mathrm{i}}$, the lower is the strain energy remaining in the structure after the removal of element i and the higher is the effect of element i on the structure; thus, the higher the importance of element i, the lower is the structural robustness when this element fails.

2.2. Parameter Analysis for Determining the Importance Coefficient of Elements

An engineering case was analysed to explore the effects of the external load, the initial prestress, and cross-sectional areas of elements on the importance coefficients of elements. This case involved a flexible cable–strut tensile structure with a Geiger-type cable dome. The span of this structure had a length of 71.2 m and a rise of 5.5 m. The aforementioned structure was composed of 16 symmetrical parts, and each part included an outer ridge cable (denoted as RC1), inner ridge cable (denoted as RC2), outer diagonal cable (denoted as DC1), inner diagonal cable (denoted as DC2), outer strut (denoted as WG1), outer hoop cable (denoted as HC1), inner top hoop cable (denoted as THC2), inner lower hoop cable (denoted as LHC2), and inner strut (denoted as WG2). The profile of the considered structure and the corresponding numbers of elements are displayed in Figure 1. The material parameters of the elements and the initial prestress are presented in

Table 1. Areas of the element sections and initial prestress values.

Element	RC1	RC2	DC1	DC2	WG1	WG2	HC1	THC2	LHC2
Area (×10−4 m2)	5.716	2.314	4.976	3.412	11.71	7.011	12.69	9.954	9.954
Prestress (kN)	188.8	80.1	116.8	108.8	−28.0	−14.1	290.5	206.1	272.6

. The inner lower hoop cable and inner upper hoop cable constituted the inner hoop cable. For simplicity, the areas of the element sections of the aforementioned two types of elements were consistent in the parameter analysis and optimisation process. The elastic modulus values of the cables and struts were 160 and 206 GPa respectively. The structure was fixed on a hinged support and bore a design load of 0.4 kN/m².

The effects of the external load, the initial prestress, and the cross-sectional areas of elements on the importance coefficients of the elements are presented in Table 2, Table 3 and Table 4, respectively. On the basis of these tables, the following inferences are obtained:

(1)

Different elements had different importance coefficients. The outer hoop cable had the highest importance coefficient, followed by the inner lower hoop cable and inner upper hoop cable. The inner strut exhibited the smallest importance coefficient.

(2)

The hoop cables exhibited the highest importance coefficients, followed by the ridge cables and diagonal cables. The struts exhibited the smallest importance coefficients.

(3)

In terms of the positions of the elements, the importance coefficients of the outer elements were higher than those of the inner elements.

Table 2. Importance coefficients of the elements under different initial prestress levels.

Element	Importance Coefficients	Prestress
		0.5 P	1.0 P	1.5 P	2.0 P
RC1	$k_1$	0.070163	0.058055	0.053604	0.051754
RC2	$k_2$	0.020178	0.016488	0.013876	0.01325
DC1	$k_3$	0.235073	0.131665	0.067967	0.036867
DC2	$k_4$	0.040101	0.027865	0.023377	0.021531
WG1	$k_5$	0.032925	0.017571	0.011399	0.008266
WG2	$k_6$	0.000098	0.000162	0.000168	0.000155
HC1	$k_7$	0.415335	0.392424	0.384673	0.379775
THC2	$k_8$	0.510334	0.318913	0.182745	0.203077
LHC2	$k_9$	0.319583	0.309017	0.200168	0.118609

Table 2. Importance coefficients of the elements under different initial prestress levels.

Element	Importance Coefficients	Prestress
		0.5 P	1.0 P	1.5 P	2.0 P
RC1	$k_1$	0.070163	0.058055	0.053604	0.051754
RC2	$k_2$	0.020178	0.016488	0.013876	0.01325
DC1	$k_3$	0.235073	0.131665	0.067967	0.036867
DC2	$k_4$	0.040101	0.027865	0.023377	0.021531
WG1	$k_5$	0.032925	0.017571	0.011399	0.008266
WG2	$k_6$	0.000098	0.000162	0.000168	0.000155
HC1	$k_7$	0.415335	0.392424	0.384673	0.379775
THC2	$k_8$	0.510334	0.318913	0.182745	0.203077
LHC2	$k_9$	0.319583	0.309017	0.200168	0.118609

Table 3. Importance coefficients of the elements under different external loads.

Element	Importance Coefficients	External Load
		0.5 F	1.0 F	1.5 F	2.0 F
RC1	$k_1$	0.058936	0.058055	0.057238	0.056325
RC2	$k_2$	0.017094	0.016488	0.015873	0.015250
DC1	$k_3$	0.129883	0.131665	0.133305	0.134860
DC2	$k_4$	0.027908	0.027865	0.027772	0.027710
WG1	$k_5$	0.018979	0.017571	0.015920	0.014031
WG2	$k_6$	0.000130	0.000162	0.000192	0.000222
HC1	$k_7$	0.381889	0.392424	0.402959	0.413494
THC2	$k_8$	0.321704	0.318913	0.316017	0.312898
LHC2	$k_9$	0.311291	0.309017	0.305002	0.300766

Table 3. Importance coefficients of the elements under different external loads.

Element	Importance Coefficients	External Load
		0.5 F	1.0 F	1.5 F	2.0 F
RC1	$k_1$	0.058936	0.058055	0.057238	0.056325
RC2	$k_2$	0.017094	0.016488	0.015873	0.015250
DC1	$k_3$	0.129883	0.131665	0.133305	0.134860
DC2	$k_4$	0.027908	0.027865	0.027772	0.027710
WG1	$k_5$	0.018979	0.017571	0.015920	0.014031
WG2	$k_6$	0.000130	0.000162	0.000192	0.000222
HC1	$k_7$	0.381889	0.392424	0.402959	0.413494
THC2	$k_8$	0.321704	0.318913	0.316017	0.312898
LHC2	$k_9$	0.311291	0.309017	0.305002	0.300766

Table 4. Importance coefficients of the elements under different areas of the element sections.

	Importance Coefficients	k1	k2	k3	k4	k5	k6	k7	k8	k9
Section Sizes
RC1	0.8 A	0.05887	0.015314	0.12173	0.026334	0.015616	0.000153	0.364302	0.296647	0.297071
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.057804	0.017396	0.139213	0.028982	0.01902	0.000169	0.413341	0.328802	0.316454
RC2	0.8 A	0.05802	0.018158	0.12705	0.026886	0.01696	0.000218	0.379508	0.300345	0.302016
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058258	0.015652	0.134901	0.028905	0.018032	0.000251	0.401221	0.291511	0.301105
DC1	0.8 A	0.055947	0.015901	0.131898	0.026784	0.016602	0.000157	0.414111	0.309294	0.298302
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.0596	0.016933	0.131173	0.028698	0.018359	0.000165	0.376069	0.325345	0.316647
DC2	0.8 A	0.058754	0.016018	0.126122	0.029171	0.017425	0.000187	0.377422	0.259439	0.293367
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.057525	0.016819	0.135547	0.026818	0.017648	0.000168	0.402469	0.360229	0.320426
WG1	0.8 A	0.057569	0.016287	0.127649	0.027517	0.017575	0.000159	0.397635	0.315285	0.307203
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058399	0.016637	0.134634	0.028202	0.017574	0.00017	0.387521	0.319196	0.307949
WG2	0.8 A	0.057499	0.016179	0.131715	0.027839	0.017671	0.000176	0.39246	0.298214	0.305824
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058563	0.016524	0.131621	0.028003	0.017487	0.000149	0.392389	0.332275	0.308337
HL1	0.8 A	0.0559	0.01576	0.115023	0.026894	0.016284	0.000161	0.420049	0.309944	0.297148
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.059647	0.017043	0.140861	0.028555	0.018584	0.000162	0.370816	0.324687	0.317635
LHC2	0.8 A	0.057737	0.016635	0.131025	0.027884	0.01751	0.000167	0.390575	0.305499	0.305819
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058158	0.016422	0.132089	0.027596	0.017603	0.000178	0.393661	0.325046	0.311724

Table 4. Importance coefficients of the elements under different areas of the element sections.

	Importance Coefficients	k1	k2	k3	k4	k5	k6	k7	k8	k9
Section Sizes
RC1	0.8 A	0.05887	0.015314	0.12173	0.026334	0.015616	0.000153	0.364302	0.296647	0.297071
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.057804	0.017396	0.139213	0.028982	0.01902	0.000169	0.413341	0.328802	0.316454
RC2	0.8 A	0.05802	0.018158	0.12705	0.026886	0.01696	0.000218	0.379508	0.300345	0.302016
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058258	0.015652	0.134901	0.028905	0.018032	0.000251	0.401221	0.291511	0.301105
DC1	0.8 A	0.055947	0.015901	0.131898	0.026784	0.016602	0.000157	0.414111	0.309294	0.298302
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.0596	0.016933	0.131173	0.028698	0.018359	0.000165	0.376069	0.325345	0.316647
DC2	0.8 A	0.058754	0.016018	0.126122	0.029171	0.017425	0.000187	0.377422	0.259439	0.293367
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.057525	0.016819	0.135547	0.026818	0.017648	0.000168	0.402469	0.360229	0.320426
WG1	0.8 A	0.057569	0.016287	0.127649	0.027517	0.017575	0.000159	0.397635	0.315285	0.307203
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058399	0.016637	0.134634	0.028202	0.017574	0.00017	0.387521	0.319196	0.307949
WG2	0.8 A	0.057499	0.016179	0.131715	0.027839	0.017671	0.000176	0.39246	0.298214	0.305824
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058563	0.016524	0.131621	0.028003	0.017487	0.000149	0.392389	0.332275	0.308337
HL1	0.8 A	0.0559	0.01576	0.115023	0.026894	0.016284	0.000161	0.420049	0.309944	0.297148
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.059647	0.017043	0.140861	0.028555	0.018584	0.000162	0.370816	0.324687	0.317635
LHC2	0.8 A	0.057737	0.016635	0.131025	0.027884	0.01751	0.000167	0.390575	0.305499	0.305819
	1.0 A	0.058055	0.016488	0.131665	0.027865	0.017571	0.000162	0.392424	0.318913	0.309017
	1.2 A	0.058158	0.016422	0.132089	0.027596	0.017603	0.000178	0.393661	0.325046	0.311724

(1)

Effects of the initial prestress on the importance coefficients of the elements

The importance coefficients of the elements were analysed under different initial prestress levels (Table 2). In Table 2, 0.5 P, 1.5 P, and 2.0 P represented 0.5, 1.5, and 2 times the initial prestress, respectively. The results presented in Table 2 indicated, as the prestress increased, that the importance coefficients of all the elements decreased by different extents. The importance coefficients of the outer diagonal cable, outer strut, and inner top hoop cable changed by −84.3%, −74.9%, and −62.9%, respectively. However, the importance coefficients of the outer hoop cables changed marginally by an average of −8.6%.

(2)

Effects of the external load on the importance coefficients of the elements

The importance coefficients of the elements were analysed under different external loads, and the results were presented in Table 3, where 0.5 F, 1.5 F, and 2.0 F represented 0.5, 1.5, and 2 times the initial external load, respectively. As the external load increased, different elements exhibited different trends in their importance coefficients. The importance coefficients of the outer ridge cable, inner ridge cable, inner diagonal cable, outer strut, inner lower hoop cable, and inner top hoop cable decreased with an increase in the external load. By contrast, the importance coefficients of the outer diagonal cable, inner strut, and outer hoop increased with an increase in the external load. When the load increased from 0.5 F to 2.0 F, the importance coefficients of the inner strut and outer strut changed by 70.8% and −26.1%, respectively. For the aforementioned increase in the load, the importance coefficients of the inner diagonal cable, inner lower hoop cable, and inner top hoop cable changed marginally by −0.7%, −2.7%, and −3.4%, respectively.

(3)

Effects of the areas of the element sections on the importance coefficients of the elements

The importance coefficients of the elements were analysed under three cross-sectional areas of the elements, namely 0.8 times the initial cross-sectional area (0.8 A), the initial cross-sectional area (1.0 A), and 1.2 times the initial cross-sectional area (1.2 A). The obtained results were presented in Table 4. The importance coefficients of different elements exhibited different trends with changes in the cross-sectional areas of the elements. For example, when the cross-sectional area of the outer ridge cable increased from 0.8 A to 1.2 A, its importance coefficient decreased gradually, whereas the importance coefficients of the other elements increased gradually. The areas of different elements had different effects on the importance coefficients of the other elements. For instance, when the cross-sectional areas of the outer ridge cable, inner ridge cable, outer diagonal cable, inner diagonal cable, outer strut, inner strut, outer hoop cable, and inner hoop cable increased from 1.0 A to 1.2 A, the importance coefficients of these elements changed by 5.3%, 2.2%, −4.3%, 2.6%, −1.3%, −0.01%, −5.8%, and 0.3%, respectively. The outer hoop cable exhibited the highest importance coefficient. Its importance coefficient decreased with decreases in the cross-sectional areas of the outer ridge cable, inner ridge cable, inner diagonal cable, inner lower hoop cable, and inner upper hoop cable and increases in the cross-sectional areas of the outer diagonal cable, outer strut, inner strut, and outer hoop cable.

In summary, different elements had different importance coefficients, and the outer hoop cable had the highest importance coefficient in this study. Moreover, the importance coefficients of different elements exhibited different trends with variations in the external load, the initial prestress, and areas of element sections.

3. Optimisation of the Element Section Design

3.1. Optimisation Model

Variations in the cross-sectional areas of various elements had different influences on the importance coefficients of the elements. In this study, the element section design was optimised on the basis of the element importance for the above-mentioned Geiger-type cable dome. An optimisation model was developed to minimise the maximum importance coefficient obtained in this study to weaken the effects of the partial failure of elements, especially important elements, on a structure and thus improve the structure’s robustness and resistance to continuous collapse. In this study, element 7 (outer hoop cable) had the highest importance coefficient; thus, the optimisation objective in this study was min k₇.

$$\{\begin{array}{c} \begin{array}{c}\min k_7=\min \frac{U_0-U_7}{U_0}\\ \\ s.t. M\le M_0\\ \end{array}\\ A_{i,\min }\le A_i\le A_{i,\max }\end{array}$$

(2)

In addition to constraints related to the equilibrium and total feasibility of a structure, three other constraints should be satisfied. First, the variation in the cross-sectional area of each element should be 0.8–1.2 times the initial cross-sectional area, and the stress of each element under load should not exceed the yield point. Second, the optimal quality of the structure $M$ should not exceed the structural quality prior to optimisation $M_0$. Third, the deformation should not exceed the allowable value, namely 1/250 times the span length.

3.2. Optimisation Algorithm

A genetic algorithm was adopted to determine the optimal section design according to the element importance, and this algorithm was written using the MATLAB software. Moreover, structural modelling as well as prefailure and postfailure internal force analysis under load were conducted using the ANSYS finite-element analysis software. The ANSYS software could be called automatically by the main program of MATLAB, and the statistics of these two software programs could be cross-read to optimise the element section design. The optimisation process involved the following steps:

(1)

The optimisation parameters of the genetic algorithm were set in MATLAB. In this study, the population size was set as 15, the encoding string length was 32, the crossover probability was 0.8, the mutation probability was 0.2, and the number of evolutionary iterations was 300.

(2)

The initial population generated by MATLAB was imported into ANSYS. Moreover, the importance coefficient k₇ was calculated for every individual in the population. The number of structural robustness indices was equal to the population size.

(3)

The calculation results obtained using ANSYS were imported into MATLAB, and the fitness values were sequenced in accordance with the value 1/k₇. The individual with the highest fitness in the initial population was selected, duplicated, crossed, and mutated for cyclic iteration calculation until the set number of iterations had been completed.

3.3. Optimisation Results

The iteration process for optimisation was illustrated in Figure 2, and the optimisation results were presented in

Table 5. Cross-sectional areas and importance coefficients of the elements before and after optimisation.

	Pre-Optimisation and Post-Optimisation	Pre-Optimisation		Post-Optimisation
Element		Area m2	Importance Coefficient k	Area m2	Importance Coefficient k
RC1		5.716 × 10−4	0.058055	4.573 × 10−4	0.061513
RC2		2.314 × 10−4	0.016488	2.036 × 10−4	0.016403
DC1		4.976 × 10−4	0.131665	5.839 × 10−4	0.121584
DC2		3.412 × 10−4	0.027865	2.730 × 10−4	0.028304
WG1		11.71 × 10−4	0.017571	12.49 × 10−4	0.016256
WG2		7.011 × 10−4	0.000162	6.357 × 10−4	0.000283
HC1		12.69 × 10−4	0.392424	14.89 × 10−4	0.311977
LHC2		9.954 × 10−4	0.318913	8.229 × 10−4	0.232345
THC2		9.954 × 10−4	0.309017	8.229 × 10−4	0.287594

. As the number of iterations increased, the importance coefficient of the outer hoop cable ($k_7$) decreased and finally levelled off to the optimal value of 0.311977 (which represented a decrease of 20.5%). In the optimisation process, the cross-sectional areas of the outer ridge cable, inner ridge cable, outer diagonal cable, inner diagonal cable, outer strut, inner strut, outer hoop cable, and inner hoop cable changed by 20.0%, −12.0%, 17.3%, −20.0%, 6.7%, −9.3%, 17.3%, and −17.3%, respectively. The effects of the cross-sectional areas of elements on the element importance, which were described in the aforementioned text, indicated that changes in the cross-sectional areas of the inner strut had the weakest effect on the importance coefficient of the outer hoop cable. Thus, the inner strut had the lowest optimisation efficiency. The overall idea of element section optimisation was to eliminate the cross-sectional areas of elements with low optimisation efficiencies and provide more optimisation space for elements with high optimisation efficiencies.

4. Optimisation of the Cable Arrangement Scheme

The outer hoop cable had the highest importance coefficient in this study. Thus, the failure of the outer hoop cable would exert the strongest effect on the structure. Consequently, the cable arrangement scheme was optimised by adding an alternative load transfer path to the outer hoop cable to reduce its importance. In this study, the outer elements of the entire structure had a Levy-type arrangement rather than a Geiger-type arrangement so that a Geiger–Levy composite cable dome was formed (Figure 3).

The numbers of outer ridge and outer diagonal cables increased from 16 to 32 after optimisation. According to the Maxwell criterion, the redundancy of the structure was $\mathrm{r}=\mathrm{b}+\mathrm{k}-3\mathrm{j}=176+3\times 16-3\times 64=32$. The redundancy of the structure considerably increased after optimisation, which increased the overall stability of the structure and enhanced its ability to resist continuous collapse. Moreover, compared with a Levy-type cable dome, the optimal structure contained fewer elements, which reduced the weight of the structure. In the optimal structure, the cross-sectional areas of the outer ridge cable and outer diagonal cable were adjusted to 0.6 times the initial cross-sectional area. Basic information on the optimised cable arrangement was presented in

Table 6. Importance coefficients of the elements before and after cable arrangement optimisation.

	Pre-Optimisation and Post-Optimisation	Area (×10−4 m2)	Prestress(KN)	Importance Coefficient of Elements k before Optimisation	Importance Coefficient of Elements k after Optimisation
Element
RC1		3.430	102.7	0.058055	0.074138
RC2		2.314	80.1	0.016488	0.017592
DC1		2.986	63.3	0.131665	0.007768
DC2		3.412	108.8	0.027865	0.028631
WG1		11.71	−28.0	0.017571	0.022302
WG2		7.011	−14.1	0.000162	0.000147
HC1		12.69	290.5	0.392424	0.339590
LHC2		9.954	206.1	0.318913	0.346835
THC2		9.954	272.6	0.309017	0.311623

.

The structural robustness achieved after optimising the cable arrangement was qualitatively analysed. The outer diagonal cable was an alternative load transfer path for the outer hoop cable. When the outer hoop cable was damaged, the outer diagonal cable could bear a part of the load previously borne by the damaged hoop cable. Moreover, the outer hoop cable was the alternative load transfer path for the outer diagonal cable. Thus, the importance coefficients of the outer hoop cable and outer diagonal cable decreased by a certain extent after optimisation. Because of the noticeable enhancement in the structural redundancy after cable arrangement optimisation, the connections between the elements in the subsystem were changed, and the overall structural reliability and robustness improved.

The importance coefficients of all the elements were recalculated (Table 6). After optimisation, the importance coefficient of the hoop cable reduced from 0.392424 to 0.33959, which represented a decrease of 13.5%. The importance coefficient of the outer diagonal cable also reduced considerably after optimisation, which indicated that the optimal outer diagonal cable and outer hoop cable were alternative load transfer paths for each other; thus, their importance coefficients reduced after optimisation. After cable arrangement optimisation, the importance coefficients of the different elements exhibited different trends, with the importance coefficient of the outer diagonal cable exhibiting the highest change after optimisation (−94.1%), followed by those of the outer ridge cable and outer strut (27.7% and 26.9% respectively). The inner upper hoop cable exhibited the lowest change in the importance coefficient after optimisation (only 0.8%). After optimisation, the element with the highest importance coefficient changed from the outer hoop cable to the inner lower hoop cable. The highest importance coefficient decreased from 0.392424 before optimisation (for the outer hoop cable) to 0.346835 after optimisation (for the inner lower hoop cable); thus, the highest importance coefficient decreased by 11.6% after optimisation, which indicated a strong optimisation effect. Consequently, cable arrangement optimisation considerably reduced the maximum importance coefficient. The optimised cable arrangement scheme offered alternative load transfer paths for key structural elements.

5. Comprehensive Optimisation of the Cable Arrangement and Element Section Design

Optimisation Model

On the basis of the cable arrangement optimisation, the genetic algorithm was used to optimise the element section design for further reducing the maximum importance coefficient; thus, the aim was to obtain the minimum value of ${\mathrm{k}}_{\max }^{\mathrm{t}}$, where ${\mathrm{k}}_{\mathrm{i}}^{\mathrm{t}}$ represented the importance coefficient of element i of the optimal structure after cable arrangement optimisation. Considering the small difference between the importance coefficients of the outer hoop cable and inner hoop cable (which comprised the inner lower hoop cable and inner upper hoop cable) and the different influences generated by changes in the cross-sectional areas of various elements on these importance coefficients, the optimisation objective of the calculation model used in this study was $\min \left\{\max \left\{k_7^{\mathrm{t}},k_8^{\mathrm{t}},k_9^{\mathrm{t}}\right\}\right\}$. The cross-sectional areas of the elements could vary from 0.8 to 1.2 times the initial cross-sectional area, and the stress under load could not exceed the yield point. The total weight of the optimal structure $M$ could not exceed the total weight of the initial structure $M_0$. The mathematical formula of the adopted calculation model was as follows:

$$\{\begin{array}{c} \begin{array}{c} \min \left\{\max \left\{k_7^t,k_8^t,k_9^t\right\}\right\}=\min \left\{\max \left\{\frac{U_0-U_7^t}{U_0},\frac{U_0-U_8^t}{U_0},\frac{U_0-U_9^t}{U_0}\right\}\right\}\\ \\ s.t. M\le M_0\\ \end{array}\\ A_{i,\min }\le A_i\le A_{i,\max }\end{array}$$

(3)

The aforementioned iterative optimisation process was illustrated in Figure 4, and the optimisation results were presented in

Table 7. Cross-sectional areas and importance coefficients of different elements before and after comprehensive optimisation.

	Pre-Optimisation & Post-Optimisation	Pre-Optimisation		Post-Optimisation
Element		Area (m2)	Importance Coefficient k	Area (m2)	Importance Coefficient k
RC1		3.430 × 10−4	0.074138	2.744 × 10−4	0.073988
RC2		2.314 × 10−4	0.017592	2.036 × 10−4	0.016918
DC1		2.986 × 10−4	0.007768	3.025 × 10−4	0.006751
DC2		3.412 × 10−4	0.028631	2.730 × 10−4	0.027974
WG1		11.71 × 10−4	0.022302	13.42 × 10−4	0.020618
WG2		7.011 × 10−4	0.000147	5.983 × 10−4	0.000266
HC1		12.69 × 10−4	0.339590	13.88 × 10−4	0.282688
LHC2		9.954 × 10−4	0.346835	7.963 × 10−4	0.246262
THC2		9.954 × 10−4	0.311623	7.963 × 10−4	0.283549

. After optimisation, the element with the maximum importance coefficient changed from the inner lower hoop cable to the inner upper hoop cable, and the maximum importance coefficient decreased from 0.346835 to 0.283549, which represented a decrease of 18.2%. The highest importance coefficient obtained after optimisation was 27.7% lower than that obtained before optimisation. In the optimisation process, different elements exhibited different trends in the importance coefficient. The cross-sectional areas of the outer ridge cable, inner ridge cable, inner diagonal cable, inner strut, and inner hoop cable decreased by 20%, 12%, 20%, 14.7%, and 20%, respectively, after the aforementioned optimisation. However, the cross-sectional areas of the outer ridge cable, outer strut, and outer hoop cable increased by 1.3%, 14.6%, and 9.4%, respectively, after the optimisation. Subsequently, the element section design was optimised under no quality constraints. After this optimisation, the maximum importance coefficient of the optimal structure decreased from 0.346835 to 0.278006 (a decrease of 19.8%). Thus, the aforementioned optimisation process resulted in a greater reduction in the maximum importance coefficient than did that conducted under quality constraints.

6. Conclusions

In this study, the design of a flexible cable–strut tensile structure was optimised on the basis of structural robustness and element importance. First, the importance coefficients of different elements of a structure were determined by comparing its structural prefailure and postfailure strain energy to reflect the structural robustness. Moreover, the effects of the external load, the initial prestress, and the cross-sectional areas of elements on the importance coefficients were analysed. Second, a genetic algorithm was used to conduct element section optimisation design and minimise the maximum of importance coefficient. Third, the cable arrangement was optimised by adding an alternative load transfer path to the outer hoop cable, which exhibited the maximum importance coefficient in this study. The arrangement of the outer elements of the structure was changed from a Geiger-type arrangement to a Levy-type arrangement. Finally, the cable arrangement and element section design were comprehensively optimised to achieve an optimal section distribution and reduce the maximum importance coefficient further. The results of this study revealed that: (1) different elements had different importance coefficients, which exhibited different trends with changes in the external load, the initial prestress, and the cross-sectional areas of elements. (2) After element section optimisation, the importance coefficient of the outer hoop cable, which had the highest importance coefficient, reduced from 0.392424 to 0.311977, which represented a decrease of 20.5%. (3) After cable arrangement optimisation, which involved adding an alternative load transfer path for the outer hoop cable, the maximum importance coefficient (i.e., the importance coefficient of the outer hoop cable) reduced from 0.392424 to 0.346835, which represented a decrease of 11.6%. (4) After the comprehensive optimisation of the cable arrangement and element section design, the maximum importance coefficient decreased from 0.346835 to 0.283549. The maximum importance coefficient obtained after the comprehensive optimisation was 27.7% lower than that of the initial structure. Thus, the structure optimisation method proposed in this paper can be used in theoretical research and engineering applications.