Optimization of Steel Consumption for Prestressed Spatial Arch-Supported Partial Single-Layer Reticulated Shells

Abstract

Steel smelting and production produces a large amount of exhaust gas, which is damaging to the environment. Prestressed spatial arch-supported partial single-layer reticulated shells (PSASPSRSs) are introduced to promote sustainable development in the construction industry. An optimization strategy based on uniform design experiments and iterations is proposed with respect to the design of PSASPSRSs. The optimization aims to reduce steel consumption as much as possible. The optimization constraint takes into account the stability coefficient, frequency, and deflection of the structures. The search space gradually shrinks around the local optimal solution and moves toward the global optimal solution during the optimization process. The optimization procedure stops when the error between local optimal solutions is less than the permitted error of 5%. The tensile force of the prestressed cable, the unified design stress ratio of the members, and the radial grid number of the single-layer reticulated shells act as optimization variables in the finite element model. The parametric analysis revealed that the radial grid number of single-layer reticulated shells significantly affected steel consumption, which was reduced by 13% in the optimized structure. The effectiveness and the practicality of the proposed optimization strategy in the initial design of complicated space grid structures are systematically illustrated.

1. Introduction

The construction industry is responsible for consuming nearly 40% of the global energy supply and emitting over 35% of the overall volume of greenhouse gases into the atmosphere (Maxineasa 2021; Ürge-Vorsatz 2020) [1,2]. Given the significant growth of the world’s population, it is expected that the consumption of construction materials will increase thanks to the need to expand the built environment. Steel is considered a main material for the construction industry, with more than 50% of the global steel supply used in construction (Santos 2017; Oluwafemi 2021) [3,4]. Steel production contributes 25% of the world’s industrial carbon emissions (Griffin 2019; Kerr 2022) [5,6]. It is therefore critically important to minimize steel consumption in the built environment in order to achieve sustainable development in the construction industry (Sudarsan 2022) [7].

To realize this purpose, researchers in the civil engineering field have achieved remarkable results. For instance, Lonetti (2014) [8] presented a design methodology to predict the optimal dimensioning of the cable system for suspension bridges to achieve the lowest steel consumption involved in the cable system and the maximal performance of the cables. Wang (2021) [9] used the finite element analysis software STAAD to optimize the section of a large thermal power plant, and the optimization results saved more than 500 tons of steel compared with the original value. Saedi (2021) [10] optimized the seismic design of steel frames with a steel shear wall system, based on a modified dolphin algorithm. The designed structures with the minimum steel consumption satisfy all the requirements in the seismic design and steel structures design code. Yang (2020) [11] proposed an improved population initialization method to minimize the steel consumption of steel truss bridges. Amico (2018) [12] proposed a computational tool that aimed to help practitioners to design material-efficient structures for multistory building frames. Al-Obaidy (2022) [13] examined the correlation between building material selection and carbon emissions and proposed a parameterization method that optimizes sustainable architectural design.

To date, several forms of steel roofs have been developed (Gasii 2017; Zhang 2020; Wang 2020) [14,15,16]. The author’s team also proposed prestressed spatial arch-supported partial single-layer reticulated shells (PSASPSRSs), which have been applied to practical engineering in Zunyi, China. This kind of novel structure is capable of achieving a load-bearing advantage and can significantly reduce the demand of the supports by using prestressed cables. In addition, the shape of the reticulated shell has a sense of rhythm, fully demonstrating the architectural aesthetics.

Toward the sustainable development of PSASPSRSs, steel consumption should be optimized in the structural design phase. However, owing to the complexity of the variables, existing optimization methods are not suitable for such complicated space grid structures (Sun 2020; Lu 2021; Kumar 2021) [17,18,19]. Therefore, this paper develops a multiparameter, multilevel optimization method that is based on the characteristics of PSASPSRSs. This method combines a uniform design method and iterations aimed at ensuring structural safety while reducing steel consumption. A PSASPSRS with 1416 nodes is used as an optimization case, where the tensile force of the prestressed cable, the unified design stress ratio of the members, and the radial grid number of the single-layer reticulated shells act as optimization variables. All models were analyzed for nonlinear buckling by using the general finite element software ANSYS. The variation law of each index with the design parameters was obtained through the parameter analysis. In conclusion, the effectiveness and the practicality of the proposed optimization strategy in the initial design of complicated space grid structures are systematically illustrated through comparisons with other works.

2. Optimization Object Statement

2.1. Building Overview

A gymnasium at Zunyi Normal University, with a seating capacity of 4000, is a multifunctional structure, whose events range from sports competitions to large-scale performances. The construction area is 7100 m², and the activity site area is 4200 m². The indoor and outdoor views of this gymnasium are shown in Figure 1. The roof is a symmetrical steel space structure with an undulating shape. The substructure is a reinforced concrete system. The gymnasium was designed in the shape of an octagon.

2.2. Composition of the Roof Structure

2.2.1. General Description

The roof structure consists of a crossed spatial arch and four partial single-layer hyperbolic paraboloid reticulated shells, as shown in Figure 2. Generally, a spatial arch has high in-plane stability but is prone to out-of-plane instability (Guo 2017; He 2020) [20,21]. For a hyperbolic paraboloid reticulated shell, owing to the presence of the negative Gaussian curvature, partial members are in tension, which improves the overall stability and stiffness of the roof (Ishakov 1999) [22]. The PSASPSRS combines the advantages of the high load-bearing capacity of an arch with the high stability of a hyperbolic paraboloid reticulated shell. The main force transfer components are the spatial arches and the double-layer reticulated shells. The single-layer reticulated shell is in the membrane stress region. Additional web members are placed among the main force transfer components and the single-layer reticulated shell to gradually change the bending stiffness. Reinforced concrete columns are used for the substructure. The prestressed cables are arranged between the supports at the top of the substructure, thus reducing the cost of the reinforced concrete columns and the foundation.

As shown in Figure 3, the maximum diagonal size of the roof structure is 82 m, and the height is 13 m. The grid of the reticulated shells is a three-way grid. The members of the reticulated shell are pipes that are connected by welded, hollow spherical balls. The pipe truss arch is formed by intersecting connections. The prestressed cables are composed of steel strands with a yield strength of 1.86 GPa. The rest of the steel is Q345 (China Architecture and Building Press 2017) [23].

2.2.2. Construction Process of Roof Structure

The PSASPSRS is divided into three parts: spatial arch, double-layer reticulated shell, and single-layer reticulated shell. The spatial arch and the double-layer reticulated shell are installed in situ first because they are the main force parts. The single-layer reticulated shell is then lifted into the installation position after being assembled on the ground. Compared to in situ installation, welding the single-layer reticulated shell on the ground reduces the weld deformation in the overall structure. To improve the construction efficiency of the PSASPSRS, the block installation method was used along with the lifting method. The construction process of PSASPSRS is shown in Figure 4. The lifting method allows for a reduction in the use of scaffolding. The prestressed cables penetrate the hollow balls of the plate rubber supports. The anchor nodes are located at the supports of the spatial arch. To avoid uneven forces in PSASPSRS after tensioning, all prestressed cables are simultaneously tensioned.

3. Proposed Structural Optimization Method

3.1. Overview

According to the features of the PSASPSRS structure, the preliminary design requires the rapid determination of the scope of the single-layer reticulated shells, the prestressing force, the grid size, and the cross section of the members. These variables influence each other, and it is difficult to distinguish whether a variable is an active or a passive variable. Constraints such as strength, deformation, and stability are difficult to express as functions. Compared with other methods, the response surface method has no strict requirements for variables and constraints (Li 2021) [24]. In addition, its operation is basically programmed. It is adequate for the multilevel optimization of these types of complex structures (Chai 2018; Datta 2020; Wang 2006) [25,26,27]. The experimental design method is used to obtain experimental results that achieve valid statistical analysis by developing efficient experimental schemes (Richard 2012) [28]. There are many reports on the optimization of a spatial structure through experimental design (Winslow 2009; Zhu 2020) [29,30].

Iteration is necessary to improve the accuracy of optimization. Modeling, calculation, and sensitivity analysis are time-consuming. It is hoped that the model will be simple and that the number of iterations will be small. Although the use of symmetry complicates the boundary constraints and boundary forces of the basic model, it speeds up the calculation. This is more obvious for iterative calculations. It is necessary to use symmetry for complex symmetric structures. Therefore, we propose a fast and efficient optimization strategy that is based on a symmetric structure, the uniform design method, and iteration.

3.2. Basic Principles of Structural Optimization

3.2.1. Structural Optimization Formulation

The structural optimization is described in Equation (1) (Christensen 2008) [31]:

$$\begin{array}{l}\mathrm{Minimize} f(\mathit{x})\\ \mathrm{Subject} \mathrm{to}\{\begin{array}{l}{y}_i(\mathit{x})\ge 0 1\le i\le r\\ {\mathit{x}}^L\le \mathit{x}\le {\mathit{x}}^U\end{array}\end{array}$$

(1)

where f(x) is the objective function, which is expressed in terms of steel consumption or cost; x is a design parameter vector; $y_i\left(\mathit{x}\right)$ represents the structural responses, such as the displacement, stress, and stability coefficient; i is the number of the structural responses; and ${\mathit{x}}^U$ and ${\mathit{x}}^L$ are the upper and lower bound vectors of x, respectively.

In Equation (1), the objective function is implicit relative to the design parameters. Therefore, evaluating the function requires calculating results at each point in the design space, resulting in a large computational burden. The response surface method (RSM) can identify optimal parameters at a lower cost. RSM enables the implicit optimization problem to substitute for the explicit optimization problem (Oudjene 2009) [32].

3.2.2. Functional Projection Theorem

If M is a complete linear space of the inner product space X, then

$$\forall \mathit{x}\in X \exists ! {\mathit{x}}_0\in M$$

(2)

That is,

$$\mathit{z}\in M^{\perp } \mathrm{and} \mathit{x}={\mathit{x}}_0+\mathit{z}$$

(3)

This theorem shows that the projection of a point to a closed convex set is exactly the best approximation element of this point for the closed convex set (Lu and Qiu 2021) [33]. Thus, it is possible to obtain the best approximation of the optimal solution on the basis of this theorem. This means that the iterative method can be used in the optimization process to gradually converge on the optimal solution.

3.2.3. Extremum Principle

If M is a linear subspace of the inner product space H, x ∈ H, and ${\mathit{x}}_0$∈ M, then

$$\Vert \mathit{x}-{\mathit{x}}_0\Vert =\underset{\mathit{y}\in M}{inf}\Vert \mathit{x}-\mathit{y}\Vert$$

(4)

where ${\mathit{x}}_0$ is the projection of x onto M. The distance from the point y to the approximation element ${\mathit{x}}_0$ is calculated with

$$d\left({\mathit{x}}_0,\mathit{y}\right)=\Vert {\mathit{x}}_0-\mathit{y}\Vert$$

(5)

The error ε is defined as

$$\epsilon =\frac{\Vert {\mathit{x}}_0-\mathit{y}\Vert }{\Vert {\mathit{x}}_0\Vert }$$

(6)

Structural optimization problems often require practical criteria for terminating iterations (Jiang 2020) [34]. Therefore, when ε is less than the allowable error [ε], the iteration is terminated, and the latest optimized solution is considered the global optimal solution.

3.3. Uniform Design

The experimental points of the uniform design are uniformly distributed in the experimental space (Fang 2000) [35]. In terms of the deviation of the uniformity of the experimental points, the uniform design is superior to the orthogonal design.

The uniform design table is expressed as U_n(q^s), where n is the number of the experiment, q is the number of parameter levels, and s is the number of parameters. The orthogonal design table is expressed as L_n(q^s) (Ma 2001) [36]. For s = 3 and q = 4, the experimental points are shown in Figure 5, where the red balls represent the experimental points.

3.4. Mirroring and Basic Block

The calculation time is related directly to the unknown quantity that existed in the finite element analysis (Mammoli 1999) [37]. The time taken to solve the equations with the usual solution is proportional to the cube of the number of unknowns (Itu 2021) [38].

The superelement method can also reduce the number of unknown quantities and speed up the analysis process (Qiu 2009) [39]. However, the loads for these types of structures are applied mainly to the top chord nodes, which must be set as the primary nodes. Therefore, the number of unknown quantities reduced by the superelement method is not obvious. The analytical methods of symmetrical structures in structural mechanics can comparatively eliminate more unknowns.

Selecting a part of the structure according to symmetry as the basic block makes the problem clearer. Compared with the original structure, the number of unknowns has been reduced, and fast calculation has been achieved (Noor 1987) [40]. In order to simplify the analysis, the nonlinear influence is not considered. Additionally, the superposition principle is available.

Symmetry includes mirroring, rotation, and translation (Antoine 2021; Richardson 2013) [41,42]. Only mirroring is considered here. According to the characteristics of the roof structure, we discuss only the mirroring of the x and y axes, that is, the one-quarter symmetric structure.

As shown in Figure 6, the original structure is composed of the same four blocks. One of the blocks, for example, the block in the first quadrant (Q. I), is defined as the basic block. The way to solve the problem is to analyze the basic block and combine the results. As long as the combined results are the same as those of the original structure, the indexes of the original structure can be obtained. The indexes include the steel consumption and the structural responses $y_i\left(\mathit{x}\right)$. Therefore, the problem comes down to determining the stiffness matrix, boundary conditions, equivalent loads, and actions of the basic block, as well as the combination coefficient of the indexes.

It is relatively easy to determine the stiffness matrix of the basic block. The elements where all the nodes are on the mirror plane appear to overlap after mirroring and need to be resolved. It is assumed that g is the number of element overlaps. Its element stiffness matrix is divided by g; then, it is consistent with the original structure after mirroring. The other elements are unique in each symmetric block after mirroring. These elements that do not overlap are not dealt with. Figure 6 shows the boundary conditions and the load decomposition of each block when load L acts on the second quadrant (Q. II). The meanings of the symbols in the figure and the description are as follows:

• Q. I, Q. II, Q. III, and Q. IV are the first, second, third, and fourth quadrants, respectively;
• L is a load of any form that can be replaced by a combination of symmetry and antisymmetry. For simplicity, complex loads are simplified during optimization. Here, the seismic action is approximately equivalent to the static load. The example here is the z-direction load;
• S indicates that only symmetric constraints exist in the boundary conditions. A indicates that only antisymmetric constraints exist in the boundary conditions.

  Table 1. Degrees of freedom corresponding to different boundary conditions.

  Type	Symmetry Plane	Boundary Condition	Degree of Freedom
  			Ux	Uy	Uz	Rotx	Roty	Rotz
  1	xoz	A	F	R	F	R	F	R
  2	xoz	S	R	F	R	F	R	F
  3	yoz	A	R	F	F	F	R	R
  4	yoz	S	F	R	R	R	F	F

  Note: S and A represent symmetric and antisymmetric boundary conditions, respectively; F and R represent freezing and releasing of degrees of freedom, respectively.

  shows the degrees of freedom corresponding to the different boundary conditions, where U_x, U_y, and U_z are the translational degrees of freedom and where Rot_x, Rot_y, and Rot_z are the rotational degrees of freedom. With the effect of the symmetry constraint, the points on the symmetry plane cannot undergo out-of-plane movement or in-plane rotation. With the antisymmetric constraint, the points on the symmetry plane cannot move in plane or rotate out of plane.

In terms of boundary conditions and loads, for a one-quarter symmetric structure, there are seven models of a basic block.

Table 2. Model classification when the basic block is in Q. I.

Load and Location in the Original Structure		Model of the Basic Blocks
		Boundary Condition		Equivalent Load	Model Type
		x-Axis	y-Axis
1.0L	Q. I	S	S	+0.25L	1
		S	A	+0.25L	2
		A	S	+0.25L	3
		A	A	+0.25L	4
	Q. II	S	S	+0.25L	1
		A	S	+0.25L	3
		S	A	−0.25L	5
		A	A	−0.25L	6
	Q. III	S	S	+0.25L	1
		S	A	−0.25L	5
		A	S	−0.25L	7
		A	A	+0.25L	4
	Q. IV	S	S	+0.25L	1
		A	S	−0.25L	7
		S	A	+0.25L	2
		A	A	−0.25L	6

shows the boundary conditions and the equivalent load of the basic block located in Q. I when the loads are in each of the four quadrants. Each working condition of the original structure is considered by quadrant. The processing of all the working conditions is the same, which is convenient for programmed operations. The combination coefficients of all the indexes are determined according to the working condition combination of the original structure.

3.5. Optimization Process

On the basis of the above basic principles, the following optimization process is proposed. The optimization strategy is shown in Figure 7.

• Step 1: The regression analysis is carried out on the uniform design experimental data, and the functional relationship between the parameters and the indexes is obtained. Subsequently, the nonlinear programming problem is solved to obtain the first optimization point ${\mathit{x}}_{\mathrm{opt}}^1$.
• Step 2: For convex set programming, a single optimization may not give the global solution. Thus, ${\mathit{x}}_{\mathrm{opt}}^1$ may be a local optimal solution, not a global solution. Here, ${\mathit{x}}_{\mathrm{opt}}^1$ is taken as the center, and with it, the parameter level combination for the second optimization is designed. Next, the new numerical experiments are carried out, and the second optimization point ${\mathit{x}}_{\mathrm{opt}}^2$ is obtained.
• Step 3: Because the second optimization is based on the first optimization result, ${\mathit{x}}_{\mathrm{opt}}^2$ is closer to the global optimal solution than ${\mathit{x}}_{\mathrm{opt}}^1$ is. If the error between ${\mathit{x}}_{\mathrm{opt}}^1$ and ${\mathit{x}}_{\mathrm{opt}}^2$ is less than the allowable value [ε], ${\mathit{x}}_{\mathrm{opt}}^2$ is regarded as the global optimization point. Otherwise, it is necessary to continue the iteration until the error ε between optimization points ${\mathit{x}}_{\mathrm{opt}}^{\mathrm{m}}$ and ${\mathit{x}}_{\mathrm{opt}}^{\mathrm{m}-1}$ is less than [ε], and the steps are the same as those from before.

The flow of the optimization process is shown in Figure 8. The main parameters are the key variables that have a significant effect on structural performance. In this study, the main parameters include the tensile force, the unified design stress ratio of members, and the radial grid number of single-layer reticulated shells. The constraints are generally listed according to the specification requirements, such as the deflection limitation, the stress of the members, and the stability coefficient of the structure (China Architecture and Building Press 2010) [43].

The finite element analysis of the basic block is achieved by using the general finite element software ANSYS. An elastic plasticity analysis is performed on the structure because of the large deformations and material nonlinearities. Initial geometric imperfection is introduced during the elastic plasticity analysis. According to the Chinese specification (China Architecture and Building Press 2010) [43], initial geometric imperfection is the first-order buckling mode obtained through a linear stability analysis under combined load conditions.

MS Excel was utilized to perform a regression analysis and establish the functional relationship between each index and the parameters. The regression functions were analyzed for extreme values by using the technical computing software Wolfram Mathematica to identify the optimization parameter combination. Next, new experimental points were selected near the latest optimization parameter combination, marking the first step in the iterative process. In each regression analysis, all experimental points within the global domain were taken into account to ensure that the regression function approximated the actual response surface. If the error between the latest two optimization results is less than allowable [ε], the global optimal solution is between the two results. The iterative process can be stopped. The solution with the minimum steel consumption is taken as the global optimal solution.

4. Case Study: Optimization of a PSASPSRS

4.1. Case Description

The optimization case involved in this study is based on an original design of PSASPSRS, which consists of 1416 nodes. The original structure contains a total of 1416 nodes. The steel consumption of the original design solution was 170 tons. The standard values of the loads for the original design are as follows: the dead load of the top chord layer is 0.3 kN/m², the live load of the top chord layer is 0.5 kN/m², the live load of the bottom chord layer is 0.1 kN/m², the wind load is 0.35 kN/m², and the temperature differences are −25 °C and 30 °C. The seismic fortification intensity is grade six, and the site is class I. Here, the seismic action is approximately equivalent to the static load.

The optimization target is the steel consumption G. The key parameters include the radial grid number k of single-layer reticulated shells, the unified design stress ratio ρ of the members of a single-layer reticulated shell, and the tensile force T of the prestressed cable. Notably, the vertical projection of a single-layer reticulated shell is triangular, and the grid number on each side is the same. Therefore, the radial grid number k can represent the scope of the single-layer reticulated shells.

4.2. First Optimization Results

The constraint conditions have the absolute value of the maximum upward deflection $w_u$, the absolute value of the maximum downward deflection $w_d$, the basic frequency f, and the stability bearing capacity coefficient λ. The number of levels for each design parameter is four. The specific values are shown in

Table 3. Levels of design parameters.

Level Number	Design Parameters
	k	ρ	T/kN
1	9	1.0	1400
2	7	0.8	1000
3	5	0.6	600
4	3	0.4	200

. The first experiment is performed according to the uniform design table U₈(4³) with a CD₂ deviation of 0.022386. The arrangement and the results of the first experiment are shown in

Table 4. Arrangement and results of the first experiment.

Experiment Number	Combination of Design Parameters			Results
	k	ρ	T	G/ton	${\mathit{w}}_{\mathit{u}}$/mm	${\mathit{w}}_{\mathit{d}}$/mm	f/Hz	λ
1	3 (4)	0.8 (2)	200 (4)	205.83	60.6	125.3	2.792	34.374
2	3 (4)	0.6 (3)	1000 (2)	206.21	45.5	101.5	2.801	32.599
3	9 (1)	0.8 (2)	1000 (2)	155.93	51.3	144.3	2.931	16.565
4	5 (3)	0.4 (4)	600 (3)	197.12	59.2	121.4	2.84	25.924
5	5 (3)	1.0 (1)	1400 (1)	194.97	36.2	91.4	2.872	25.665
6	9 (1)	0.6 (3)	200 (4)	166.48	66.6	150.4	2.892	22.214
7	7 (2)	0.2 (4)	1400 (1)	197.63	35.6	86.3	2.793	24.375
8	7 (2)	1.0 (1)	600 (3)	171.96	58.3	150.4	2.959	23.598

Note: (1)–(4) indicate the level number of each parameter.

.

To improve the accuracy of the response surface function and satisfy the minimum experiment number, a second-order polynomial regression model is chosen:

$$f(\mathit{x})=a+{\displaystyle \sum _{i=1}^n{b}_i{\mathit{x}}_i}+{\displaystyle \sum _{i=1}^n{c}_i{\mathit{x}}_i^2}$$

(7)

where a, b_i, and c_i are the parameters to be determined, which can be obtained with interpolation techniques. The first regression analysis is carried out, and the response surface functions of the experimental indexes are obtained, as follows:

$$G=222.206839385+2.5737857141k-0.0372375324T-16.93571429{\rho }^2-0.81873213k^2+0.000025968T^2$$

(8)

$$w_u=56.9957756+2.284222156k-0.005188426245T-0.251055{\rho }^2-0.12125225k^2-0.0000117188T^2$$

(9)

$$w_d=100.925178612-0.63142857255k+0.0415668432T+21.42857143{\rho }^2+0.526785714k^2-0.0000501563T^2$$

(10)

$$f=2.5979676225+0.03836517921k+0.00020021635T+0.11144645{\rho }^2-0.00167732k^2-0.00000014194T^2$$

(11)

$$\lambda =46.290204021-3.0830946334k-0.01208093118T-1.244107144{\rho }^2+0.07316831k^2+0.0000062011T^2$$

(12)

The significance test of each response surface function of the first regression is shown in

Table 5. Significance test of first regression.

Regression Data	Experimental Indexes
	G/ton	${\mathit{w}}_{\mathit{u}}$/mm	${\mathit{w}}_{\mathit{d}}$/mm	f/Hz	λ
Correlation coefficient Re	0.9993	0.9996	0.9992	0.998	0.9865
Standard error σ	1.85	0.77	2.66	0.011	2.011
F-statistic	126.91	203.81	111.71	41.99	6.04
Significance F	0.0678	0.0471	0.0723	0.117	0.3017
Significance order of design parameters	T2, T, ρ, k2, k, ρ2	k, T2, k2, ρ, ρ2, T	T2, T, k2, ρ2, ρ, k	T2, T, k, k2, ρ, ρ2	k, k2, T, ρ, ρ2, T2

. The correlation coefficients R_e of all the experimental indexes are close to 1, which indicates that each response surface has a good approximation at the experimental point. From the perspective of significance F, it is still necessary to add experimental points to improve the significance of each response surface function.

A conditional extremum problem is shown in Equation (13). The constraints are determined by the specification JGJ 7-2010 (China Architecture and Building Press 2010) [43]. L_s is the span of the structure.

$$\begin{array}{l}\mathrm{Minimize} G(k,\rho ,T)\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}{w}_u\le L_s/400 \\ w_d\le L_s/400\\ \lambda \ge 4.2\\ 0<\rho <1 \\ 3\le k\le 9 , k\in \mathrm{int}\\ 200\le T\le 1400\end{array}\end{array}$$

(13)

The first optimization parameter combination ${\mathit{x}}_{\mathrm{opt}}^1$ = {$k_1^{*}$ = 9, ${\rho }_1^{*}$ = 1.0, $T_1^{*}$ = 717} is obtained by solving the nonlinear programming problem. The finite element model is established by using the ${\mathit{x}}_{\mathrm{opt}}^1$, and next, the experimental values (EV) for each index are calculated. The reliability of the regression function can be verified by comparing the error between the EV and the regression values (RVs). The first optimization result and its verification are shown in Figure 9. The error of $w_u$ is the largest, at 6.4%.

4.3. Second Optimization Results

Eight experimental points centered on ${\mathit{x}}_{\mathrm{opt}}^1$ are added for the second uniform experiment. The arrangement and the results of the second experiment are shown in

Table 6. Arrangement and results of the second experiment.

Experiment Number	Combination of Design Parameters			Results
	k	ρ	T	G/ton	${\mathit{w}}_{\mathit{u}}$/mm	${\mathit{w}}_{\mathit{d}}$/mm	f/Hz	λ
9	9 (4)	0.8 (2)	1117 (4)	157.12	47.5	139.1	2.933	16.438
10	9 (4)	1.0 (3)	417 (2)	150.95	67.3	178.1	2.988	19.079
11	5 (1)	0.8 (2)	417 (2)	190.02	61.3	126.2	2.889	26.767
12	9 (3)	1.2 (4)	717 (3)	149.57	61.7	169.7	2.979	17.534
13	9 (3)	0.6 (1)	117 (1)	167.09	67.8	152.2	2.894	23.392
14	5 (1)	1.0 (3)	1117 (4)	192.35	43.7	103.4	2.886	25.661
15	7 (2)	1.2 (4)	117 (1)	173.76	67.2	162.3	2.953	25.995
16	7 (2)	0.6 (1)	717 (3)	177.46	56.9	124.5	2.92	23.668

.

The second regression analysis is performed. The latest regression functions of the experimental indexes are as follows:

$$G=244.2206-42.91234887\rho -1.210138247k-0.0327222T+13.993592{\rho }^2-0.5235892k^2+0.00002824T^2$$

(14)

$$w_u=66.01077189-15.39301402\rho +0.48685078k-0.003659186T+10.89603276{\rho }^2+0.039135737k^2-0.0000129881T^2$$

(15)

$$w_d=131.1305238-29.50468542\rho -7.475622554k+0.024924735T+40.72827896{\rho }^2+1.129982787k^2-0.0000399546T^2$$

(16)

$$f=2.449312779+0.297964356\rho +0.064348607k+0.000157440T-0.103681109{\rho }^2-0.003668456k^2-0.000000120775T^2$$

(17)

$$\lambda =48.30469076-6.03712289\rho -2.868501999k-0.012844939T+2.491732205{\rho }^2+0.056141092k^2+0.00000642788T^2$$

(18)

The significance test of the second regression is shown in

Table 7. Significance test of the second regression.

Regression Data	Experimental Indexes
	G/ton	${\mathit{w}}_{\mathit{u}}$/mm	${\mathit{w}}_{\mathit{d}}$/mm	f/Hz	λ
Correlation coefficient Re	0.9974	0.9951	0.9878	0.9986	0.9813
Standard error σ	1.79	1.38	5.54	0.012	1.259
F-statistic	295.72	251.16	120.63	64.99	39.83
Significance F	8.17 × 10−10	1.62 × 10−8	7.06 × 10−7	7.71 × 10−7	6.13 × 10−6
Significance order of design parameters	T2, T, k2, ρ, ρ2, k	T2, ρ2, ρ, T, k, k2	T2, k2, ρ2, T, k, ρ	k, T2, T, k2, ρ, ρ2	T, T2 k, ρ, k2, ρ2

. Compared with the results of the first significance test, the standard error σ of the response surface function decreases, while the F-statistic increases. This implies that the validity of the response surface is increased and that the predicted results are credible.

Similarly, the second optimization parameter combination ${\mathit{x}}_{\mathrm{opt}}^2$ = {$k_2^{*}$ = 9, ${\rho }_2^{*}$ = 1.0, $T_2^{*}$ = 687} is obtained by solving the conditional extremum. The second optimization result and its verification are shown in Figure 10. The error of λ is the largest, at 1.12%.

The definition of the convergence criterion is as follows:

$$\frac{\Vert {\mathit{x}}^{(j+1)}-{\mathit{x}}^{(j)}\Vert }{\Vert {\mathit{x}}^{(j)}\Vert }<\left[\epsilon \right]$$

(19)

where x^(j) is the j-th optimization parameter combination and [ε] is the allowable error value, such as 5%. The error between ${\mathit{x}}_{\mathrm{opt}}^1$ and ${\mathit{x}}_{\mathrm{opt}}^2$ is 4.2%, which meets the requirements. This case has only one iteration, it has a total of 16 finite element models, and the regression-predicted results are in good agreement with the numerical experimental values. In addition, Figure 11 shows the vertical deformation ω of the PSASPSRS structure when the parameter combinations are ${\mathit{x}}_{\mathrm{opt}}^1$ and ${\mathit{x}}_{\mathrm{opt}}^2$. There is no jump phenomenon at the position of the maximum displacement point. Therefore, the small reduction in the prestress has no significant effect on the deformation of the PSASPSRS structure. The smaller prestress can reduce mechanical energy consumption and worker costs during construction.

For the response surface function of the steel consumption G, the second-order partial differential of at least one parameter is greater than zero. Thus, there is an optimization parameter combination in a set composed of the design parameters to minimize the response function value. Figure 12 illustrates the variation of the optimization parameter combination for the steel consumption G with the increasing number of iterations. The two curves represent the edges of the response surfaces where the optimal values of k and ρ are unchanged. The function response value at ${\mathit{x}}_{\mathrm{opt}}^1$ is smaller than that at ${\mathit{x}}_{\mathrm{opt}}^2$. This proves that ${\mathit{x}}_{\mathrm{opt}}^1$ is still closer to the global optimal solution in the domain composed of constraints despite the addition of experimental points around ${\mathit{x}}_{\mathrm{opt}}^1$.

4.4. FE Computational Efficiency and Accuracy

All the experimental results were solved by a computer with an Intel(R) Xeon(R) E-2226G CPU and a 3.40 GHz processor. For this case, the computation time is significantly affected by the radial grid number of single-layer reticulated shells because the total degrees of freedom significantly increase as the radial grid number of single-layer reticulated shells decreases, which directly increases the workload of solving the governing FE equations. In addition, the finite element calculation time based on one-quarter basic block was compared with that based on the whole structure. The results show that the finite element calculation based on one-quarter basic block can save nearly two-thirds of the time compared with the overall analysis, which indirectly improves the efficiency of the initial optimization design of the PSASPSRS structure.

Furthermore, on the basis of the optimization objective G and four constraints, we explored the error between two finite element calculation methods to verify the reliability of the FE calculation based on the one-quarter basic block, as shown in Figure 13. For all the experimental indexes, the average error between the two FE calculation methods is less than 5%, which is acceptable. The reasons for the minor errors will be explored in further studies. In summary, the applied FE calculation method has ideal efficiency and accuracy. Therefore, it is suitable for one-quarter symmetric PSASPSRS structures.

4.5. Parametric Analysis

In addition to optimization, the regression functions also determine how the experimental indexes are affected by the parameters. According to Equations (14)–(18), the responses of the experimental indexes to k and T are obtained when ρ is 0.2, 0.6, and 1.0 (Figure 14). The variation law of each index with the design parameters is summarized as follows:

• k is the main parameter affecting G. G significantly decreases with the increase in k. A smaller k means more weight for the double-layer reticulated shells, which leads to an increase in the web and lower chord bars. When k < 5 or k > 8, G is positively correlated with T. G gradually decreases with the increase i ρ because a larger ρ indicates a more adequate use of the material.
• $w_u$ usually occurs at the supports, and $w_u$ is mainly affected by T. When k remains constant, $w_u$ decreases with the increasing T. This may be due to the higher prestress amplitude’s limiting the deformation of the supports. In contrast, $w_u$ remains stable when k changes.
• k has a great effect on $w_d$. The overall stiffness of the roof structure increases when the part of the double-layer reticulated shells is enlarged and reduces the internal force difference of the members. The load is uniformly transferred to the support system, reflecting better force performance. Larger prestress can offset the structural deformation caused by more external loads. When T > 1200 kN and k < 4, $w_d$ can reach the minimum. $w_d$ is positively correlated with ρ.
• When T remains unchanged, f increases with the increase in k. According to structural dynamics, it is known that frequency is negatively correlated with mass and positively correlated with stiffness. It can be seen that in this case, the increase in k causes a decrease in mass, which has a greater effect on frequency than the decrease in stiffness has on frequency. When k remains unchanged, f does not significantly change with the increase in T. There is a positive correlation between f and ρ, meaning it is possible that the change in ρ causes a change in structural stiffness.
• λ has no obvious change with the increase in T. With the increase in k, λ gradually decreases. This may be because the overall stiffness is negatively affected, thus increasing the deformation of the roof structure to deviate from the balanced position. The FE model shows that when k < 5, the structure is controlled by the overall instability. When k > 5, the structure is controlled by the local instability.

4.6. Comparison of Optimization Methods

To verify the effectiveness and practicality of the proposed optimization method, this section compares it with similar optimization methods. In the field of aeronautical engineering, it is crucial to reduce the mass of the structure. For instance, Hu (2020) [44] proposed a fully parametric optimization design aiming to adapt it to continuous structural changes and shapes while optimizing for a lighter weight of the wing. Importantly, the central composite design (CCD) and the second-order approximation polynomial are used to model the flight load response. Fang (2020) [45] proposed a response surface method that is based on uniform design (UD) experiments and weighted least squares (WLSs) to improve the accuracy of the limit state function during a reliability analysis. Each approximate optimal point will be used for a new sample set to improve the accuracy of the response surface. In addition, the authors use a solid propellant grain in the rocket motor as an example to verify its effectiveness. For the PSASPSRS case in this study, the optimization results of the above two methods and a comparison of those results with the results of the proposed optimization method are shown in

Table 8. Optimization results of the PSASPSRS case via different methods.

Methods	Optimization Results				Sample Amount	The Error of G (Relative to the Proposed Method)
	k	ρ	T/kN	G/ton
Proposed method	9	1.0	687	149.59	16	0
Hu 2020 [44]	9	1.0	689	148.67	26	0.6%
Fang 2020 [45]	9	1.0	682	152.11	13	1.7%

, and the following conclusions can be drawn:

• The difference in sample amount suggests that the uniform design effectively fills the experimental space with only about half the samples of the central composite design. Under the same FEA conditions, the proposed method in this study can save significant design time and costs. However, the method proposed by Hu (2020) [44] further reduces the steel consumption by 0.6% compared to the proposed method. It can be seen that the response surface in (Fang 2020) [45] has a better approximation to the global optimum solution, which may be due to the application of the regression fitting formulation with endogenous interaction terms.
• The method proposed in this paper uses a larger number of samples compared with the method in (Fang 2020) [45]. This is because only the local optimization points are added to the current sample set for the next iteration. This method does not pursue the fitting accuracy for the whole space, thus speeding up the optimization. However, the approximation accuracy is limited because of the lack of sample sets near the optimization points. As a result, the steel consumption of the PSASPSRS structure designed by Fang (2020) [45] is 1.7% higher than that of the proposed method.
• All errors of the steel consumption G are less than 2%, indicating the effectiveness of the proposed method for the initial design of the PSASPSRS, and the optimization result is acceptable.

5. Conclusions

The PSASPSRS not only has excellent load-bearing capacity and stability but also fully demonstrates the architectural aesthetics of its shape. It is a favorable choice to apply the PSASPSRS in the roof structure of an urban gymnasium. For promoting sustainable development in the construction industry, an optimization strategy based on the uniform design method and iterations is proposed with respect to the design of the PSASPSRS.

This paper applied the proposed optimization strategy to a PSASPSRS case. In total, 16 finite element models were built by one iteration. The search space gradually shrank around the local optimal solution and moved toward the global optimal solution during the optimization process. The optimization procedure stopped when the error between local optimal solutions was less than the allowable error. The optimized structure saves 13% of steel consumption compared with the original design and shows that the proposed optimization strategy is suitable for the steel consumption optimization of PSASPSRS. Utilizing mirror technology for finite element analysis enhances the efficiency of the overall optimization process. A parameter analysis indicates that the radial grid number of single-layer reticulated shells has the most significant impact on steel consumption. A comparison of it with previous studies reveals the proposed optimization strategy demonstrates ideal efficiency and accuracy.

The main advantage of this optimization strategy is its ability to save time and costs. However, the level of automation of this method requires improvement, such as determining new upper and lower bounds for design parameters during the iterative process. Additionally, it is essential to study the robustness of the optimization strategy, which is critical for solving practical problems. Given the constraints, such as deformation, frequency, and stability during optimization, the optimized structure can be regarded as a suitable starting point for further detailed structural analysis. In summary, this method is expected to have wide applicability to complex space structures.