Multi-Objective Optimisation and Deformation Analysis of Double-System Composite Guideway Based on NSGA-II

Abstract

To study the optimal design of the section of the double-system composite guideway under the economic, steel consumption, and carbon emission characteristics, this paper introduced the multi-objective constrained optimisation model, which was established by the non-dominated sorting genetic algorithm II. In addition, the finite element model was established to further analyse the optimised section’s deformation and summarise the rail girder’s deformation law under different loads. The results showed that compared with the original design scheme, the optimised scheme can effectively reduce carbon emission during the construction of the double-system composite guideway, by 23.67% for Scheme I and 42.03% for Scheme II. On the other hand, steel had the largest share in the economic targets of the three design options, accounting for about 75% to 88.5% of the total cost. Concrete had the highest share of carbon emissions, ranging from 90% to 95% of the total carbon emissions. The distribution patterns of horizontal and vertical deformations in the three design options were independent of the load type as well as the load magnitude, but the vertical deformations were related to the load type, especially the self-weight load. The conclusions of this paper aim to fill the gap in the theoretical study of section optimisation of the double-system composite guideway and lay the theoretical foundation for developing the multi-system monorail transportation system.

1. Introduction

In recent years, the rapid development of urban rail transit systems has led to significant advancements in the integrated construction of transportation systems, particularly those utilising railway–highway bridges [1,2]. Building on this progress, researchers at Beijing Jiaotong University [3,4,5] have further explored three major monorail systems—straddle-type, suspended, and magnetic levitation—as primary research platforms, and proposed novel transportation frameworks, such as the railway–highway corridor system and multi-system monorail transport. Song, P. [6] pioneered research on multi-system monorail transport by investigating the local stress behaviour of dual-system monorail track beams. Following this, scholars such as Hu, T. [7], Sun, H. [8], and Liu, X. [9] conducted extensive studies on the standardisation of dual-system monorail beam templates, dynamic performance, and turnout structures, laying a robust foundation for the development of multi-system monorail transportation. Since 2020, research in this field has deepened, leading to the initiation of group standards projects, including the “Multi-System Monorail Transit Design Specification” and “Multi-System Monorail Construction and Acceptance Code”. In 2022, the project titled “Design Calculation Theory and Experimental Research on Multi-system Traffic Multipurpose Rail Beam” was funded by the National Natural Science Foundation of China. As a result, integrating urban rail transit systems has emerged as a critical challenge, driving the development of the double-system composite guideway (DSCG) research based on existing straddle-type pre-stressed concrete (PC) rail guideways and suspended steel guideways.

In their research on PC composite guideway beams, Liu, S. [10] and Zhu, E. [11] highlighted that the current cross-sectional design of composite structures is relatively simplistic, typically relying on engineering experience or basic superposition methods. This approach often limits the full utilisation of the material properties. In 2001, Ma, B et al. [12] optimised the design of urban prefabricated steel–concrete composite girders using existing design specifications and drawings, employing a more traditional optimisation method. This underscores that engineering optimisation methods based on mathematical algorithms had not yet received significant attention in the early stages of structural optimisation research. However, with the advancement of computer technology, intelligent optimisation algorithms capable of addressing complex mathematical problems have evolved, enabling the potential for intelligent structural optimisation. In 1994, Srinivas, N. and Deb, K. introduced the non-dominated sorting genetic algorithm (NSGA), an extension of the traditional genetic algorithm. The NSGA could rapidly locate Pareto frontiers and preserve population diversity. Nonetheless, it had limitations, such as high time complexity in non-dominated sorting, lack of elite strategy support, and the manual requirement to set sharing parameters. To address these issues, Deb, K. [13] later proposed NSGA-II, an improved version incorporating the elite strategy. NSGA-II resolved the primary shortcomings of NSGA and provided a practical approach to optimising the cross-sections of engineering structures characterised by multi-parameter and multi-objective problems.

After introducing NSGA-II, its superior performance in addressing multi-objective optimisation problems led to its rapid and widespread adoption across various fields. Scholars such as Senouci, A. [14], Pedro [15], and Junior, F. [16] applied genetic algorithms to optimise engineering structures, successfully deriving optimal solutions under different objective functions. However, these studies primarily focused on the optimisation of individual components. For cross-section optimisation, Sharafi, P. [17] conducted a study on the topology optimisation of beam sections and developed a procedure applicable to engineering structures. While this method optimised cross-sectional details, it did not address the overall structural design. Using Python, Yin, W. [18] created an intelligent optimisation algorithm capable of optimising the structural capacity of any cross-section at any location. Wu, W. [19] explored the factors affecting interfacial slip in composite beams, focusing on improving crack resistance and developing an optimisation algorithm to enhance the crack resistance of continuous composite beams. Despite these advancements, research on optimising multiple cross-sections of the DSCG remains limited [20,21,22]. With the rapid growth of multi-system monorail transportation, optimising the cross-section design of multi-system combined guideway girders has emerged as a critical challenge.

To address the challenges in the cross-section optimisation of the DSCG, this paper proposes a multi-objective optimisation model that incorporates the NSGA-II algorithm, with the economy, steel consumption, and carbon emissions as the primary optimisation goals. Previous research has not sufficiently explored cross-section optimisation for multi-system combined guideways, nor has it systematically considered the trade-off between economic and environmental objectives. To fill this gap, this study develops a constrained multi-objective optimisation model comprising 16 unknown variables and 3 objective functions. The relationships and potential linear characteristics among the three indicators are verified through Pareto frontier analysis. By leveraging the NSGA-II algorithm, an optimal design scheme for the DSCG is obtained. To further validate the differences in structural performance between the proposed optimised model and the original design, this paper also establishes a finite element model of the DSCG. The lateral, vertical, and longitudinal deformation patterns under varying loads for pre- and post-optimisation schemes are analysed, revealing key performance distinctions. The findings demonstrate that the optimised design significantly reduces carbon emissions while enhancing economic performance and minimising steel usage. Moreover, finite element analysis confirms that the optimised guideway exhibits improved cross-sectional stiffness and reduced deformation under load, validating the advantages of the proposed optimisation approach.

This paper verifies the practical application of the NSGA-II intelligent algorithm in the cross-section optimisation of multi-standard combination guideways. The computational model quantifies the relationships between economy, steel consumption, and carbon emissions and addresses the cross-section optimisation problem across different locations within the same structure. This study broadens the use of the NSGA-II algorithm in engineering applications, providing an enhanced solution to the challenges of cross-section optimisation in dual-mode composite structures. Significantly, the proposed methodology is not restricted to guideways with an L = 30 m span, nor is it limited to the DSCG optimisation. The concepts and approaches outlined in this research can be broadly applied to cross-section optimisation challenges in other engineering structures. By integrating the intelligent optimisation algorithm with finite element analysis, this study demonstrates how optimised design can effectively reduce carbon emissions and material usage while still meeting the structural stiffness and deformation control requirements. Through comprehensive analysis and validation, the proposed optimisation model offers a scientific framework for future guideway designs under multi-objective and multi-variable conditions. Moreover, it contributes to the sustainable development of green transport infrastructure, laying a solid foundation for environmentally friendly engineering practices.

2. The DSCG Multi-Objective Optimisation Model

The upper straddle rail structure of the DSCG adopts the PC guideway, and the lower suspended rail structure adopts the steel guideway, as shown in Figure 1. Under vehicle loading, the deformation of the upper PC guideway will be restrained by the bottom steel guideway. That is one of the advantages of the DSCG over a single guideway in increasing the section stiffness to reduce the deformation. However, if the design of the DSCG section is a simple combination, it cannot fully utilise the material’s stress performance. That is one of the reasons why we need to optimise its section.

2.1. Optimisation of Model Parameter Settings

The guideway span studied in this paper was L = 30 m. The deformation in the longitudinal direction of the guideway was dominated by linear variation. The location of the variable section was set according to the shear and bending moment envelope diagram of the supported beam. The variable section location of the subject of this paper was based on the variable section location of the straddle 25 m straight guideway [23]. The parameters of the suspended guideway section were related to the section dimensions of the guideway in the Guanggu Wuhan monorail line. Regarding section setting, there were three main sections: the support section (Section 1-1), transition section (Section 2-2), and mid-span section (Section 3-3). The exact location and size distribution of the sections are shown in Figure 2. The transition section of the change was between Sections 2-2 and 3-3. According to Figure 3b, it can be seen that the main difference between Section 2-2 and Section 3-3 is based on the shape of the steel guideway section inside the concrete guideway. That was the key location to be optimised by the optimisation algorithm in this paper. The section parameters of the DSCG mainly included two main categories: the thickness of the steel plate and the shape of the steel girder section.

According to the dimensions in [24] on the section of the straddle guideway, it is known that the positions of the bogie guide and stable wheels of the straddle vehicle are as shown in Figure 3. The guide and stable wheels are located within the intervals [125, 400] and [955, 1385] from the top of the guideway. The specific range of values of the design variables of the section to be optimised is shown in

Table 1. Range of values of design variables for optimised sections (unit: mm).

Variable	Meaning	Range of Values	Variable	Meaning	Range of Values
bc1i	Concrete internal steel beam bottom plate width	[600, 850]	tc3	The thickness of the top side plate of the steel beam inside the condensate	[20, 36]
bc2i	Width of the top slab of the steel beam inside concrete	[100, 660]	tc4	The thickness of the top plate of the steel girder inside the condensate	[20, 36]
hc1i	The height of the upper side slab of the steel beam inside the concrete	[100, 1000], [250, 1000]	ts1	The thickness of the lower side plate of the steel beam	[20, 36]
hc2i	The height of the lower side plate of the steel beam inside the concrete	[100, 300], [000, 300]	ts2	The thickness of the bottom plate at the lower side of the steel beam	[20, 36]
tc1	The thickness of the side plate of a steel beam in concrete	[20, 36]	ts3	The thickness of the lower side plate of the steel girder	[20, 36]
tc2	The thickness of the top plate of the middle part of the steel beam inside the concrete	[20, 36]	ts4	The thickness of the top plate at the centre side of the steel girder	[20, 36]

.

At the same time, it was considered that the research in this paper does not focus on the role of the reinforcing plate of the bottom suspended steel guideway. Therefore, the fixing spacing was taken as 1.5 m, and the thickness of the steel plate was 12 mm [25]. The sectional dimensions of the reinforcing plate and the three-dimensional illustration are shown in Figure 4. It should be noted that because of the combination of the DSCG straddle and suspended guideway, the top of the reinforcing plate was set in a planar form, different from the structure of the suspended guideway alone.

The design of the model parameters, mainly including the design of the PC guideway cross-section, the steel girder cross-section, and the thickness of the steel plate, is optimised. The guideway shape’s overall design needs to be analysed in conjunction with the form of the upper and lower vehicle bogies. The overall section dimensions were not modified. In addition, the optimisation concept of the DSCG section model is the same, and this algorithm can still be applied after modifying the section parameters.

2.2. Establishment of Multi-Objective Optimisation Function

The optimisation model’s objective function focuses on three key objectives: economic efficiency, steel consumption, and carbon emissions. It is well understood that identifying a single optimal solution for multi-objective optimisation problems presents significant challenges. Therefore, the main aim of this paper is to determine the optimal sectional design for the DSCG that maximises each objective function. Simultaneously, the study seeks to identify the most reasonable parameter configuration for the DSCG section, ensuring an effective balance between economic, material, and environmental factors.

2.2.1. Calculation of Quantities

Based on the characteristics of the DSCG during production and construction, as well as the sub-projects involving concrete, reinforcement, and formwork, the corresponding formulas for calculating the quantities of work are provided in Equations (1) to (12). When calculating these quantities, it is essential to account for the material loss rate, uniformly represented by w.

(1)

Calculation of the volume of concrete

$$V_c=(1+{\omega }_c)(A{c}_1{L}_1+V{c}_2+A{c}_3{L}_3)$$

(1)

$$A{c}_1=0.85\times 1.5-(0.42+0.54)\times 0.06$$

(2)

$$V{c}_2=[2A{c}_1-(0.85h_{c21}+b_{c21}{h}_{c11})-(0.85h_{c22}+b_{c22}{h}_{c12})]{\displaystyle \frac{L_2}{2}}$$

(3)

$$A{c}_3=A{c}_1-0.85h_{c22}-b_{c22}{h}_{c12}$$

(4)

where V_c is the concrete volume, A_c1 and A_c3 are the areas of Sections 1-1 and 3-3, V_c2 is the volume of concrete in the transition section, and w_c is the concrete loss rate.

(2)

Calculation of steel consumption

$$Q_s=(1+{\omega }_s)(V_{s1}+V_{s2}+V_{s3}+V_{s4})\times {\rho }_1$$

(5)

$$V_{s1}=2(L_2+L_3)((b_{c12}-b_{c22})t_{c1}+b_{c22}{t}_{c2}+2h_{c22}{t}_{c3}+2h_{c12}{t}_{c4})$$

(6)

$$V_{s2}=\{[(b_{c11}-b_{c21})-(b_{c12}-b_{c22})]t_{c1}+(b_{c21}-b_{c22})t_{c2}+2(h_{c21}-h_{c22})t_{c3}+2(h_{c11}-h_{c12})t_{c4}\}\times L_2$$

(7)

$$V_{s3}=30\times [0.44\times 2t_{s2}+(0.3+0.78+2t_{s3})\times t_{s4}+1.5\times 2t_{s3}+(0.1-t_{s2})\times 2t_{s1}]$$

(8)

$$V_{s4}=21\times 2\times (1.56+0.24)\times (0.15\times 0.15-0.13\times 0.13)$$

(9)

where Q_s is the mass of steel, ${\rho }_1$ is the steel capacity weight, V_s1 is the volume of steel with a section of mid-span Section 3-3, V_s2 is the volume of the transition section steel beam, V_s3 is the volume of the bottom suspended beam, V_s4 is the volume of the reinforcing plate, and w_s is the steel loss rate.

(3)

Calculation of the area required for steel formwork

$$S_f={\displaystyle \frac{(1+{\omega }_2)(3850+4\times 60\sqrt{2})}{1000}}L$$

(10)

where S_f is the area of the steel mould plate, w₂ is the steel mould plate loss rate, L is the total length of the guideway, and L = (L₁ + L₂ + L₃) × 2 = 30 m.

(4)

Calculation of the amount of ordinary steel bars

For calculating the cost of ordinary and pre-stressed reinforcement, the total mass of ordinary reinforcement can be estimated from the volumetric steel content of the PC guideway. In this case, the integrated steel content rate in the concrete structure is 0.38 t/m³, and the mass of ordinary steel reinforcement is shown in Equation (11):

$$Q_{ss}=0.38(1+{\omega }_3)(2A_{c1}\times L_1+2V_{c2}+2A_{c3}\times L_3)$$

(11)

where Q_ss is the mass of ordinary rebar and w₃ is the loss rate of ordinary rebar.

(5)

Calculation of the amount of pre-stressed steel reinforcement

The pre-stressed arrangement scheme adopted the design scheme of extracorporeal pre-stressed bundles adopted in [26]. The steel strand is 1860, with six bundles of 1 × 7 strands on each side and twelve bundles arranged symmetrically. The nominal diameter of the steel strand is 15.2 mm. The specific calculation formula is shown in Equation (12):

$$Q_p=7.8(1+{\omega }_4)\times \pi D^{2}NL$$

(12)

where Q_p is the mass of pre-stressed steel, N is the number of strand roots, D is the nominal diameter of the strand, and w₄ is the pre-stressed steel loss rate.

2.2.2. Objective Function of Economy

The DSCG was constructed using prefabricated assembly technology, and the cost case of each sub-cost was superimposed to obtain the overall economic objective function. Therefore, the objective function of the overall cost of the guideway is expressed in Equation (13):

$$C(x)=C_s(x)+C_c(x)+{\displaystyle \sum _{i=1}^4{C}_i(x)}$$

(13)

where C(x) is the overall cost of the DSCG, C_s(x) and C_c(x) are the steel material cost and concrete material cost, and C₁(x), C₂(x), C₃(x), and C₄(x) are the cost of the DSCG transportation, the cost of steel formwork, the cost of ordinary steel reinforcement, and the cost of pre-stressed steel reinforcement. Considering the above quantities and multiplying them by the unit cost of the sub-projects, Equation (13) can be transformed into Equation (14):

$$C(x)=V_c{c}_c+Q_s{c}_s+M_s{c}_{1s}+V{c}_{1c}+S_f{c}_2+Q_{ss}{c}_3+Q_p{c}_4$$

(14)

where c_c is the unit price of concrete material, c_s is the unit price of steel material, c_1c and c_1s are the unit prices for transportation of concrete and steel materials, c₂ is the unit price of steel formwork, c₃ is the unit price for plain steel reinforcement, c₄ is the unit price of the 1860 steel strand, and M_s is the total mass of steel transported.

According to the market survey data [27,28], the researched price of each part was summarised, as shown in

Table 2. Prices of each part in the cost calculation.

cs/(yuan·t−1)	cc/(yuan·m−3)	c1s/(yuan·t−1)	c1c/(yuan·m−3)	c2/(yuan·m−2)	c3/(yuan·t−1)	c4/(yuan·t−1)
14,462	1200	260	566	56	8269	4299

.

2.2.3. Objective Function for Steel Consumption

The objective function of steel usage of the guideway is shown in Equation (5) to Equation (9). The amount of steel used for steel beams involves operations such as transportation, processing, and welding, and it is also necessary to consider the loss rate during processing.

2.2.4. Objective Function for Carbon Emission

Regarding setting the carbon emission objective function, it is necessary to consider the whole process of material production, transportation, and construction and adopt the system boundary of “from cradle to site” to calculate carbon emissions [29].

(1)

Calculation of physical carbon emissions

Physical carbon emissions can be calculated by superposing the sub-projects, as shown in Equation (15):

$$E_{tot}=V_c{e}_c+Q_s{e}_s+Q_{\mathrm{ss}}{e}_{ss}+S_f{e}_f+Q_p{e}_p$$

(15)

where E_tot is the total physical and chemical carbon emissions, e_c is the comprehensive carbon emission coefficients of concrete sub-projects, e_s is the comprehensive carbon emission coefficients of steel girder projects, e_ss is the comprehensive carbon emission coefficients of steel reinforcement sub-projects, e_f is the comprehensive carbon emission coefficients of formwork sub-projects, and e_p is the comprehensive carbon emission coefficients of pre-stressed sub-projects.

(2)

Selection of comprehensive carbon emission factor

When calculating the comprehensive carbon emission coefficient, the concrete sub-project includes the production, transportation, pouring, and maintenance of concrete. The steel guideway project involves producing, transporting, welding, and moulding steel guideways. Reinforcing steel sub-projects include the production, transportation, processing, and tying of reinforcing steel. The formwork sub-project provides for the production of formwork (according to the turnover frequency and loss rate), transportation, processing, and installation. The pre-stressed sub-project includes the production, transportation, tensioning, and anchoring of pre-stressed steel bars. The specific value of the integrated carbon emission factor [27] is shown in

Table 3. Summary of the value of the comprehensive carbon emission factor of unit sub-projects.

Sub-Works	Material	Symbolic	Carbon Emission Factor Value	Unit
Concrete	C60	ec	491	kgCO2e/m3
Steel beams	Q235	es	5.96	kgCO2e/kg
Ordinary reinforcing steel	HRB400	ess1	2.56	kgCO2e/kg
Hoop reinforcement	HPB300	ess2	2.51	kgCO2e/kg
Pre-stressed reinforcement	1860 steel strand	ef	3.57	kgCO2e/kg
Steel formwork	Q235	ep	4.96	kgCO2e/m2

.

It is important to note that in the original text, “e_ss” represents the comprehensive carbon emission factor for the sub-projects involving ordinary steel reinforcement. However, given the distinction in materials between hoop reinforcement and longitudinal reinforcement, the notation is further refined for clarity. Specifically, “e_ss1” is introduced to denote the comprehensive carbon emission factor for the longitudinal reinforcement sub-project, while “e_ss2” represents the comprehensive carbon emission factor for the hoop reinforcement sub-project.

2.3. The Vehicle Problem of the Optimisation Model

The upper straddle guideway of the DSCG is a composite structure, and the lower suspended guideway is a steel structure. Therefore, the DSCG must meet the straddle and suspended guideway’s deformation and construction requirements. In the constraints of the monorail guideway model, the bending capacity of the standard section, the shear capacity, the deflection deformation, and the construction requirements are usually used as the constraints of the structure.

2.3.1. The DSCG Deflection Requirements

In the specification [30], it is stipulated that under the action of static live vehicle load, the vertical deflection of the guideway should not exceed 1/800 of its span, which is also satisfied by Equation (16). Relative to the specification [31], it can be seen that under the action of static live vehicle load, the vertical deflection of a supported beam should not exceed 1/1000 of the span, so the vertical deformation of the DSCG can be satisfied as Equation (16):

$$f_{01}={\displaystyle \sum _{i=1}^8\frac{P_j{b}_{ji}(3L^2-4b_{0i}^2)}{24E_c{I}_{0c}}}\le {\displaystyle \frac{L}{800}}$$

(16)

where f₀₁ is the deflection of the guideway under concentrated load, b_ji is the distance from the loading point to the beam end, i = 1, 2, 3, 4, P_j is the symmetric concentrated load, taking the axle weight of straddle vehicle P₁ = 110 kN [11] and suspended vehicle P₂ = 41.24 kN, j = 1,2 [25], E_c is the modulus of elasticity of concrete, and I_0c is the moment of inertia of the composited section converted to concrete. For Equation (16), the formula includes all concentrated load effects, including straddle and suspended vehicles. Also, considering the most unfavourable state of vehicle loading, symmetrical loading of two car sets is used, respectively. The most unfavourable loading is illustrated, as shown in Figure 5.

2.3.2. Flexural Capacity Requirement of Standard Section

Concrete structures have a high compressive capacity. Therefore, it should be considered in the DSCG design to set the neutral axis of the composited section within the concrete section. Equation (17) needs to be satisfied:

$$y_{sc}\le 1.5-h_{1i}-h_{2i}$$

(17)

where y_sc is the distance of the neutral axis of the guideway section from the top.

In addition, according to the code for the design of composite structures [32], the relevant provisions for the calculation of the bending capacity of the section according to the elastic design method are known. It is necessary to take the bending capacity calculation formula of the upbeat section of the composite guideway under the action of complete connection. It should be able to meet Equation (18):

$$M\le f_{cd}b{y}_{sc}y$$

(18)

where M is the design value of the bending moment to which the section is subjected, b is the width of the guideway, f_cd is the design value of concrete compressive strength, and y is the distance from the composited stress of the steel guideway section to the composited stress of the section of the concrete compression zone.

2.3.3. Shear Capacity Requirement

When designing a composited guideway following the elastic design method, the shear capacity must follow Equation (19):

$$V_b\le 2h_{c1i}{t}_{c3}{f}_{av}$$

(19)

where V_b is the design value of shear force, and f_av is the design value of shear strength of the steel beam web. By referring to Figure 5 and incorporating knowledge of structural mechanics, it is easy to derive the distribution of the bending moment and shear force of the DSCG under the action of vehicle and self-weight loads.

In addition, we must check the DSCG sections and items, which mainly include the shear force value of Section 2-2 and the bending moment value of Section 3-3. When checking the shear force of Section 2-2, to ensure that the composite guideway has sufficient shear capacity, the shear force value at the location of the support is taken, as in Equation (20). When checking the bending moment of Section 3-3, the value of the bending moment at the centre of the span is taken to ensure that the composited guideway has sufficient moment resistance. The bending moment and shear force values are calculated, as shown in Equations (20) and (21):

$$M_{{\displaystyle \frac{L}{2}}}={\displaystyle \frac{1}{8}}q{L}^2+{\displaystyle \sum _{j=1}^2{\displaystyle \sum _{i=1}^4{P}_j{b}_{ji}}}$$

(20)

$$V_0=4(P_1+P_2)=608.96 \mathrm{k}\mathrm{N}$$

(21)

$$q=({\displaystyle \frac{2400Q_c}{1+{\omega }_c}}+{\displaystyle \frac{Q_s}{1+{\omega }_s}}+{\displaystyle \frac{Q_{ss}}{1+{\omega }_3}}+{\displaystyle \frac{Q_p}{1+{\omega }_4}})\times {\displaystyle \frac{g}{L}}$$

(22)

where q is the self-weight of the DSCG structure, calculated as shown in Equation (22), g is the gravitational acceleration, taken as 9.8 m/s², and 2400 is the concrete capacity weight.

2.3.4. Constructional Requirements

For the construction requirements of the DSCG, mainly the thickness of concrete at the top and sides of the guideway section needs to be limited. The setting of ordinary reinforcement and pre-stressed reinforcement in the guideway can be set according to the structural requirements of the straddle structure.

(1)

Thickness of concrete at the top of the section

The DSCG differs from the traditional bridge structure, which consists of carrying and operating structures. To prevent localised damage to the guideway, we draw on the provisions of the design code for composited structures for structural slab supports and floor structures. The top of the steel guideway should be provided with pegs, the length of the pegs $l_s\ge 80$ mm, and the distance of the top of the pegs from the top surface of the guideway $d_s\ge 50$ mm. Therefore, the steel beam’s top surface dimensions from the guideway’s top surface should satisfy Equation (23):

$${\mathrm{d}}_{\mathrm{t}}\ge l_s+l_s^{\prime }+d_s$$

(23)

where d_t is the thickness of concrete at the top of the section and $l_s^{\prime }$ is the thickness of the nut.

(2)

Thickness of concrete at the side of the section

The loads acting on the guide and stable wheel exist on the side of the guideway. Considering the stiffness factor, the steel guideway sides must also be arranged with spigot connectors:

$${\displaystyle \frac{b_{1i}-b_{2i}}{2}}\ge l_s+l_s^{\prime }+20$$

(24)

In Equation (24), considering there is no vehicle loading at the dummy opening location, ensuring a sufficient thickness of the protective layer from the peg’s top cap to the guideway’s surface is necessary. Therefore, the side dimensions that should be satisfied in Equation (23) should also be confident in Equation (24), which is limited to the size requirements.

2.4. Establishment of the Multi-Objective Optimisation Model

For the DSCG multi-objective optimisation model, the main objective functions to be considered are economy, steel consumption, and carbon emissions, totalling three objective functions. The corresponding independent variables mainly include the design variables of the sixteen optimised sections in Table 1. In engineering, the design variables of girder sections are mostly connected. Their feasible solutions are generally discretised and distributed, and the objective functions are no longer continuous and differentiable. The DSCG section parameter optimisation problem is described in Equation (25):

$$\left\{\begin{array}{l}x=\left[b_{c1i},b_{c2i},h_{c1i},h_{c2i},t_{c1},t_{c2},t_{c3},t_{c4},t_{s1},t_{s2},t_{s3},t_{s4}\right]\\ \hspace{1em}\hspace{1em}i=1,2\\ \min \left[C(x),Q_s(x),E_{tot}(x)\right]\\ s.t. M\le f_{cd}b{y}_{sc}y\\ \hspace{1em}\hspace{1em}{V}_b\le 2h_{c1i}{t}_{c3}{f}_{av}\\ \hspace{1em}\hspace{1em}{\displaystyle \sum _{i=1}^8\frac{P_j{b}_{ji}(3L^2-4b_{0i}^2)}{24E_c{I}_{0c}}}\le {\displaystyle \frac{L}{800}}\\ \hspace{1em}\hspace{1em}{d}_t\ge l_s+l_s^{\prime }+d_s\\ \hspace{1em}\hspace{1em}{\displaystyle \frac{b_{1i}-b_{2i}}{2}}\ge l_s+l_s^{\prime }+20\end{array}\right.$$

(25)

From Equation (24), the optimisation model is constrained with three objective functions and sixteen variables. NSGA-II is a widely used multi-objective genetic algorithm whose core steps include initialisation, evaluation, fast non-dominated sorting, congestion calculation, selection, crossover, mutation, and elite retention. NSGA-II is an improved version of the traditional genetic algorithm capable of solving multi-objective constrained optimisation problems. Based on the conventional genetic algorithm, Deb, K. [13] improved the NSGA algorithm in 2002 and proposed the NSGA-II algorithm. NSGA-II reduces the complexity of non-dominated sorting and introduces the elite strategy mechanism, significantly improving the solution speed and the accuracy of the optimal solution. In addition, NSGA-II introduces a crowding degree and crowding degree comparison operator, which overcomes the problem of NSGA needing to specify the sharing parameters artificially. The use of the crowding degree as the comparison criterion enables the individuals of the population in the quasi-Pareto domain to be uniformly dispersed in the Pareto domain, which ensures the diversity of the population. Unlike multi-objective optimisation problems, finding the optimal solution to satisfy all objective functions is generally tricky. Because there is a specific connection between each objective function, we will discretise the optimisation search of Equation (25) in the subsequent study and propose different optimisation schemes according to other objective functions.

3. Implementation of the DSCG-Constrained Multi-Objective Optimisation Model and Analysis of Results

NSGA-II is a multi-objective optimisation algorithm mainly used to solve problems with multiple conflicting objective functions, where a single optimal solution may not exist, and a series of non-inferior solutions, i.e., Pareto optimal solutions, exist. Next, we focus on the DSCG computational optimisation model based on the NSGA-II algorithm.

3.1. Reasons for Using NSGA-II Intelligent Algorithm and Model Building

3.1.1. Reasons for Using NSGA-II Intelligent Algorithm

The primary rationale for selecting the NSGA-II model in this study is rooted in the alignment between the intricate nature of the DSCG cross-sectional optimization and the problem-solving capabilities inherent to the NSGA-II algorithm’s computational framework. Given that our research subject involves sixteen unknown variables, three objective functions, and various constraints, it falls within the domain of constrained multi-objective optimization—a challenge that traditional single-objective optimization techniques cannot adequately address. This necessitates the application of a sophisticated multi-objective optimization algorithm to achieve the optimal design of the DSCG cross-sections. One key advantage of NSGA-II is its incorporation of a non-dominated sorting mechanism, enabling the generation of a series of Pareto optimal solutions. These solutions allow us to evaluate the optimality across different objective functions comprehensively. Our study’s primary objective functions include economic viability, steel consumption, and carbon emissions. By leveraging the characteristics of these objective functions, we can select solutions from the Pareto optimal set that balance these factors effectively. Furthermore, NSGA-II ensures the uniform distribution of the solution set within the objective space via congestion distance calculations. This approach preserves the diversity of the population, thereby ensuring precise identification of the optimal solution’s location throughout the solving process and avoiding entrapment in local optima. This characteristic of the NSGA-II intelligent algorithm significantly enhances the efficiency of searching for the optimal solution amidst the sixteen unknown parameters. In conclusion, the NSGA-II algorithm exhibits robust flexibility and maintains population diversity, making it well suited for handling multi-objective constrained optimization problems. Its ability to systematically track the optimal solution across multiple objectives underscores our decision to employ the NSGA-II algorithm for the DSCG model optimization.

3.1.2. NSGA-II Modelling

The primary solution process of NSGA-II is illustrated in Figure 6. Deb K’s algorithm research results are the primary reference for solving the objective function. When running the NSGA-II calculation program, some specific parameters need to be set in advance, and the primary setting parameters include population size, chromosome length, maximum evolutionary generations, crossover probability, and mutation probability. Among them, the crossover and mutation probability greatly influence the running efficiency and convergence of the genetic algorithm. Thus, selecting the appropriate crossover and mutation probability is crucial [30]. The improvement of the adaptive genetic algorithm aims to improve the compatibility with the advantage of the adaptive genetic operator, which can maintain a significant crossover probability and mutation probability when the individual fitness is lower than the average fitness. Conversely, the genetic algorithm will choose smaller crossover probabilities.

For multi-objective optimisation problems, no solution typically can fully satisfy all objective functions simultaneously. Hence, the Pareto optimal solution concept is introduced to describe a dynamic equilibrium process in which optimal solutions are found across multiple objectives. In the NSGA-II algorithm, rules of non-dominated sorting are applied, allowing the solution set to be ranked based on robust and weak domination relationships. The ranking process used in this study follows a hierarchical approach: Within a solution set, the Pareto rank of the non-dominated solutions is assigned as 1. Once these non-dominated solutions are removed from the set, the next set of non-dominated solutions is assigned a Pareto rank of 2, and this process continues until all solutions in the set are ranked accordingly.

Additionally, since multi-objective functions usually lack a definitive optimal solution, further selection of the best solution is based on the crowding degree within the Pareto rank one solution set. This ensures that the chosen solution balances the different objectives efficiently. For the multi-objective function, the Pareto frontier of Equation (18), obtained when the population size p = 100 and the number of iterations g = 200, is illustrated in Figure 7. This shows a specific correlation between the three objective function targets, but the optimal solution cannot be taken simultaneously. Moreover, the figure has both positive and irrelevant relationships, and the relationship between the functions is intricate. The linear correlation coefficient between the economy target and the steel consumption target is R = 0.99, and the two targets have a high linear relationship, indicating that steel consumption will directly lead to an increase in the level of the economy. There is no significant linear relationship between carbon emission and steel consumption targets, with a linear correlation coefficient R = 0.03. There is no significant linear relationship between carbon emissions and economic targets, with a linear correlation coefficient of R = 0. It can be seen that carbon emission is not directly related to economy and steel consumption, and it is not reasonable to evaluate carbon emission targets simply by the level of economy and steel consumption.

3.2. Result Analysis of the DSCG Optimisation Model

In the NSGA-II algorithm, the number of populations and iterations critically influences the objective function to find the optimal solution. To more accurately solve the optimal solutions of the three objective functions in this paper, the reasonable number of iterations and population size will be further determined next. These data are used to analyse and adjust the optimal solutions.

3.2.1. Determination of a Reasonable Number of Iterations

To further analyse the section optimisation data, a programming study with different numbers of iterations was purposely conducted for the population size p = 100 and with the economy target as the primary research object. The results are summarised in

Table 4. Summary of the values of independent variables and the objective function under population size p = 100 and different iteration times (unit: mm).

	Objective Function
	C/yuan	Qs/t	Etot/kg
p = 100, g = 100	8.37 × 105	59.40	1.50 × 104
p = 100, g = 150	8.33 × 105	59.20	1.52 × 104
p = 100, g = 200	8.73 × 105	62	1.57 × 104
p = 100, g = 250	8.58 × 105	60.60	1.23 × 104
p = 100, g = 300	8.28 × 105	58.90	1.53 × 104

.

The data in the table indicate that the values of the objective function are different under different iterations when the population size is the same. At the same time, considering the role of random factors, we need to select the data reasonably under different iteration times. In addition, although the data for each target are different, their difference is not apparent. The optimal value of the economy target falls between 8.28 × 10⁵ yuan and 8.73 × 10⁵ yuan. The optimal value of the steel consumption target is between 58.9 t and 62 t, and the carbon emission target is between 1.23 × 10⁴ kg and 1.57 × 10⁴ kg. Based on the data relationships of the three targets, it can be seen that this paper takes the data of the optimisation of the scheme with the number of populations p = 100 and the number of iterations g = 200 for analysis. In addition, in the actual engineering application, composited with the characteristics of the bending moment and shear force applied to the supported structural section of the guideway, the design scheme with a smaller concrete section area in the span and a larger concrete section area in the support is generally selected. The above data also reflect this feature. The sixteen independent variables of the composited guideway under the action of different iteration times change relatively little. However, the corresponding data must be rounded off for practical engineering construction to align with real-world applications.

3.2.2. Adjustment and Determination of the Optimal Program

The next step involves further adjusting the parameters of the optimal program to determine a reasonable program for the DSCG. According to the data relationship in Figure 7, it can be seen that the economy target and the steel usage target are in a robust linear relationship, and they are considered optimal simultaneously. Therefore, there are two optimal programs for the DSCG and the corresponding data settings for the two programs, as shown in

Table 5. Summary of adjusted material usage and required price for the optimal solution (unit: mm).

		Objective Function
		C/yuan	Qs/t	Etot/kg
Optimised scheme	Scheme I	8.73 × 105	62	1.57 × 104
	Scheme II	1.12 × 106	78.50	1.20 × 104
Optimised and adjusted scheme	Original scheme	8.39 × 105	59.50	2.07 × 104
	Scheme I	8.85 × 105	62.90	1.58 × 104
	Scheme II	1.12 × 106	78.40	1.20 × 104

. In the optimisation scheme with population size p = 100 and g = 200, Scheme I is the arrangement scheme when the economic and steel usage targets are optimal. Scheme II is the arrangement scheme when the carbon emission target is optimal, and the adjusted parameter data are shown in Table 5. We designate the scheme without interface optimisation to analyse the optimisation effect further. That is the scheme with a Section 1-1 in all sections, as the original scheme.

Based on the data presented in Table 5, it can be seen that the variation of the independent variables is slight for both design options. The value of steel beam thickness ranges from 20 to 36, and the same thickness of the steel plate in the actual project is favourable to the manufacturing work. Regarding the value of the objective function indicators, the change in the independent variable leads to a slight change in the objective function, and the specific performance of the adjusted objective function indicators is summarised in

Table 6. Data from the pre- and post-optimisation schemes and the original scheme are compared.

Scheme	Indicators	Data	Percentage
Scheme I	Economy/yuan	8.85 × 105	5.48%
	Carbon emission/kg	1.58 × 104	−23.67%
	Steel consumption/t	62.9	5.71%
Scheme II	Economy/yuan	1.12 × 106	33.49%
	Carbon emission/kg	1.20 × 104	−42.03%
	Steel consumption/t	78.4	31.76%

. Among them, the principle of calculating the percentage in Table 6 is as follows:

$${\displaystyle \frac{\mathrm{Optimised}\mathrm{programme}\mathrm{data}-\mathrm{Original}\mathrm{programme}\mathrm{data}}{\mathrm{Original}\mathrm{programme}\mathrm{data}}}\text{×}100\%$$

Based on the data in Table 6, Scheme I demonstrates a 5.48% improvement in economic performance, reaching 8.85 × 10⁵ yuan, with a 5.71% increase in steel usage, reaching 62.9 tonnes. However, carbon emissions are reduced by 23.67%, decreasing to 1.58 × 10⁴ kg. In contrast, Scheme II exhibits a more substantial economic improvement of 33.49%, with financial performance rising to 1.12 × 10⁶ yuan. Steel consumption increases by 31.76%, reaching 78.4 tonnes, while carbon emissions see a marked reduction of 42.03%, dropping to 1.2 × 10⁴ kg. In summary, Scheme I shows moderate improvements in economic performance and steel usage, and a significant decrease in carbon emissions. On the other hand, Scheme II presents more pronounced enhancements in both economy and steel consumption, alongside a more substantial reduction in carbon emissions. Both optimisation schemes focus on enhancing economic performance and steel usage while reducing carbon emissions.

The optimised solution reflects increased economy, steel consumption, and decreased carbon emissions. According to the parameter data, the steel guideway near the support is smaller than the mid-span section. Based on the cross-section data, it can be seen that the cross-section of the steel girder in the vicinity of the support is small in comparison to the mid-span cross-section. Also, the area of the steel guideway cross-section at the mid-span section is limited, considering the constraints. The specific optimised cross-section parameters are shown in Figure 8.

3.2.3. Summary of Material Usage and Price of Sub-Projects of the Optimal Scheme

The adjusted optimal scheme, the summary of material dosage, and the price of each sub-project are shown in

Table 7. Summary of material usage and required price of the adjusted optimal solution.

	Type	Quantity	C/yuan	Qs/t	Etot/kg
Original scheme	Vc/m3	36.52	4.82 × 104		1.97 × 104
	Qs/t	43.71	6.32 × 105	43.71	2.61 × 102
	Sf/m2	138.25	7.74 × 103		6.86 × 102
	Qss/t	15.27	1.26 × 105	15.27	39.10
	Qp/t	0.56	2.41 × 103	0.56	2.00
	Transportation		2.28 × 104
Scheme I	Vc/m3	30.01	3.60 × 104		1.47 × 104
	Qs/t	49.57	7.17 × 105	49.57	2.95 × 102
	Sf/m2	138.25	7.74 × 103		6.86 × 102
	Qss/t	12.73	1.05 × 105	12.73	32.60
	Qp/t	0.56	2.41 × 103	0.56	2.00
	Transportation		1.70 × 104
Scheme II	Vc/m3	22.11	2.65 × 104		1.09 × 104
	Qs/t	68.33	9.88 × 105	68.33	4.07 × 102
	Sf/m2	138.25	7.74 × 103		6.86 × 102
	Qss/t	9.53	7.88 × 104	9.53	24.40
	Qp/t	0.56	2.41 × 103	0.56	2.00
	Transportation		1.25 × 104

. There is no change in the material dosage of steel formwork and pre-stressed reinforcement, and the optimisation of the scheme is mainly reflected in the reduction of the dosage of concrete and ordinary steel reinforcement and the increase of the steel dosage. In addition, the transportation expense is primarily related to the weight of the guideway, and the overall weight of the optimised guideway is reduced, as is the corresponding transportation expense. Specifically, Scheme I is reduced by 5.80 × 10³ yuan, 25% lower than the original plan, while Scheme II is reduced by 1.03 × 10³ yuan, 45% lower than the original scheme. A graphical representation of the relationship between the various data is drawn to analyse the relationship between the sub-data, as shown in Figure 9.

According to Figure 9, it can be seen that steel accounts for the most significant economic target for the three schemes, between about 75% and 88.5% of the total amount spent. The optimisation of the section will further increase the amount of steel used and the amount of steel spent. This is followed by plain steel reinforcement, which accounts for about 7% to 15% of the total cost. Optimisation of the section reduces the amount of concrete used, and the amount of money spent on plain steel reinforcement is reduced in parallel. Regarding the steel consumption target, the steel beams account for about 73% to 87% of the total steel consumption in the three schemes. Optimisation of the section increases the amount of steel used for the steel guideway and decreases the amount of ordinary reinforcement, while the amount of pre-stressed reinforcement does not change with the optimisation of the section. Considering the carbon emission target, concrete has the highest share of carbon emissions, between 90% and 95%. The optimisation of the section plays a role in reducing the carbon emissions of concrete, but the effect could be more precise. In addition, the optimal solution adjustment focuses on the concrete and steel beam section adjustment. To further determine the feasibility of the optimisation scheme, it needs to be verified and analysed by combining theoretical and experimental data.

4. Validation and Analysis of the DSCG Optimisation Model

The primary objective of this section is to assess the performance of the guideway’s structural scheme following optimisation using the NSGA-II algorithm, specifically in terms of the load-bearing and deformation characteristics. The data obtained through the equivalent cross-section method and results from the finite element model serve to verify the differences in structural performance, particularly in force distribution, between the optimised and pre-optimised schemes. This comparison will highlight the improvements achieved by optimising structural stiffness, deformation control, and overall efficiency under various loading conditions [33].

4.1. Validation and Analysis of Theoretical Data

To further validate the DSCG section data before and after optimisation, this subsection summarises the calculation results of the effective stiffness and deflection of the DSCG by converting the section and combining it with the design scheme of the optimised section in the above section, as shown in Table 7. The DCSG section was uniformly converted to a concrete section in the theoretical calculation. Among them, the effective stiffness of the mid-span section of the guideway in the original scheme was 4.00 × 10¹⁰ Pa. The effective stiffness of the spanning centre section of the guideway in Scheme I was 5.14 × 10¹⁰ Pa. The effective stiffness of the mid-span section of the guideway in Scheme II was 8.51 × 10¹⁰ Pa. The data were further brought into Equation (26) to obtain the deflection deformation data in the guideway span under straddle and suspended vehicle action. The deflection of the guideway under gravity load can be calculated using the mechanics of materials method, as in Equation (26):

$$f_{02}={\displaystyle \frac{5q{L}^4}{384E_c{I}_{0c}}}$$

(26)

where f₀₂ is the deflection of the guideway under uniform load and q is the load set of self-weight load in the unit area.

According to

Table 8. Summary of vertical deflections of the guideway calculated using the converted section method (unit: mm).

Scheme	Straddle	Suspended	Self-Weight	Total	Normative Limits (L/800)
Original scheme	8.21	3.02	12.85	24.08	37.50
Scheme I	6.42	2.40	8.94	17.76	37.50
Scheme II	3.91	1.40	5.45	10.76	37.50

, it can be seen that the optimisation for the DSCG mid-span interface enhanced the effective stiffness of the section. The enhancement of effective stiffness was about 128.48% for Scheme I and 212.63% for Scheme II. In addition to this, optimising the section reduced the self-weight of the guideway, which is one of the reasons for the minor deflection of the guideway under uniform load. It should be noted that the assumption premise of the commutative section method was that the composited section conforms to the flat section assumption, so the influence factor of the composited interface slip was not considered in the theoretical calculation. Of course, the commutative section method also did not consider the influence of the pre-stressed effect on the structure, and the whole structure was assumed to be the same section in the calculation. Such an assumption led to significant errors in the results, especially for variable section guideway structures.

4.2. Finite Element Simulation and Verification

To further evaluate the design solutions generated by the NSGA-II optimisation model, a finite element model of the original the DSCG and two optimised test models were developed using Abaqus for numerical analysis. The objective was to investigate the guideway’s internal force and deformation patterns before and after optimisation. In multi-objective optimisation, finite element analysis (FEA) serves as a critical support tool for validating the performance of each optimised design iteration, providing essential feedback for refining the optimisation process. Through FEA, the impact of various design variables on structural behaviour can be thoroughly assessed, enabling a deeper understanding of the performance of models based on different optimisation schemes.

4.2.1. Establishment of the Finite Element Model

The upper concrete and lower steel guideways were modelled using 3D solid units. Hoop, plain, and pre-stressing bars were modelled using line units. To avoid the stress concentration problem under loading, which leads to abnormal operation, infinite stiffness elastomers were set as load pads. At the same time, considering that all the DSCG structures are symmetric structures with mid-span sections, the model was built by symmetric modelling. The summary of the property settings of each material is shown in

Table 9. Summary of parameters for each type of material.

Serial Number	Material	Model	Sectional Area (m2)	Modulus of Elasticity (MPa)	Poisson’s Ratio	Mass Density (kg/m3)
1	Concrete	C60	1.22	3.60 × 104	0.17	2400
2	Steel beams	HPB345	0.05	2.10 × 105	0.30	7800
3	Longitudinal reinforcement	HRB400	0.20	2.00 × 105	0.30	7800
4	Hoop reinforcement	HPB235	0.11	2.10 × 105	0.30	7800
5	Pre-stressing tendons	1860 steel strand	0.18 × 10−3	1.95 × 105	0.30	7800
6	Rigid pads	-	-	2.10 × 106	0.30	7800

. The configuration of pre-stressing reinforcement was in the form of a 30 m straddle linear guideway arrangement, with 6 bundles of 1 × 7 strands on each side and 12 bundles symmetrically arranged. The nominal diameter of the steel strand was 15.2 mm, and the tensioning control stress was 1141.7 MPa. The strand was selected as a 1860 steel strand, and the temperature linear expansion coefficient was about 1.15 × 10⁻⁵/°C. The principle of temperature reduction applied the pre-stressing force. The pre-stressing force was applied using the principle of cooling [34]. The pre-stressing reinforcement was cooled to make it shrink to realise the application of the pre-stressing force. In addition, to avoid excessive deformation of the bottom structure affecting the DSCG finite element model data, the load application locations were all placed at the top of the concrete beams.

The model boundary conditions adopted simple support boundary constraints, and the constraints on the degrees of freedom of the support mainly included U1, U2, and UR3, and the symmetric section was selected as ZSYMM (U3 = UR1 = UR2 = 0). U1, U2, and U3 denote the degrees of freedom along the coordinate axes, x-axis, y-axis, and z-axis, respectively. UR1, UR2, and UR3 denote the rotational degrees of freedom along the coordinate axes, x-axis, y-axis, and z-axis, respectively. The hexahedral mesh will produce deformation in meshing because the optimised model was not a regular model. Therefore, tetrahedral mesh modelling was used for complex section models, and the hexahedral mesh approach was used for simple models. The concrete mesh properties were C3D8R, and the steel beam mesh properties were T3D2—the composited interface used a Tie connection for interface binding.

4.2.2. Intrinsic Model of Concrete

The converted section method used an elastic model, different from the intrinsic model of concrete materials in real engineering. To further establish the finite element model to reveal the vertical deformation law of the DSCG, the plasticity parameter of the intrinsic concrete model used damage-plasticity as the selection criterion. The uniaxial compressive stress–vehicle relationship of concrete adopted the expression proposed by Guo, Z. [35]. The data model of plasticity parameters is shown in Figure 10. The plasticity model parameters of the steel reinforcement were set as Young Stress as 2.1 × 10⁸ and Poisson’s ratio as 0.

4.3. Vertical Deformation (U2) Analysis of the DSCG Finite Element Model

The vertical deformation cloud diagrams of the three schemes under the total load are shown in Figure 11. It should be noted that the vertical deformation data under load are the overall deformation data of the DSCG. The guideway deflection was primarily based on the data at the bottom of the upper concrete beam. The cloud diagram is presented in Figure 11d. According to the data in Figure 11, the overall deformation distribution pattern of the DSCG was similar. The maximum vertical deformation in the span of the original scheme was 1.78 × 10² m. The maximum vertical deformation in the span of Scheme I was 1.83 × 10⁻² m. The maximum vertical deformation in the span of Scheme II was 1.49 × 10⁻² m. It can be seen that the optimisation of the section did not reduce the vertical deformation of the guideway as a whole, as calculated for the converted section.

On the contrary, replacing some concrete with steel guideways in the DSCG may have increased the vertical deformation of the guideway, an excellent example of which is the higher vertical deformation in Scheme I than in the original scheme. The vertical deformation of the guideway in Scheme II was less than the original design. At the same time, considering that the bottom steel guideway deformed more under load, we further extracted the vertical deformation data of the PC guideway in different schemes, as shown in

Table 10. Summary of vertical deformation of the DSCG superstructure by the finite element method (unit: mm).

Scheme	Straddle	Suspended	Self-Weight	Total	Normative Limits (L/800)
Original scheme	3.780	1.40	11.87	17.04	37.50
Scheme I	2.95	1.09	13.52	17.55	37.50
Scheme II	2.91	1.07	10.53	14.51	37.50

.

According to the data in Table 9, the deflections of the three schemes under the total load could satisfy the code limit of 37.50 mm. Among them, the mid-span deflection of the composited interface of Scheme I was 17.55 mm, which was slightly higher than that of the original scheme of 17.04 mm. The mid-span deflection of the composited interface of Scheme II was 14.51 mm, which was lower than that of the original scheme of 17.04 mm. However, the vertical deflection of the composited interface under different loads was mainly related to the load type. The vertical deflection of the original DSCG design was lower than that of the two optimised solutions, indicating that optimising the section improved the mid-span section stiffness of the guideway. However, the vertical deformation under self-weight load was the largest in Scheme I, at 13.52 mm. The vertical deformation of Scheme II was the smallest, at 10.53 mm. It can be seen that the effects of different design schemes on the vertical deformation of the DSCG composited interface were mainly reflected in the action of self-weight load. Because the steel guideway section replaced the concrete section, the structural self-weight of the DSCG was reduced.

4.4. Transverse (U1) and Longitudinal (U3) Deformation Analysis

Under load, the lateral deformation of the DSCG showed different deformation trends. The cloud diagram of the lateral deformation of the upper PC guideway under load is shown in Figure 12. To further analyse the development pattern of the original design scheme and the two optimised schemes regarding transverse deformation, we extracted the deformation data of the top of the guideway at points A and B.

As seen from Figure 12 and

Table 11. Summary of transverse (U1) deformation of the DSCG.

Scheme	Position of Deformation	Straddle	Suspended	Self-Weight	Total
Original scheme	A	−2.17 × 10−5	−5.80 × 10−6	−3.28 × 10−5	−6.03 × 10−5
	B	2.14 × 10−5	5.85 × 10−6	5.81 × 10−5	8.54 × 10−5
Scheme I	A	−3.81 × 10−5	−1.13 × 10−5	−4.16 × 10−4	−4.66 × 10−4
	B	9.57 × 10−6	1.03 × 10−6	4.58 × 10−4	4.69 × 10−4
Scheme II	A	−6.87 × 10−5	−2.56 × 10−5	−3.09 × 10−4	−4.04 × 10−4
	B	1.92 × 10−5	7.39 × 10−6	3.26 × 10−4	3.53 × 10−4

, the original design scheme was slightly different from the optimised scheme in terms of transverse deformation. The transverse deformation at the bottom edge of Scheme II and the deformation at the top of the PC guideway were also not negligible. The amplitude of lateral deformation of the original scheme was lower than that of the two optimised design schemes, in which the deformation value of Scheme II was smaller than that of Scheme I. Under the total load, the maximum deformation value at point A of the three schemes was found for Scheme I, which reached −4.66 × 10⁻⁴ m. The minimum value was found for the original scheme, which was −4.66 × 10⁻⁴ m. The minimum value for the original scheme was −6.03 × 10⁻⁵ m. The maximum deformation value at point A was 7.72 times higher than the minimum value. The maximum deformation value at point B of the three schemes was found for Scheme I, which reached 4.69 × 10⁻⁴ m. The minimum value was found for the original scheme, which was 8.54 × 10⁻⁵ m. The maximum deformation value at point B was 5.49 times higher than the minimum value. The trend of data distribution under different loads was the same as that under total load. The scheme with the most significant value of transverse deformation was Scheme I, and the scheme with the smallest deformation value was the original scheme. It can be seen that the optimisation of the section increased the transverse deformation of the interface, mainly because the transverse stiffness of the concrete section was greater than the transverse stiffness of the composited interface.

The DSCG was optimised longitudinally using a variable section, and the specific interface changes could be found in the previous section on the rules for setting the cross-section parameters. Similar to the transverse deformation, the guideway’s longitudinal deformation also showed different development laws under various loads. Figure 13 shows the cloud diagram of the longitudinal deformation of the upper PC guideway under load. To further analyse the development law of the original design scheme and the two optimised schemes in terms of longitudinal deformation, we extracted the deformation data of the top of the guideway at points A and B and summarised them in Table 10. It should be noted that the definitions of A and B are different from the transverse deformation cloud positions, and the positions are defined according to the distribution of the cloud maxima, which can be shown in the finite element model cloud diagram.

Figure 13 and

Table 12. Summary of transverse (U3) deformation of the DSCG by the finite element method.

Scheme	Position of Deformation	Straddle	Suspended	Self-Weight	Total
Original scheme	A	−3.16 × 10−4	−1.14 × 10−4	−9.51 × 10−4	−1.38 × 10−3
	B	2.66 × 10−4	9.79 × 10−5	8.31 × 10−4	1.20 × 10−3
Scheme I	A	−1.15 × 10−4	−4.15 × 10−5	−4.73 × 10−4	−6.30 × 10−4
	B	2.92 × 10−4	1.08 × 10−4	1.30 × 10−3	1.70 × 10−3
Scheme II	A	−1.17 × 10−4	−4.19 × 10−5	−3.74 × 10−4	−5.32 × 10−4
	B	2.84 × 10−4	1.04 × 10−4	1.00 × 10−3	1.39 × 10−3

demonstrate that the original design scheme differed from the optimised longitudinal deformation scheme. Specifically, in the distribution of the maximum value of longitudinal displacement at the bottom of the PC guideway, the original design scheme had the maximum value distributed in the end section of the guideway. The distribution location of the interface’s maximum value of the optimised scheme was Section 3-3, which is the end position of the transition region of the longitudinal section change. The maximum values of longitudinal displacements at the top of the PC guideway were all distributed at the end position of the track girder. In addition, the magnitude of longitudinal deformation at the bottom of the original scheme, the DSCG, was higher than that of the two optimised schemes.

The original scheme had the most significant value at point A, at −1.38 × 10⁻³ m. The smallest value was at point A of Scheme II, which was −5.32 × 10⁻⁴ m. The value at point A of the original scheme was 2.60 times higher than that at point A of Scheme II. Point B of Scheme I had the highest value, at 1.70 × 10⁻³ m. The original scheme point B had the smallest value of 1.20 × 10⁻³ m. The value at point B of Scheme I was 1.42 times higher than that at point B of the original scheme.

The trend of data distribution under different loads was the same as that under total load. The solution with the most significant value of longitudinal deformation at point A at the bottom of the guideway was the original solution, and the solution with the most considerable value of longitudinal deformation at point B at the top of the guideway was Scheme I. It can be seen that the section optimisation increased the overall longitudinal stiffness of the composited interface of the guideway. The distribution pattern of longitudinal deformation of the optimised design scheme differed from that of the original design scheme, and the maximum value was lower than that of the original design scheme. However, the optimisation of the section did not lead to changes in the distribution pattern of the longitudinal displacement at the top of the guideway, and the maximum value of the two optimised design solutions was more significant than that of the original design solution.

5. Conclusions and Outlook

5.1. Conclusions

This paper presented a constrained model with three objective functions and sixteen variables to be optimised. The structural optimisation design of the DSCG was completed with the help of the NSGA-II intelligent algorithm. The research methodology and results could be applied to the section optimisation design of pre-stressed concrete, steel, and steel-hybrid structures. The main research conclusions of this paper were as follows:

(1)

Based on NSGA-II, the optimisation calculation of the DSCG structure with constraints involving sixteen variables and three objective functions was realised. In the Pareto frontier distribution trend of the optimisation model, the linear correlation coefficient between the DSCG economy target and the steel consumption target was R = 0.99, which had a high linear relationship. There was no apparent linear relationship between carbon emission and steel consumption targets, with a linear correlation coefficient R = 0.03. However, there was no linear relationship between carbon emission and economic targets, with a linear correlation coefficient R = 0.

(2)

According to the characteristics of the distribution of the multi-objective function Pareto frontier (p = 100, g = 200), we obtained a program that could represent the optimisation of economic and steel consumption targets and the optimisation of carbon emission targets. Regarding economic and steel consumption targets, the optimised section in Scheme I slightly increased by about 5.5%, compared to the original scheme. The carbon emission target decreased more, by about 23.67%. The optimised section in Scheme II slightly increased economy and steel consumption compared with the original scheme, by about 32%. The carbon emission target was reduced by about 42.03%. The optimised scheme explicitly reflected increased economy and steel consumption but decreased carbon emissions.

(3)

Steel accounted for the most significant economic targets of the three options, accounting for about 75% to 88.5% of the total cost. That was followed by plain steel, which accounted for between 7% and 15% of the total spend. The steel consumption of the lower steel guideway of the DSCG in the schemes ranged from about 73% to 87% of the total steel consumption. The section optimisation increased the steel used in the steel beams and decreased the standard steel reinforcement. Concrete had the highest share of carbon emissions, between about 90% and 95%. The optimisation of the section played a role in reducing the carbon emission of concrete, although the effect was not very obvious.

(4)

Based on the equivalent section method, the improvement of the effective stiffness of Scheme I was about 128.48%, and the improvement of the effective stiffness of Scheme II was about 212.63%, but with a significant error. Based on the finite element analysis method, optimising the section increased the guideway’s longitudinal and transverse deformation, significantly changing the law of the longitudinal slip distribution at the DSCG interface. The distribution pattern of transverse and longitudinal deformations in the three design alternatives was unrelated to the loads’ type and magnitude. However, the nature of such deformation was different from vertical deformation. The vertical deformation in the three design options was directly related to the type of loading, especially under self-weight.

5.2. Outlook

This research significantly advances the field of multi-objective optimisation in civil engineering in several ways: (1) The study offered a novel approach to optimising complex civil engineering structures by incorporating the NSGA-II algorithm. Unlike traditional methods that rely on empirical data or single-objective optimisation, NSGA-II efficiently handles multiple conflicting objectives—such as economy, steel usage, and carbon emissions. This makes multi-objective balancing in civil engineering more scientific and practical. (2) By employing NSGA-II, this study provided balanced design solutions, allowing engineers greater flexibility in making trade-offs between objectives. Instead of limiting design decisions to a single solution, this method enables engineers to select the most suitable design based on practical needs and constraints. (3) Incorporation of environmental and economic factors This research moved beyond traditional optimisation focused on cost by including environmental factors, such as carbon emission reduction. This broadens the scope of civil engineering optimisation to address sustainability, marking a significant step forward in the field. (4) Considering NSGA-II optimisation with FEA enhanced the study’s credibility by verifying the structural performance of the optimised designs. This integration improved the reliability of the results and ensured that the optimised designs were feasible for practical applications, reinforcing the importance of performance validation in multi-objective optimisation. (5) Although the primary focus of this study was on the DSCG, the proposed optimisation approach can be applied to a wide range of civil engineering structures. The methods demonstrated here can be adapted to different cross-sections, spans, and structural types, thereby expanding the utility of intelligent optimisation algorithms across civil engineering projects. (6) The study offers a practical guide for incorporating intelligent optimisation algorithms into real-world engineering design. Integrating MATLAB programming and NSGA-II will enhance engineers’ understanding and capability to tackle multi-objective optimisation challenges, bridging the gap between theory and practice.

In conclusion, by integrating NSGA-II, finite element analysis, and multi-objective optimisation, this research advances these methods’ theoretical and practical applications in civil engineering. It provides valuable tools for optimising complex structures and laying a foundation for the future use of intelligent algorithms in green design and material optimisation. The study contributes to the growing understanding of balancing multiple objectives in structural design, supporting the development of more sustainable and efficient engineering solutions.