Optimization Design for Steel Trusses Based on a Genetic Algorithm

Abstract

Steel trusses are widely utilized in engineering structures, and their optimization is essential for enhancing structural performance and reducing material consumption. Existing optimization methods for steel trusses predominantly rely on the trial-and-error method, which is not only inefficient but also inaccurate. Therefore, this study focused on the optimization of steel trusses using an efficient and accurate optimization methodology. Based on a genetic algorithm and the finite element method, both mono- and multi-parameter optimization designs for steel trusses were executed, an applicable optimization design method for steel trusses was established, and corresponding optimization design programs were developed. The analysis demonstrates that the proposed optimization method effectively optimizes truss height and member cross-section, leading to a significant reduction in material consumption. Compared to the traditional trial-and-error method, the proposed optimization method exhibits adequate calculation accuracy and superior optimization efficiency, thereby providing a robust theoretical foundation for the engineering design of steel trusses.

1. Introduction

Steel trusses are widely used in practical engineering owing to their superior mechanical behaviors. In addition, steel trusses also possess the advantage of convenient transportation and easy disassembly. Thus, they could also be used in temporary buildings. The load-carrying capacity of a steel truss can be affected by various parameters, such as the truss height and member section dimensions. In order to improve capacity or decrease material consumption, it is essential to optimize the design of steel trusses.

In the 1960s, Schmit [1] developed a mathematical model for the optimal design of elastic structures under multiple working conditions and proposed the use of mathematical programming methods for problem solving. To address structural optimization problems with multiple constraints, various optimization methods, such as linear programming [2], gradient projection [3], feasible directions algorithm [4], and penalty function [5], were applied. However, these optimization methods were based on a mathematical programming theory, and their computation models were under ideal conditions, thus making them challenging to solve and compute in practical engineering applications. To find simpler and more efficient optimization methods, researchers began to explore new approaches. In 1964, Dorml et al. [6] employed a novel method to optimize the structural configuration of steel trusses with the aim of minimizing the self-weight. In 1969, Venkayyaol et al. [7] proposed the optimal criteria method, which enhanced computational efficiency by selecting the iteration form of design variables based on established optimality criteria. To achieve a high working performance of the compliant mechanism, Huynh et al. [8] proposed a gray relational analysis method for its optimization. Each optimization method has its advantages and disadvantages. To overcome the shortcomings of each method, researchers have attempted to combine two or more optimization methods to improve accuracy and efficiency during optimization. Nguyen et al. [9] employed a hybrid algorithm that combines fuzzy logic and an adaptive neuro-fuzzy inference system to optimize a robot arm with multiple objectives.

In recent years, researchers have begun to pay more attention to the combination of truss structure optimization and intelligent algorithms. Suzuki [10] studied a Markov chain analysis of genetic algorithms, particularly for a variety called a modified elitist strategy. By comparing it with the standard genetic algorithm, the results showed that the improved algorithm had a better convergence rate and accuracy. Hasançebi et al. [11] employed seven metaheuristic algorithms to optimize the design of pin-jointed structures, and evaluated and compared their convergence rates and reliabilities. Based on a particle swarm optimizer, harmony search optimization, and simple genetic algorithms, Carbas et al. [12] conducted a study on the optimum design of real-size plane trusses. Serpik et al. [13] proposed a new metaheuristic algorithm that combines truss discrete size and shape optimization with intermediate value search objectives using working search inspired strategies. Celso et al. [14] introduced an adaptive penalty function mechanism to optimize the design of truss structures. Their research results indicated that the improved algorithm was highly effective. To reduce construction costs, Kurniawan et al. [15] proposed a genetic algorithm that maximizes structural performance within the allowable stress range. Nguyen-Van et al. [16] proposed a novel optimization algorithm that integrates hybrid differential evolution and symbiotic organism search optimization algorithms to optimize the size and shape of truss structures under multi-frequency constraints. Azad et al. [17] used a new heuristic algorithm to investigate the optimization of a large-scale truss, and their methodology solved the truss optimization problem with complex design variables. Inspired by the traffic volume on a dual carriageway, Kumar et al. [18] proposed a novel multi-objective passing vehicle search algorithm, which exhibits superior convergence behavior and optimization performance compared to four commonly used multi-objective optimization algorithms. Baykasoğlu et al. [19] proposed a weighted superposition attraction–repulsion (WSAR) algorithm to address the high dimensionality, nonlinearity, and non-convexity issues in truss structure optimization. Jawad et al. [20] proposed an innovative dragonfly optimization algorithm inspired by the static and dynamic clustering behaviors of dragonflies in nature. Based on the actual water circulation behavior of rivers and precipitation, Eid et al. [21] proposed a new metaheuristic algorithm named the spiral water cycle algorithm. Piereran et al. [22] proposed an improved algorithm based on the Tinkerbell map chaotic sequence, which takes advantage of the grouped subdivision characteristic of an algorithm population to harmonize discrete and continuous probabilities.

The aforementioned research provides ideas and methodologies to optimize steel trusses. Nevertheless, it is worth noting that the studies of steel truss optimization mentioned above are primarily focused on mono-parameter cases, i.e., there is only one variable during the optimization process. However, the behavior of steel trusses can be affected by a series of different variables, such as truss height, member section, etc. Thus, it is imperative to conduct multi-parameter optimization to minimize the material consumption of trusses. To address this gap in research, a genetic algorithm was developed in this study using the Python language and combined with the finite element software ABAQUS to perform the structural optimization of a truss. Firstly, the optimization strategy was elaborated in detail. Secondly, the proposed optimization method and the traditional trial-and-error method were employed to optimize the truss, and the optimized results of both methods were compared to verify the reliability and accuracy of the proposed optimization method. Finally, both mono-parameter and multi-parameter optimization analyses were performed and compared. The results demonstrate that material consumption can be significantly decreased using the proposed optimization analyses, especially when using the multi-parameter optimization analysis.

2. Proposed Method

As mentioned above, this study focused on mono-objective optimization with the sole objective of minimizing the structural weight, using the truss height and cross-sectional height of truss members as the optimization variables. The optimization was performed using a genetic algorithm, which includes mono- and multi-parameter optimization. Given that the optimization strategies for the mono- and multi-parameter cases are identical, the mono-parameter optimization case is taken as an example to elaborate on the optimization strategy in detail.

2.1. Optimization Constraint Model

The aim of the steel truss optimization is to minimize material consumption without decreasing the load-carrying capacity. Thus, the objective function can be written as Equation (1):

$$\min W={\displaystyle \sum _{i=1}^n{\rho }_i{A}_i{L}_i}$$

(1)

where W is the truss mass; n is the total number of members in the steel truss; and ${\rho }_i$, $A_i$, and $L_i$ denote the steel density, member cross-sectional area, and member length, respectively. Equation (1) can be simplified to Equation (2) considering that the densities of all members are the same:

$$\min V={\displaystyle \sum _{i=1}^n{A}_i{L}_i}$$

(2)

where V is the total volume of all members.

For the truss optimization in this work, the truss height and member sectional heights were selected to be the optimization variables. Thus, the optimization variable X can be expressed using Equation (3):

$$X={[H,h_1,h_2,\dots h_i\dots ,h_n]}^T$$

(3)

where $H$ is the steel truss height, and $h_i$ is the cross-sectional height of each truss member.

For the design of a steel truss, the member stress and truss deflection should not exceed the allowable magnitude to ensure structural safety. Thus, the truss optimization should satisfy the constraint conditions shown in Equation (4):

$$\left\{\begin{array}{l}\left|u\right|\le [u]\\ \left|{\sigma }_i\right|\le [{\sigma }_i],i=1,2,\dots ,n\end{array}\right.$$

(4)

where $u$ and $[u]$ are the calculated deflection and allowable deflection of the truss, respectively, and ${\sigma }_i$ and $[{\sigma }_i]$ are the calculated member stress and allowable stress of the truss structure, respectively.

2.2. Optimization Methodology

2.2.1. Optimization Procedure

Compared with other algorithms, a genetic algorithm has the advantages of global search, parallel computing, and strong adaptability [23]. Thus, the steel truss optimization problem was solved by using the combination of a genetic algorithm and finite element analysis (FEA) in this work. The search space of the optimization problem is thought to be a kind of population. According to the simulation of heredity and mutation in the evolution process, new solutions are generated constantly, and the solutions with a high fitness are adopted for further evolution. This study employed the Python language for parametric FEA, in which the modeling, solving, and post-processing processes of the structure were all scripted into ABAQUS files. Additionally, a main program for the genetic algorithm was developed to implement these ABAQUS files and optimize the steel truss. The procedure for the steel truss optimization is shown in Figure 1.

As shown in Figure 1, the first step of truss optimization is parameter setting. Relevant optimization parameters, including the encoding scheme, population size, evolutionary generations, and evolutionary termination criterion, are set in this step. The second step is population initialization, in which the initial population is randomly generated according to the population size and encoding scheme. The third step is fitness calculation. In this step, the parameter information for each individual is written into the finite element script, and ABAQUS is invoked to calculate the stress and deflection of the truss. The resulting stress and deflection are then fed back to the algorithm program to evaluate the fitness and examine if any constraints have been violated. Individuals that violate the constraint conditions are penalized using a penalty function. The next step is selection, recombination, and mutation. The aims of this step are to generate new individuals and to obtain a new population from the generated individuals. More details about this step will be introduced in Section 2.2.2, Section 2.2.3 and Section 2.2.4. The last step is evolutionary termination judgment. The aim of this step is to judge whether the results satisfy the termination criterion.

2.2.2. Selection Operators

At the beginning of each iteration, it is essential to use a selection operator to select individuals from the population. The selected individuals are the seeds of the next generation. The probability of individuals being selected is associated with their fitness, implying that individuals with a higher fitness are more likely to be selected. The selection operators for mono-parameter optimization and multi-parameter optimization are different.

A tournament selection operator [24] is adopted for mono-parameter optimization because it could improve the convergence speed of the optimization methodology and reduce the computational complexity. The basic idea for the tournament selection operator is to randomly select a certain number of individuals, and the best individuals can be obtained by comparing all individuals with each other. The best individuals are used as the seeds for the next generation.

An elite tournament selection operator [25] is adopted for multi-parameter optimization. In contrast to the traditional tournament selection operator used in mono-parameter optimization, the individual selection of an elite tournament selection operator is not random; thus, the elite tournament selection operator can ensure that the most excellent individual is selected. Consequently, the fitness of the population can be improved.

2.2.3. Recombination Operators

After obtaining the seeds of a new generation of population, the recombination operator is employed to generate new individuals that do not exist in the original population. In this study, the two-point crossover recombination operator was used, which aims to select the intersection point of two chromosomes and exchange the gene segments between the two intersections, and, thus, two new offspring chromosomes are generated. The two-point crossover recombination operator is shown in Figure 2, where numbers 1–8 with the blue background represent the gene segment code for Parent 1’s chromosome, while letters a–h with the yellow background represent the gene segment code for Parent 2’s chromosome. The detailed steps are listed as follows:

(1)

Select two chromosomes as the parents randomly.

(2)

Select two intersections in the chromosomes randomly.

(3)

Exchange the gene segments between the two intersections and obtain two new offspring chromosomes.

(4)

Input the new offspring chromosomes into the population of the next generation.

(5)

Repeat the above steps until obtaining enough offspring chromosomes.

2.2.4. Mutation Operators

During the iteration process, it is possible for each individual to have the same genes, but these genes are not the global optimal solution. In this case, the recombination operation will not change the genotype of the individuals and cannot generate new individuals, causing the algorithm to get stuck in a local optimal solution. In contrast, mutation operation can generate new individuals through mutation, thereby increasing the algorithm’s ability to optimize globally.

The mutation operator used in this study is the Gaussian mutation operator, which works by randomly perturbing a certain component of an individual’s genes according to a Gaussian distribution to achieve mutation. The specific implementation method is to first determine the gene component that needs to be mutated, and then generate a random number from a Gaussian distribution with the gene component’s value as the mean. The random number is then added to the gene component’s value to obtain the mutated individual. The advantage of the Gaussian mutation operator is that it can control the degree of mutation by adjusting the standard deviation of the Gaussian distribution, thus achieving a better balance between exploration and exploitation.

For the constrained optimization problem addressed in this study, the mutation value of each gene component is checked when performing the Gaussian mutation; if the mutated value exceeds the constraint range, it is adjusted to be within the constraint range.

3. Optimization Methodology Validation

3.1. Validation Model

In order to validate the proposed optimization methodology, a truss model was established, as shown in Figure 3. The span L and height H of the truss are 43,200 mm and 5150 mm. The length L1 of the chords, including the top chords and bottom chords, is 7200 mm. A concentrated load P with the value of 156 kN was applied to each top node of the truss. The material of all the members is Q355 steel with a yielding stress of 355 MPa. The corresponding Young’s modulus and Poisson’s ratio are 206 GPa and 0.3, respectively. The steel used is assumed to be an ideal elastic–plastic material in this study. In the modeling of the truss, to consider the influence of the member section on the bearing capacity of the truss, the beam element B33 was selected as the element type, rather than the truss element, which only considers the influence of the section area. Since the connection between the beam elements is rigid, in order to simulate the hinge of the truss structure, the ‘release’ command in ABAQUS was used to release the bending moment of the beam end.

There are 23 members of the truss, which are numbered from 1 to 23 in Figure 3. For the convenience of the latter expression, the truss members are divided into five types, including the top chord, bottom chord, diagonal middle web member, diagonal edge web member, and vertical web member. Among them, the cross-section of the diagonal edge web member is a double-box-shaped section (see Figure 4), and the other members all have an H-shaped section. The member number and cross-section are shown in

Table 1. Classification of members in the large-span steel truss. Member numbers correspond to numbers 1–23 in Figure 3.

Member Type	Member Number	Cross-Section
Top chord	1, 2, 3, 4, 5, and 6	H600 × 300 × 25 × 16
Bottom chord	18, 19, 20, 21, 22, and 23	H600 × 300 × 20 × 14
Diagonal web member	9, 11, 13, and 15	H350 × 300 × 14 × 12
Diagonal edge web member	7 and 17	600 × 300 × 20 × 14
Vertical web member	8, 10, 12, 14, and 16	H350 × 300 × 14 × 12

. The cross-section in Table 1 is expressed as “sectional height × sectional width × flange thickness×web thickness”.

3.2. Optimization Method Verification

Based on the model mentioned in Section 3.1, the height of the large-span steel truss was optimized and analyzed. Comparing the calculated result from the optimization algorithm with that from the trial-and-error method [26], the accuracy of the optimization algorithm presented in this study was verified.

The height of the large-span steel truss involved in the optimization analysis refers to the distance between the top-chord axis and the bottom-chord axis. Additionally, the optimization convergence process of truss height and volume is shown in Figure 5a, in which the horizontal axis indicates the evolutionary generations, the left vertical axis indicates the truss height, and the right vertical axis indicates the truss volume. In this optimization analysis, the genetic algorithm uses real-number encoding, with the population size being set to 8, the limit of evolutionary generations being set to 50, and the recombination probability being set to 0.7. From Figure 5a, it can be observed that the optimization of the truss height converges at around generation 18, with the optimal truss height being 4557 mm and the corresponding minimum truss volume being 3.3025 m³. Compared to the initial model shown in Section 3.1, the truss height has been reduced by 11.5% and the truss volume has been reduced by 2.4%.

To examine the effectiveness of the optimization algorithm in handling constraints, the iteration curves of the truss deflection and rod stress were plotted, as shown in Figure 5b. From Figure 5b, it can be observed that the rod stress and truss deflection are within the allowable range, and the truss deflection and rod stress stabilize near the limit values. This indicates that the optimized structure has fully utilized its performance under the constraint conditions, demonstrating the effectiveness of the constraint-handling approach used in this study.

To verify the accuracy of the optimization results based on the genetic algorithm, this study employed the trial-and-error method for comparison. Specifically, a series of large-span steel truss models with different truss heights were established using the trial-and-error method, and FEA was performed sequentially using the ABAQUS software to determine the truss height that satisfied the truss deflection and rod stress. In this study, a total of 800 models were analyzed via the trial-and-error method, where the truss height ranged from 2000 mm to 10,000 mm while the other parameters remained consistent with the model presented in Section 3.1. The analysis results are shown in Figure 6. When the truss deflection reaches the allowable deflection, the truss height is 4520 mm, but the rod stress exceeds the limit. When the rod stress reaches the allowable stress, the truss height is 4560 mm, and the truss deflection is still under the allowable deflection. Therefore, the height of the large-span steel truss obtained using the trial-and-error method is 4560 mm.

Comparing the calculation results of Figure 5 and Figure 6, it can be observed that the truss height obtained via the trial-and-error method is 4560 mm, while the optimal truss height obtained using the proposed method is 4557 mm. The error between the two is only about 0.07%, indicating that the optimization results obtained using the proposed method are close to those obtained using the trial-and-error method, which validates the accuracy of the optimization based on the genetic algorithm. However, the trial-and-error method required 800 calculations to obtain the optimal result, while the proposed method required only 18 calculations, indicating that the proposed optimization method has a significant advantage in terms of efficiency. Therefore, in engineering practice, using a genetic algorithm for the optimization design of large-span steel trusses has better practicality and feasibility.

4. Mono-Parameter Optimizations

As mentioned above, this study utilized a genetic algorithm to optimize a large-span steel truss. To evaluate the contribution of various truss components to the overall optimized design, the truss members are divided into the top chord, bottom chord, diagonal web member, diagonal edge web member, and vertical web member. Then, the mono-parameter optimizations of the truss were carried out with the section height of various truss components as the optimization variable and the volume of the truss as the optimization objective.

4.1. Chord Optimizations

4.1.1. Top Chord

When optimizing the cross-section of the truss top chord, the section height of the top chord is taken as the optimization variable. The section of the top chord is a welded H-shaped section. To prevent local buckling of the rod, the flange width $b$ of the top chord is set to half of the section height $h$, and the ratio of the flange width $b$ to the flange thickness $t_1$ is $b/t_1=12$, while the ratio of the web height $h_0$ to the web thickness $t_w$ is $h_0/t_w=40$. The other parameters of the steel truss are identical to the model described in Section 3.1. The genetic algorithm was employed with real number encoding and the optimization variables ranging from 100 mm to 1000 mm. The population size was set to 8, the limit of evolutionary generations was set to 50, and the recombination probability was set to 0.7.

Figure 7a illustrates the process of the optimization, revealing that the genetic algorithm converges to the optimal result at the 25th generation, with a truss volume of 3.2414 m³ and a section height of the top chord of 545 mm. Compared to the initial model, the section height of the top chord is reduced by 9.2%, and the truss volume is reduced by 4.2%. The optimization paths of the truss deflection and rod stress are shown in Figure 7b. It can be observed that both the truss deflection and rod stress eventually converge to their respective limit values, indicating that the optimized structure has fully utilized its performance while satisfying the given constraints.

4.1.2. Bottom Chord

When optimizing the cross-section of the truss bottom chord, the section height of the bottom chord is taken as the optimization variable. The section of the bottom chord is a welded H-shaped section. To avoid local buckling of the rod, the flange width $b$ of the bottom chord is half of the cross-section height $h$, the ratio of the flange width $b$ to the flange thickness $t_1$ is $b/t_1=15$, and the ratio of the web height $h_0$ to the web thickness $t_w$ is $h_0/t_w=40$. The other parameters of the steel truss are identical to the model in Section 3.1. The encoding of real number was used in the genetic algorithm, with the optimization variables ranging from 100 mm to 1000 mm, the population size being set to 8, the limit of evolutionary generations being set to 50 and the recombination probability being set to 0.7.

The process of the optimization is shown in Figure 8a, and it is found that the genetic algorithm converges to the optimum at the 41st generation. The volume of the optimum truss and the cross-section height of the bottom chord are 2.817 m³ and 331 mm, respectively, which have reduced by 44.8% in volume and by 16.7% in cross-section height when compared to the initial model. Figure 8b displays the optimization paths for both truss deflection and rod stress. Although both parameters do not approach their limits, further analysis reveals that when the cross-section of the bottom chord is 330 mm, the truss deflection reaches 102.1636 mm and the rod stress exceeds the buckling stress at 363.49 MPa, thereby violating the constraint conditions. Thus, the optimized structure obtained from the converged results represents the optimal solution under the given constraints, indicating that the truss performance has been fully utilized.

4.2. Web Member Optimizations

4.2.1. Diagonal Web Member

When optimizing the cross-section of the truss diagonal web member, the section height of the diagonal web member is taken as the optimization variable. The section of the diagonal web member is a welded H-shaped section. To avoid local buckling of the rod, the ratio of the cross-section height of the diagonal web member $h$ to the flange width $b$ is $h/b=6/7$; the ratio of the flange width $b$ to the flange thickness $t_1$ is $b/t_1=21$; and the ratio of the web height $h_0$ to the web thickness $t_w$ is $h_0/t_w=30$. The other parameters of the steel truss are identical to the model described in Section 3.1. The encoding of real number was used in the genetic algorithm, with the optimization variables ranging from 100 mm to 1000 mm, the population size being set to 8, the limit of evolutionary generations being set to 50, and the recombination probability being set to 0.7.

The process of the optimization is shown in Figure 9a. It can be observed that the genetic algorithm converges to the optimal result at the 22nd generation, with a truss volume of 3.347 m³ and a cross-section height of 330 mm. Compared to the initial model, the cross-section height of the diagonal web member is reduced by 5.7%, and the truss volume is reduced by 1.1%. The optimization paths of the truss deflection and rod stress are illustrated in Figure 9b. It can be observed that both the truss deflection and member stress eventually converge to their respective limit values. This indicates that the optimized structure has fully utilized its performance under the constraint conditions.

4.2.2. Diagonal Edge Web Member

When optimizing the cross-section of the truss diagonal edge web member, the section height of the diagonal edge web member is taken as the optimization variable. The section of the diagonal edge web member is a double-box-shaped section, as shown in Figure 4. To prevent local buckling of the rod, the flange width of the diagonal edge web member $b$ is half of the cross-section height $h$, the ratio of the cross-section height $h$ to the thickness of the box wall $t_1$ is $h/t_1=30$, and the ratio of the cross-section height $h$ to the thickness of the spacer plate $t_2$ is $h/t_2=40$. The other parameters of the steel truss are identical to the model described in Section 3.1. The encoding of real number was used in the genetic algorithm, with the optimization variables ranging from 100 mm to 1000 mm, the population size being set to 8, the limit of evolutionary generations being set to 50, and the recombination probability being set to 0.7.

The process of the optimization is shown in Figure 10a. It can be observed that the genetic algorithm converges to the optimal result at the 21st generation, with a truss volume of 2.9858 m³ and cross-section height of 401 mm. Compared to the initial model, the cross-section height of the diagonal edge web member reduces by 33.2%, and the truss volume reduces by 11.7%. The optimization paths for truss deflection and rod stress are shown in Figure 10b. It is evident that both deflection and stress converge to their respective limit values. This suggests that the performance of the truss has been fully exploited under the constraint conditions.

4.2.3. Vertical Web Member

When optimizing the cross-section of the truss vertical web member, the section height of the vertical web member is taken as the optimization variable. The section of the diagonal web member is a welded H-shaped section. To avoid local buckling of the rod, the ratio of the cross-section height $h$ to the flange width $b$ is $h/b=6/7$, the ratio of the flange width $b$ to the flange thickness $t_1$ is $b/t_1=21$, and the ratio of the web height $h_0$ to the web thickness $t_w$ is $h_0/t_w=30$. The other parameters of the steel truss are identical to the model described in Section 3.1. The encoding of real number was used in the genetic algorithm, with the optimization variables ranging from 100 mm to 1000 mm, the population size being set to 8, the limit of evolutionary generations being set to 50, and the recombination probability being set to 0.7.

The process of the optimization is plotted in Figure 11a. It is found that the genetic algorithm converges to the optimum at the 25th generation, when the truss volume and cross-section height of the vertical web member are 3.2387 m³ and 246 mm, respectively. Compared to the initial model, the cross-section height reduces by 29.7% and the total volume of the truss reduces by 4.3%. The optimization paths of truss deflection and rod stress are depicted in Figure 11b. It is apparent that neither truss deflection nor member stress approaches the limit values. However, upon analyzing the optimization process, it is discovered that when the cross-section height of the vertical web member is 245 mm, the truss deflection is 98.1401 mm and the rod stress is 365.8363 MPa, which exceed the buckling stress and do not satisfy the constraint conditions. Therefore, the optimized structure obtained from the converged results is the optimum solution under the constraint conditions, indicating that the performance of the truss has been fully utilized.

5. Multi-Parameter Optimizations

The cross-section heights of the chords (including the top chord and the bottom chord) and the web members (including the diagonal web member, the diagonal edge web member, and the vertical web member) were set as the optimization variables in the multi-parameter optimizations of the long-span steel truss. The truss volume was set as the optimization target. In addition, the optimization paths of truss deflection and rod stress were considered. With the multi-parameter optimizations, the optimal cross-section heights of the chords and web members could be obtained.

5.1. Chord Optimizations

The cross-section heights of the top chord and the bottom chord were set as the optimization variables. To avoid local buckling of the rods, the cross-section parameters of the top chord and the bottom chord were the same as those presented in Section 4.1. When initializing the population using the genetic algorithm, the encoding of real number and the elite tournament selection operator were chosen. Additionally, the optimization variables were set to range from 100 mm to 1000 mm, with a population number of 16, a generation limit of 50 generations, and a recombination probability of 0.7. The optimization results are shown in Figure 12a,b, in which the cross-section height of the top chord reaches the optimum value of 574 mm at the 24th generation. At the 30th generation, the cross-section height of the bottom chord reaches the optimum value of 332 mm, with the smallest volume of 2.7519 m³. Compared to the initial modal, the cross-section height of the top chord has been reduced by 4.3%, the cross-section height of the bottom chord has been reduced by 44.7%, and the truss volume has been reduced by 18.7%. The comparison results between the mono- and the multi-parameter optimizations are listed in

Table 2. Comparison results of mono- and multi-parameter optimizations of the chords.

	Mono-Parameter		Multi-Parameter
	Top Chord	Bottom Chord	Top Chord	Bottom Chord
Truss volume (m3)	3.2414	2.817	2.7519
Cross-section height (mm)	545	331	574	332
Truss optimization (%)	4.2	16.7	18.7

.

As shown in Table 2, the multi-parameter optimization has better volume optimization compared to the mono-parameter optimization for the top chord, but the difference for the bottom chord is small. In addition, there is no significant change in the optimal height of the cross-section for the bottom chord when comparing the multi-parameter optimization to the mono-parameter optimization, indicating that the bottom chord has a greater influence on the multi-parameter optimization.

The optimization paths of truss deflection and rod stress are shown in Figure 12c. It is found that the truss deflection and rod stress approach their limit values, which means that the performance of the truss has been fully exploited under the constraint conditions.

5.2. Web Member Optimizations

The cross-section heights of the diagonal web member, the diagonal edge web member, and the vertical web member were set as the optimization variables. To avoid the appearance of local buckling of the web members, the cross-section parameters of the three kinds of web members were set to be the same as the parameters described in Section 4.2. With the encoding of real number and the application of the elite tournament selection operator, the optimization variables ranged from 100 mm to 1000 mm, the population size was set to 24, the generation was limited to 50 generations, and the recombination probability was set to 0.7. The process of the optimization is shown in Figure 13a,b, in which the cross-section height of the diagonal web member reaches the optimum value of 332 mm at the 23rd generation. The cross-section height of the diagonal edge web member reaches the optimum value of 401 mm at the 32nd generation. Additionally, the vertical web member reaches the optimum cross-section height of 417 mm at the 44th generation. Finally, the truss volume has reduced to 2.5043 m³, which is a reduction of 25.9% compared to the initial model. Additionally, the cross-section height of the diagonal web member has reduced by 5.1%, that of the diagonal edge web member has reduced by 33.2%, and that of the vertical web member has reduced by 19.2%. The comparison results between the mono- and the multi-parameter optimization are listed in

Table 3. Comparison results of the mono- and multi-parameter optimizations of the web members.

	Mono-Parameter Optimization
	Diagonal Web	Diagonal Edge Web	Vertical Web
Truss height (m3)	3.347	2.9858	3.2387
Cross-section height (mm)	330	401	246
Volume optimization (%)	5.7	11.7	4.3
	Multi-parameter optimization
	Diagonal web	Diagonal edge web	Vertical web
Truss height (m3)	2.5043
Cross-section height (mm)	332	401	417
Volume optimization (%)	25.9

.

As shown in Table 3, for the web members, the multi-parameter optimization is more effective than the mono-parameter optimization. Additionally, the optimization paths of truss deflection and rod stress are shown in Figure 13c. It is found that the truss deflection and rod stress approach their limit values, which means that the performance of the truss has been fully exploited under the constraint conditions.

6. Conclusions

In this study, based on a genetic algorithm and the finite element method, the optimization design method of a steel truss structure was investigated, and the optimization design of various parameters, such as truss height and rod size, was carried out. The main conclusions are as follows:

(1) A mono-parameter optimization design method was proposed, with the truss height or member section size being selected as the optimization variable. The accuracy of this method was verified through a comparative analysis with the trial-and-error method. The mono-parameter optimization design of steel truss structures was conducted using the proposed method, demonstrating that significant material savings could be achieved through this approach.

(2) Based on the mono-parameter optimization design method, a multi-parameter optimization design method was proposed considering both truss height and member section size as the optimization variables. The proposed multi-parameter optimization design method was applied to optimize the design of steel truss structures, showing that the proposed improved genetic algorithm provides an efficient and robust technique for obtaining an optimum solution to the multi-parameter design problem of steel trusses. The results also show that the multi-parameter optimization design can further reduce material consumption compared to the mono-parameter optimization design. Therefore, in practical engineering, the proposed optimization method can be used for multi-parameter optimization design of trusses to achieve the lowest engineering cost while ensuring structural safety.

Although multi-parameter optimization can achieve further material reduction compared to mono-parameter optimization, this study only focused on the optimization of planar truss structures. Although the simultaneous optimization of multiple parameters is achieved in this work, the optimization objective is limited to one. In future research, a multi-objective and multi-parameter optimization design method can be developed and applied to optimize other structural forms beyond steel trusses.