Sensitivity of optimal double-layer grid designs to geometrical imperfections and geometric nonlinearity conditions in the analysis phase

Abstract

This study focuses on exploring the effects of geometrical imperfections and different analysis methods on the optimum design of Double-Layer Grids (DLGs), as used in the construction industry. A total of 12 notable meta-heuristics are assessed and contrasted, and as a result, the Slime Mold Algorithm is identified as the most effective approach for size optimization of DLGs. To evaluate the influence of geometric imperfections and nonlinearity on the optimal design of real-size DLGs, the optimization process is carried out by considering and disregarding geometric nonlinearity while incorporating three distinct forms of geometrical imperfections, namely local imperfections, global imperfections, and combinations of both. In light of the uncertain nature of geometrical imperfections, probabilistic distributions are used to define these imperfections randomly in direction and magnitude. The results demonstrate that it is necessary to account for these imperfections to obtain an optimal solution. It’s worth noting that structural imperfections can increase the maximum stress ratio by up to 70%. The analysis also reveals that the initial curvature of members has a more significant impact on the optimal design of structures than the nodal installation error, indicating the need for greater attention to local imperfection issues in space structure construction.

Graphical abstract

Keywords

double-layer grid / sizing optimization / metaheuristic algorithms / geometrical imperfections / analysis approach

1 Introduction

Over the past few decades, there has been a growing interest in the utilization of space structures due to their impressive structural capabilities, including their resilience and lightness, their ability to construct on a large scale and cover sizable spans, and their pleasing aesthetic design. One type of space structure that has gained popularity is the Double-Layer Grid (DLG), which is widely utilized for covering various structures such as exhibition pavilions, assembly halls, swimming pools, hangars, and churches. Additionally, DLGs are appropriate for numerous industrial buildings that require extensive, unobstructed spaces. DLGs consist of two flat layers arranged parallel to each other and spaced apart by a specific distance. The layers are constructed using frameworks of interconnected members, which are then linked together by vertical and inclined web members.

Optimization has been one of the topics of interest in many fields and this has encouraged many researchers to address various optimization problems and provide efficient algorithms and procedures to solve them. Optimal design of structures can be mentioned as one of the most practical optimization problems. In this regard, some researchers have addressed the sizing [1,2], shape [3,4], and topology [5,6] optimizations of structures under specific conditions. The optimal structure is both economical and considers the necessary robust structural performance for the project.

Metaheuristic optimization methods and gradient-based optimization methods are two distinct approaches to solving optimization problems. Gradient-based methods rely on gradient information to explore the solution space near the starting point and converge quickly to local optima [7–9]. However, they may not be able to find the global optimal solution and may face difficulties in dealing with non-convex objective functions or constraints. On the other hand, metaheuristic methods are stochastic and use random search or simulation to explore the solution space, making them more suitable for handling complex problems with non-convex objective functions or constraints.

Metaheuristic Algorithms (MAs) have emerged as a popular choice for optimization problems due to their superior performance compared to deterministic algorithms. These algorithms initiate the optimization process by generating solutions randomly, without requiring any gradient information. This characteristic makes them well-suited for practical nonlinear problems that lack derivative information. By sampling the extensive search space in a specific manner, MAs search for nearby optimal solutions to the problem. Therefore, MAs are primarily utilized for optimization problems involving constraint engineering, such as optimizing structural weight. Researchers have employed MAs such as the Asymmetric Genetic Algorithm (AGA) [10], Artificial Bee Colony algorithm (ABC) [3], Enhanced Colliding Bodies Optimization (ECBO) [11], Coyote Optimization Algorithm (COA) [12], and hybrid differential evolution and symbiotic organism search algorithm (HDS) [13]) to tackle such problems.

Several researchers have studied the impact of geometrical imperfections that arise during the construction process, which can adversely affect structural performance. The focus of this research is to deal with the imperfections that arise at the outset, such as the curvature of individual members and installation errors at the nodes, which are classified as local and global imperfections, respectively. Global imperfections have been analyzed using shape optimization in order to identify the worst imperfection pattern [14]. They can either be defined based on buckling modes [15–18] or considered as constant values [19]. Deterministic imperfections are typically employed in structural analysis, while real-world imperfections are inherently random. Therefore, in order to achieve more realistic results, imperfections should be generated randomly. In previous studies, various functions (e.g., linear function [17], quadratic function [18], and sinuous half-wave function [20,21]) have been used as shape definition functions to apply initial curvature to individual members. Among these functions, the sine half-wave function would be a more intelligent choice regarding its relatively high adaptation with the first buckling mode of the Euler column.

The objective of this research is to investigate the influence of both geometric imperfections and diverse analysis methods on the minimization of structural weight in DLGs. The process of optimal design takes into account engineering constraints, encompassing stress ratios, nodal displacements, and buckling criteria. In these optimal designs, the cross-sectional area of the structure is often reduced, so that the mentioned constraints approach their upper limits. Consequently, potential deviations in the execution of these optimized designs can significantly impact the structural integrity. To address this, optimization schemes employing various analysis approaches and accounting for geometric imperfections are applied to three real-size DLGs [22], each consisting of 520, 672, and 800 bar members.

2 Methodology

This study aims to explore the influence of geometric imperfections and the choice of analysis method in the optimal design of DLGs. The research steps are outlined in Fig.1.

In this study the optimization algorithms are coded in MATLAB R2021a [23] and the structures are analyzed using Sap2000 v23.3.1 [24].

2.1 Optimization approaches

Metaheuristic optimization algorithms represent a popular approach to solving intricate optimization problems. These algorithms are heuristic in nature, implying that they do not guarantee discovering the optimal solution; rather, they prioritize finding a satisfactory solution within a reasonable timeframe.

A significant advantage of metaheuristics lies in their versatility in tackling a broad spectrum of optimization problems. They can handle nonlinear, non-convex objective functions, and effectively manage constraints. This versatility positions them as robust tools for various applications, including numerous real-world engineering optimization challenges.

2.1.1 Performance evaluation of optimization algorithms

To determine the most efficient algorithm in structural weight optimization problems, 12 new and competitive MAs are subjected to a comparison in the sizing optimization problem of the 520-bar DLG. The MAs used for comparison are: Cheetah Optimizer (CO) [25], ECBO [26], Artificial Hummingbird Algorithm (AHA) [27], Slime Mold Algorithm (SMA) [28], African Vultures Optimization Algorithm (AVOA) [29], Chef-based Optimization Algorithm (CBOA) [30], COOT algorithm [31], Equilibrium Optimizer (EO) [32], Gorilla Troops Optimizer (GTO) [33], Marine Predators Algorithm (MPA) [34], Arithmetic Optimization Algorithm (AOA) [35], Wild Horse Optimizer (WHO) [36]. Fig.2(a) shows a three-dimensional (3D) view of the mentioned DLG, which is a square on a larger square and contains 520 members and 165 nodes. The bottom layer is simply supported at the nodes specified in Fig.2(b). Each top layer joint shown in Fig.2(d) is subjected to a concentrated vertical load of 46 kN. The span and height are equal to 40 and 3 m, respectively. All connections are assumed to be ball jointed.

2.1.2 Slime mold algorithm framework

As SMA achieves the best optimal design and has the best performance in terms of the average optimized weight and standard deviation (STD) on average weights, it can be considered to be the most efficient algorithm. SMA proposed in Ref. [28], is inspired by the slime mold behavior during the search for nutrition in nature. It is composed of a unique mathematical model that uses adaptive weights to simulate the positive and negative feedback behavior by the slime mold in the face of nutrients during the search process to access optimal food source. The SMA creates an optimal search scheme with an appropriate balance between the exploration and exploitation phases and the ability to avoid local optimum trapping during rapid convergence.

As slime mold can approach food sources according to the odor concentration in the air, the approaching behavior of slime mold is simulated by mathematical formulas based on this mechanism. The following formulations are specified to obtain updated slime mold position while searching for the food source. The contraction mode of mold is imitated as follows:

(1)

(2)a=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(-\left(\frac{t}{\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{\_}t}\right)+1\right),

where \overrightarrow{X} and \overrightarrow{X_{\mathrm{b}}} represent the slime mold location and the individual location with the highest odor concentration currently found, respectively; \overrightarrow{X_{\mathrm{A}}} and \overrightarrow{X_{\mathrm{B}}} are defined as two random individuals chosen from slime mold; W represents the total weight of slime mold; \overrightarrow{vb} is a parameter with a range of \left[-a,a\right], where a is given by Eq. (2); \overrightarrow{vc} is a parameter whose value tends to zero linearly; t represents the current iteration; r is introduced as a random value in the range of [0,1], and the parameters p and W are defined by the following equations:

(3)p=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|S\left(i\right)-DF\right|,

(4)\begin{array}{rl}& \overrightarrow{W\left(smellIndex\left(i\right)\right)}\\ =& \{\begin{array}{ll}1+r\cdot \log \left({\displaystyle \frac{bF-S\left(i\right)}{bF-wF}}+1\right),& \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},\\ 1-r\cdot \log \left({\displaystyle \frac{bF-S\left(i\right)}{bF-wF}}+1\right),& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s},\end{array}\end{array}

(5)smellIndex=sort\left(S\right),

where i ∈ 1,2,…,nS(i) signifies the fitness of X, and DF is introduced as the best fitness derived from all iterations; condition indicates that the parameter S(i) belongs to the first half of the population; r is introduced as a random value in the interval of [0,1]; bF and wF represent the optimal fitness value and the worst fitness value derived from the current iterative process, respectively; and smellIndex denotes the array of sorted fitness values.

To update the position of slime mold the following equation is given:

where LB and UB denote the boundaries of the search allowed range; z can be set as different values according to specific problems; rand and r signify the random value in the range of [0,1], expressing the search process uncertainty and leading to searching solution space randomly in any direction. Thus, the algorithm can find the optimum solution from the entire allowed search space. The parameters \overrightarrow{vb} and \overrightarrow{vc} oscillate during the search process in the interval of \left[-a,a\right] and \left[-1,1\right], respectively, and ultimately tend to zero with the increase of iterations. For more details refer to Ref. [28]. The pseudo code of the SMA is shown in . The flowchart of SMA is depicted in Fig.3.

2.2 Geometrical imperfections

Geometrical imperfections arising during the construction process and their adverse effect on the structure performance are inevitable in real-world structures. It is noteworthy that the nature of structural imperfections is typically random in the real world. Inclusion of random initial imperfections in structural optimization problems is necessary to obtain realistic and reliable results. The present study addresses the initial imperfections in the form of curvature of individual members and nodal installation error as the local and global imperfections, respectively. To this end, 10% of the nodal positions and members are randomly considered for application of global and local imperfections. The magnitude and directions of the imperfections are randomly defined by the uniform distribution at the beginning of each independent run of the optimization algorithm. It is noteworthy that these values are kept constant for all iterations in each run.

2.2.1 Initial curvature of the member

The curved members can be modeled by a sequence of straight elements. In this study, each member is discretized into 20 finite elements. As shown in Fig.4, a local coordinate system is arranged to set the local imperfection. The x-axis is along the member length, and the y-axis is perpendicular to the x-axis. The curved member’s shape is assumed to be a half-sine curve y=a\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\text{π}x}{L}}\right) [20,37–39], where a is the amplitude of curvature and L is the length of the member. The imperfection magnitude of each member is randomly defined using a uniform distribution in an interval with a maximum value of L/100. The curvature direction (θ) is the local system’s rotation around the x-axis as shown in Fig.4. The parameter θ for each member is a random constant from the uniform distribution from 0 to 2π rad.

2.2.2 Nodal installation error

Nodal installation error is defined randomly by uniform distribution for each joint. For this purpose, ∆x, ∆y, and ∆z are introduced as random representative parameters of the nodal position imperfections in the x, y, and z directions of the global coordinates system, respectively. The magnitude of nodal position imperfection of each joint is measured by E_{max}=\sqrt{{\mathrm{\Delta }x}^2+{\mathrm{\Delta }y}^2+{\mathrm{\Delta }\mathit{\text{z}}}^2} [39] with a maximum allowed value of l/1000, where l is the structural span. Also, ∆x, ∆y, and ∆z are generated from the uniform distribution as follows:

(7)\mathrm{\Delta }x\in U(-E_{max},E_{max}),

(8)\mathrm{\Delta }y\in U\left(-\sqrt{{E_{max}}^2-{\mathrm{\Delta }x}^2},\sqrt{{E_{max}}^2-{\mathrm{\Delta }x}^2}\right),

(9)\mathrm{\Delta }\mathit{\text{z}}\in U\left(-\sqrt{{E_{max}}^2-{\mathrm{\Delta }x}^2-{\mathrm{\Delta }y}^2},\sqrt{{E_{max}}^2-{\mathrm{\Delta }x}^2-{\mathrm{\Delta }y}^2}\right).

The random parameters ​​were added to the ideal nodal coordinates to apply nodal installation error.

3 Numerical results

To demonstrate the effect of geometrical imperfection and various analytical approaches on the optimal design of DLGs, three square-on-square DLGs, formed of 520, 672, and 800 elements, are studied and optimized. For this purpose, the optimization problem’s analysis process is performed by individually adopting the structure’s linear and geometrically nonlinear behavior. Moreover, the geometrical imperfection sensitivity of the structures is assessed by employing different imperfections consideration schemes.

The design variables, namely the cross-sectional areas of the bar elements, are chosen from the AISC Pipe Shape v15.0 list of steel pipe sections (see Appendix). Stress and slenderness ratio constraints adhere to AISC 360-16 LRFD standards [38], while a displacement constraint is imposed on all structural nodes, with a maximum value of span/360. For all examples, a span of 40 m × 40 m is taken and the height is considered equal to 3 m. All connections are considered to be ball jointed. The material properties such as the modulus of elasticity, the yield stress, and the density of steel are considered as 205 GPa, 248.2 MPa, and 7833.413 kg/m³, respectively.

3.1 Results of optimization algorithms

All 12 algorithms introduced in the previous section, are applied under the same conditions. Cross-sectional areas of the members as design variables are categorized into 20 groups that are selected from the list of the steel pipe sections from AISC Pipe Shape v15.0 [40]. All the compared algorithms run independently 30 times with a population of 20 particles and 500 iterations as the terminal condition. For the sake of investigating the algorithms’ performance in the competitive experiment, Average results (AVG), Best result (BEST), and STD are employed to assess the obtained results.

The results obtained by the algorithms are summarized in Tab.1. The data show that SMA achieves the best optimal design (i.e., 547325 N) and has the best performance in terms of the average optimized weight and STD on average weights, which are 553513 and 9134 N, respectively. A low STD with a superior average shows a reliable and stable performance in solving this optimization problem. Therefore, the SMA method can be used with high confidence for DLG optimum design problems.

3.2 Comparison of linear and nonlinear analysis

The investigated 520-bar DLG is formed of 520 members and 165 nodes (Fig.2). A concentrated vertical load of 46 kN is applied on each top layer joint. Cross-sectional areas of the members, as design variables, are categorized into 20 groups. The optimization results, achieved using SMA, of 520-bar DLG are summarized in Tab.2. The optimization schemes, using linear and geometrically nonlinear analysis considering the ideal shape of the structure, find optimized weights of 547326 and 547903 N, respectively, over 5 independent runs. Notably, the optimal result obtained from the nonlinear analysis is almost 0.1% heavier. Fig.5 shows the convergence histories. The required number of functional evaluations (FEs) to achieve the best design, using linear and nonlinear analysis are 9500 and 9820 times, respectively.

The 3D view of the 672-bar DLG is illustrated in Fig.6(a). This structure contains 205 nodes and simple support conditions are employed for the bottom layer at the nodes, as shown in Fig.6(b). Concentrated vertical loads of 30 kN are applied on all top layer joints specified in Fig.6(d). The members are grouped into 22 categories to have the same cross-sectional area.

The data found by 672-bar DLG optimization over 5 independent runs are summarized in Tab.3. The best design obtained by using linear analysis has a weight equal to 488356 N, while the best result obtained by adopting geometric nonlinearity in the analysis is almost 1.2% heavier (i.e., 494482 N). Fig.7 depicts convergence histories. It is of note that the optimization schemes based on linear and nonlinear analyses require 9460 and 9780 FEs to achieve the best design, respectively.

The 3D view of the DLG formed of 800 members and 221 nodes, is illustrated in Fig.8(a). The bottom layer is simply supported at the nodes that are specified in Fig.8(b). Concentrated vertical loads of 30 kN are applied to all top layer joints shown in Fig.8(d). The cross-sectional areas of the bar members are categorized into 24 groups as design variables.

The optimized weights obtained after 5 independent runs of the algorithm are reported in Tab.4. It is noteworthy that the best design obtained using nonlinear analysis (i.e., 532059 N) is almost 4.15 heavier than the best result of the linear analysis (i.e., 510843 N). Convergence histories are shown in Fig.9. The required number of FEs to achieve the best design, by the optimization schemes based on linear and nonlinear analyses are 9580 analyses in both cases.

Comparing the optimized weights of the case studies obtained using linear and geometrically nonlinear analyses (Fig.10) it emerges that there is a positive correlation between the size of the optimized structure and its sensitivity to the analysis approach. Thus the more extensive the structure size is, when adopting geometric nonlinearity in the analysis phase of optimization, the greater the number of resistant elements the structure requires to satisfy the constraints and reach the desirable performance; therefore, more structural weight increase takes place.

3.3 Performance of linear-designed optimal double-layer grids under geometrically nonlinear analysis

The performance of linear-designed optimal structures is subjected to nonlinear analysis, and the obtained data are summarized in Tab.5. The maximum stress ratios and maximum nodal displacement values of linearly designed 520, 672, and 800 DLGs, generally increase with increasing size, under nonlinear analysis. However, the increase in values is negligible when linear analysis is adopted. The obtained optimal design may not satisfy the constraints of allowable stress ratio and displacement. This issue will lead to failure.

The performance of linear-designed optimal structures is subjected to nonlinear analysis, and the obtained data are summarized in Tab.5. The stress ratios and nodal displacement values of linearly designed 520, 672, and 800 DLGs typically show an increase when subjected to nonlinear analysis. Even though the rise in values is minor, choosing linear analysis may lead to a design that does not meet the constraints for allowable stress ratio and displacement, ultimately resulting in failure based on nonlinear analysis results.

3.4 Imperfection sensitivity of optimal design of double-layer grids

In this sector, the effect of random imperfections on linear and nonlinear-designed optimum DLGs is addressed. The imperfections are applied to structure in local, global, and the combination of local and global, forms. The local and global imperfections are assumed to be initial curvature of individual members and nodal installation error, respectively. Each of the forms of imperfection are run five times individually, which means that for each structure, the imperfections (local, global, or both) are applied on structures five times with random values and random locations. Tab.6 shows the maximum stress ratio and the maximum nodal displacement of optimal designed DLGs without imperfections.

The influence of imperfections on optimal designed 520-bar DLG performance is summarized in Tab.7 and Tab.8. The reported data indicate an average increase of 47% in the maximum stress ratio of the linear-designed structure in the case of combination of local and global imperfections. In comparison, the maximum nodal displacement growth rate is almost 6%. That means the impact of imperfections on the stress ratio is eight times that on the nodal displacement in this case. In addition, according to reported data, the adverse effect of the combination of local and global imperfection on the stress ratio is 11 times more than that of nodal displacement in the nonlinear-designed structure.

Moreover, it is observed that the maximum stress ratio of the linear-designed structure increases by an average of 38% in the presence of local imperfections. The increase rate related to global imperfections is almost 6.5%. Also, the maximum stress ratio of the nonlinear-designed structure increases by an average of 52% and 4.5% in the effect of local and global imperfections, respectively. It demonstrates that the local imperfections have several times more adverse effects than the global imperfections on both the nonlinear and linear-designed structures.

Data for the 672-bar DLG are reported in Tab.9 and Tab.10. The results indicate that the maximum stress ratio and maximum nodal displacement increase by 67% and 4% on average, respectively, due to imposing the combination of global and local imperfections on the linear-designed structure. These values for the nonlinear-designed are 64% and 3.5% on average. For the 672-bar DLG, like the 500-bar DLG, the adverse impact of local imperfections is substantially more than global imperfections on structure efficiency. Also, as a result of applying imperfections, the maximum stress ratio is much more affected than the maximum nodal displacement.

Tab.11 and Tab.12 show the effect of imperfections on the optimal designed 800-bar DLG. The data demonstrate that the maximum stress ratio of the linear-designed structure increases on average from 0.9967 to 2.7552 (i.e., 176%) in the effect of imposing the combination of local and global imperfections, while the maximum nodal displacement increases from 7.86 to 7.93 cm (i.e., 0.9%). In addition, the maximum stress ratio and nodal displacement of the nonlinear-designed structure, respectively, have average growth rates of 63% and 6% when applying the combination of local and global imperfections. It can be concluded from the results that as the number of DLG elements increases (with the same probability of imperfection), the maximum stress ratio is more affected than is the nodal displacement.

In general, the results of this section can be summarized in the following points.

1) Local imperfections have more adverse effects than global imperfections on the nonlinear and linear-designed optimal DLGs.

2) The adverse effect of imperfections on the maximum nodal displacement is noticeably less than that on the maximum stress ratio.

3) Generally, imperfections have a more destructive effect in linear designs. Therefore, using nonlinear analysis in the optimal design process of DLGs is preferable.

4) In DLGs, a greater number of elements usually increase the number of imperfections, and the structure then shows more sensitivity to these imperfections. This point should be considered when designing the topology of DLGs.

4 Conclusions

Interest in structural optimization has increased in recent years, primarily focused on minimizing structural weight to reduce construction costs and achieve competitive solutions. The optimization process is significantly influenced by the geometry of the considered shape and the analysis approach, which are crucial factors affecting the performance of the derived solutions. Here, the effect of geometrical imperfections and various analysis approaches on the optimal design of DLGs is investigated. To obtain optimal designs, an efficient method is necessary; therefore, the present study evaluates and compares the efficiency of 12 prominent MAs for structural weight optimization. The results demonstrate that the SMA algorithm consistently delivers superior and more reliable outcomes in the context of structural optimization when compared to its counterparts. As a result, we recommend its application for future research endeavors in the field of optimal structural design.

Optimal designs obtained through nonlinear analysis tend to be heavier compared to the optimum designs obtained through linear analysis. Moreover, the sensitivity of optimal designs to the analysis approach has a positive correlation with the size of the structure. Consequently, in light of the findings from this study, it is imperative to employ nonlinear analysis in the optimal design process of DLGs.

The obtained results indicate that imperfections have a negative impact on the optimal design of structures, resulting in the potential violation of design constraints. Thus, it is essential to consider possible geometrical imperfections in the ideal shape of the structure to provide reliable optimal designs. Furthermore, the sensitivity of DLGs’ optimal design to imperfections is more significant in the case of stress ratio of members than in the case of nodal displacements. Imperfections could cause the violation of the allowable maximum stress ratio. In contrast, nodal displacements and deflections of imperfect elements usually have satisfactory values. It is worth noting that local imperfections, such as the initial curvature of individual members, have a more significant adverse effect on structural performance than global imperfections, such as nodal installation errors. Therefore, greater attention should be given to addressing local imperfection issues in the construction of space structures.

References

[1] Carbas S, Artar M. Comparative seismic design optimization of spatial steel dome structures through three recent metaheuristic algorithms. Frontiers of Structural and Civil Engineering, 2022, 16(1): 57–74

[2] Fakhimi R, Shahabsafa M, Lei W, He S, Martins J R R A, Terlaky T, Zuluaga L F. Discrete multi-load truss sizing optimization: Model analysis and computational experiments. Optimization and Engineering, 2022, 23(3): 1559–1585

[3] Jawad F K, Ozturk C, Dansheng W, Mahmood M, Al-Azzawi O, Al-Jemely A. Sizing and layout optimization of truss structures with artificial bee colony algorithm. Structures, 2021, 30: 546–559

[4] Liu W, Xu L, Zhu S, Li L, Liu F, Xiong Z. Shape optimization of aluminium alloy spherical reticulated shells considering nonlinearities. Frontiers of Structural and Civil Engineering, 2022, 16(12): 1565–1580

[5] Paulino D M, Leonel E D. Topology optimization and geometric nonlinear modeling using positional finite elements. Optimization and Engineering, 2021, 23: 1439–1469

[6] Stoiber N, Kromoser B. Topology optimization in concrete construction: A systematic review on numerical and experimental investigations. Structural and Multidisciplinary Optimization, 2021, 64(4): 1725–1749

[7] Vu-Bac N, Duong T X, Lahmer T, Zhuang X, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455

[8] Vu-Bac N, Duong T X, Lahmer T, Areias P, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 480–510

[9] Vu-Bac N, Rabczuk T, Park H S, Fu X, Zhuang X. A NURBS-based inverse analysis of swelling induced morphing of thin stimuli-responsive polymer gels. Computer Methods in Applied Mechanics and Engineering, 2022, 397: 115049

[10] Es-Haghi M S, Shishegaran A, Rabczuk T. Evaluation of a novel asymmetric genetic algorithm to optimize the structural design of 3D regular and irregular steel frames. Frontiers of Structural and Civil Engineering, 2020, 14(5): 1110–1130

[11] Al-Bazoon M, Arora J S. Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads and metaheuristic algorithms. Optimization and Engineering, 2022, 23(2): 643–687

[12] Pierezan J, dos Santos Coelho L, Cocco Mariani V, Hochsteiner de Vasconcelos Segundo E, Prayogo D. Chaotic coyote algorithm applied to truss optimization problems. Computers & Structures, 2021, 242: 106353

[13] Nguyen-Van S, Nguyen K T, Luong V H, Lee S, Lieu Q X. A novel hybrid differential evolution and symbiotic organism search algorithm for size and shape optimization of truss structures under multiple frequency constraints. Expert Systems with Applications, 2021, 184: 115534

[14] Tomei V, Grande E, Imbimbo M. Influence of geometric imperfections on the efficacy of optimization approaches for grid-shells. Engineering Structures, 2021, 228: 111502

[15] Bruno L, Sassone M, Venuti F. Effects of the equivalent geometric nodal imperfections on the stability of single layer grid shells. Engineering Structures, 2016, 112: 184–199

[16] Cai J, Gu L, Xu Y, Feng J, Zhang J. Nonlinear stability analysis of hybrid grid shells. International Journal of Structural Stability and Dynamics, 2013, 13(1): 1350006

[17] Guo J. Research on distribution and magnitude of initial geometrical imperfection affecting stability of suspen-dome. Advanced Steel Construction, 2011, 7(4): 344–358

[18] Liu H, Zhang W, Yuan H. Structural stability analysis of single-layer reticulated shells with stochastic imperfections. Engineering Structures, 2016, 124: 473–479

[19] Madah H, Amir O. Truss optimization with buckling considerations using geometrically nonlinear beam modeling. Computers & Structures, 2017, 192: 233–247

[20] Li H, Taniguchi Y. Load-carrying capacity of semi-rigid double-layer grid structures with initial crookedness of member. Engineering Structures, 2019, 184: 421–433

[21] Madah H, Amir O. Concurrent structural optimization of buckling-resistant trusses and their initial imperfections. International Journal of Solids and Structures, 2019, 162: 244–258

[22] KavehAIlchi GhazaanM. Meta-Heuristic Algorithms for Optimal Design of Real-Size Structures. Cham: Springer Cham, 2018

[23] MATLAB. Version 9.10.0. R2021a. Natick, MA: The MathWorks Inc., 2021

[24] Sap2000. Version 23.3.1. Berkeley, CA: Computers and Structures, Inc., 2023

[25] Akbari M A, Zare M, Azizipanah-Abarghooee R, Mirjalili S, Deriche M. The cheetah optimizer: A nature-inspired metaheuristic algorithm for large-scale optimization problems. Scientific Reports, 2022, 12(1): 10953

[26] Kaveh A, Ilchi Ghazaan M. Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Advances in Engineering Software, 2014, 77: 66–75

[27] Zhao W, Wang L, Mirjalili S. Artificial hummingbird algorithm: A new bio-inspired optimizer with its engineering applications. Computer Methods in Applied Mechanics and Engineering, 2022, 388: 114194

[28] Li S, Chen H, Wang M, Heidari A A, Mirjalili S. Slime mould algorithm: A new method for stochastic optimization. Future Generation Computer Systems, 2020, 111: 300–323

[29] Abdollahzadeh B, Gharehchopogh F S, Mirjalili S. African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems. Computers & Industrial Engineering, 2021, 158: 107408

[30] Trojovská E, Dehghani M. A new human-based metahurestic optimization method based on mimicking cooking training. Scientific Reports, 2022, 12(1): 14861

[31] Naruei I, Keynia F. A new optimization method based on COOT bird natural life model. Expert Systems with Applications, 2021, 183: 115352

[32] Faramarzi A, Heidarinejad M, Stephens B, Mirjalili S. Equilibrium optimizer: A novel optimization algorithm. Knowledge-Based Systems, 2020, 191: 105190

[33] Abdollahzadeh B, Soleimanian Gharehchopogh F, Mirjalili S. Artificial gorilla troops optimizer: A new nature-inspired metaheuristic algorithm for global optimization problems. International Journal of Intelligent Systems, 2021, 36(10): 5887–5958

[34] Faramarzi A, Heidarinejad M, Mirjalili S, Gandomi A H. Marine predators algorithm: A nature-inspired metaheuristic. Expert Systems with Applications, 2020, 152: 113377

[35] Abualigah L, Diabat A, Mirjalili S, Abd Elaziz M, Gandomi A H. The arithmetic optimization algorithm. Computer Methods in Applied Mechanics and Engineering, 2021, 376: 113609

[36] NarueiIKeyniaF. Wild horse optimizer: A new meta-heuristic algorithm for solving engineering optimization problems. Engineering with Computers, 2022, 38(S 4): 3025–3056

[37] Kalina M. Stability problems of pyramidal von Mises planar trusses with geometrical imperfection. International Journal of Theoretical and Applied Mechanics, 2016, 1: 118–123

[38] Santana M, Gonçalves P, Silveira R. Stability and load capacity of an elasto-plastic pyramidal truss. International Journal of Solids and Structures, 2019, 171: 158–173

[39] Zhao Z W, Liu H Q, Liang B, Yan R Z. Influence of random geometrical imperfection on the stability of single-layer reticulated domes with semi-rigid connection. Advanced Steel Construction, 2019, 15(1): 93–99

[40] AISCShapes Database. Version 15.0. Chicago, IL: AISC. 2017

RIGHTS & PERMISSIONS

Higher Education Press