Shape optimization of aluminium alloy spherical reticulated shells considering nonlinearities

Abstract

This study proposes a shape optimization method for K6 aluminum alloy spherical reticulated shells with gusset joints, considering geometric, material, and joint stiffness nonlinearities. The optimization procedure adopts a genetic algorithm in which the elastoplastic non-linear buckling load is selected as the objective function to be maximized. By confinement of the adjustment range of the controlling points, optimization results have enabled a path toward achieving a larger elastoplastic non-linear buckling load without changing the macroscopic shape of the structure. A numerical example is provided to demonstrate the effectiveness of the proposed method. In addition, the variation in structural performance during optimization is illustrated. Through parametric analysis, practical design tables containing the parameters of the optimized shape are obtained for aluminum alloy spherical shells with common geometric parameters. To explore the effect of material nonlinearity, the optimal shapes obtained based on considering and not considering material non-linear objective functions, the elastoplastic and elastic non-linear buckling loads, are compared.

Graphical abstract

Keywords

shape optimization / aluminum alloy / spherical reticulated shell / non-linear buckling / material nonlinearity / genetic algorithm

1 Introduction

With the rapid development of cultural and sports undertakings, spherical reticulated shell structures are commonly utilized in various large-space sites, such as sports venues, owing to their simple and beautiful shape, flexible span, and economic rationality. Compared with steel, aluminum alloys are lightweight, corrosion-resistant, and convenient for processing. In recent years, a number of large-span aluminum alloy reticulated shells have been applied to many landmark buildings, and the usage of the reticulated shell demonstrates a rapid development trend [1]. However, current research on aluminum alloy reticulated shells is not as thorough and systematic as that of steel reticulated shells, as there are significant differences in material properties and joint types that need to be investigated.

Nowadays, it is possible to study the stability of complex large-span reticulated shell structures using finite element (FE) software and modern computer technology. In the early years when the computational efficiency is low, many researchers considered only the linear behavior of material for simplification of computation [2]. Although considering the material elasticity is reasonable because most reticulated shells are thin-walled, existing research indicates that some members have reached the elastoplastic stage before the overall instability of the structure, which influenced the overall buckling capacity [3]. Therefore, a non-linear buckling analysis that considers both geometric and material nonlinearity can best reflect the actual force distribution and evaluate the buckling capacity of the structure.

Researchers have proposed various approaches for increasing the reliability of reticulated shell structures. Broadly, there are three optimization approaches: shape optimization, sizing optimization, and topology optimization. These approaches are meant to achieve the desired objects, such as material volume, mass, strain energy, displacements, stiffness, stress, and buckling loads. In most studies on reticulated shell optimization, the shape optimization of structures is considered to be more significant than the sizing optimization. A slight change in the structural shape can significantly affect the load-bearing capacity.

Shape optimization mainly consists of parameter-dependent and parameter-free optimizations. Conversely, the parameter-dependent method parameterizes a reticulated shell structure using profile parameters such as height and thickness, parametric surface, or design elements in advance [4]. For example, a form-finding approach was introduced in Ref. [5], and the slope of the target function was obtained using a Stiffness Sensitivity Tensor to perform a mathematical analysis utilizing the Generalised Reduced Gradient method. By combining parameters of interest as the design parameters, users can obtain the structure of maximum buckling load. Rombouts et al. [6] increased the overall stiffness of the gridshell with respect to multiple load cases to indirectly improve the carrying capacity. The optimized shape changes with the loading condition. Results show that the suggested method outperforms the classical form-finding procedure. Zhao et al. [7] proposed an improved form-finding method considering spherical and cylindrical shell structures, where the FE model was used to evaluate the member bending moment, which was set as the objective function to be optimized. The above methods have merits but also demerits, such as limited spatial discretization, joint stiffness not considered, and inefficiency in finding the global optima in constraint space.

On the other hand, there are parameter-free approaches represented by node-related methods. Ding et al. [8] presented a novel node-shifting approach for shape optimization of unstrained reticulated spatial structures. Tsavdaridis et al. [9] chose the nodal z-coordinates as variables, and the effect of the joint semi-rigidity was investigated. By varying the joint stiffness, the effects of the semi-rigid joints on the shape optimization were evaluated. The node-related method makes the structural adjustment flexible but may cause the jagged surface shape of the optimized structure. Accordingly, using a parametric continuous curve to represent the structural profile, such as a spline, the smoothness of the optimized surface can be significantly improved with little mechanical sacrifice. Besides, it can also reduce the number of variables that need to be optimized.

As the construction capability improves, large-scale reticulated structures become more prevalent. The corresponding in-depth research is also accompanied by the increase of the problem dimension, referred to as the curse of dimensionality. Many scholars have made efforts to resolve this problem. For example, the gradient-based method adopted by Vu-Bac et al. [10,11] can handle a large number of design parameters since it has a determined update direction towards the optimal solution. An alternative is a gradient-free method such as the particle swarm optimization adopted by Shaaban et al. [12–15]. Generally, the meta-heuristic algorithms developed based on natural laws are gradient-free optimization methods. Notably, the gradient-based method and meta-heuristics both have pros and cons. Specifically, the gradient-based method requires significant computational cost regarding highly non-linear objective functions, while meta-heuristics cannot handle problems with a large number of design variables and can easily get trapped into a local optimum.

To ensure the robustness of the buckling capacity of the shell with the optimized shape, the potential impact of initial imperfections such as joint deviations as well as the nonlinearities should be taken into consideration. Among them, geometrical nonlinearity should be the most important issue for designing reticulated structures. Reitinger and Ramm [16] considered geometrically non-linear analyses, including stability behavior and imperfection sensitivity, to optimize the structure. Results demonstrated by the numerical example that the stability behavior was greatly improved. Lagaros and Papadopoulos [17] proposed an optimization method that considers random geometric, material and thickness imperfections. The challenge of this approach is that stability is very sensitive to those imperfections. Firl and Bletzinger [18] considered a similar problem of geometrically non-linear mechanics. They developed a structural optimization technique that integrates FE-based parametrization with non-linear kinematics to optimize the geometry of thin shell structures. The proposed method can be used for a wide range of optimization strategies. Tomei et al. [19] designed a schedule to investigate the effect of geometric imperfections on the efficacy of different optimization approaches for grid shells. However, the impact of imperfection on buckling capacity was not included in their studies. Furthermore, material nonlinearity was not covered in their nonlinearity considerations.

It appears from the investigations mentioned above that most attention has been paid to the aspects of shape optimization of shell structures in the early design stage; thus, detailed design conditions are undetermined, and there are no appearance constraints, which leads to various unattractive optimization appearances. This leads to difficulties in using many novel innovations for shape-determined applications. Of course, if architects and structural engineers can jointly find the optimal spatial surface shape at the conceptual scheme stage, the efficiency would be maximized from both the aesthetic and structural standpoints. But in the traditional design process, architects would select the structure shape in advance. On this basis, if the reliability of the reticulated shell structure can be improved as much as possible without affecting the macroscopic shape, it can be a favorable solution for both sides. This provides the breakthrough point to realize that a shape-constraint reticulated shell structure can be used with higher safety. Moreover, there have been few investigations on the mechanical properties of aluminum alloy reticulated structures before and after shape optimization, that is, the mechanism of optimization, which limits the application of shape optimizing if there have particular shape constraints. Therefore, a method that addresses all the problems as mentioned above is highly desirable.

In this paper, based on the above considerations, we take a step toward filling these gaps. We propose a shape optimization method using the genetic algorithm (GA) to enhance the buckling capacity of K6 aluminum alloy reticulated shells, considering geometric, material, and joint stiffness nonlinearities. In Section 2, the shape optimization method, including the FE model considering nonlinearities, objective function, design variables, and the optimization algorithm, are described. In Section 3, a numerical example is provided to illustrate the efficiency of the proposed method. The effects of the design parameters are analyzed, based on which practical design tables of the optimized design variables are obtained. In Section 4, the optimization mechanism is investigated by comparing the mechanical behavior of the shells before and after optimization.

2 Shape optimization method

2.1 Finite element model

The FE software ANSYS was used to establish the numerical model and conduct structural analysis. This paper mainly concerns the buckling behavior of aluminum alloy single-layer reticulated shells with gusset joints. As shown in Fig.1, the aluminum alloy gusset joint consists of several H-shaped beam members and two circular gusset plates. Note that the H-shaped beam members and gusset plates are connected using bolted connections and the height of the members needs to be identical within the whole structure.

The shell model was created in the ANSYS software using the three-dimensional element BEAM188 from the ANSYS unit library. In the FE model, the Timoshenko beam element BEAM188 was adopted to simulate the members and joint zones in K6 aluminum alloy spherical shells, as illustrated in Fig.2. The BEAM188 element is able to consider the shear effect, and both nodes at the end of the element have seven degrees of freedom (DOFs), with the warping DOF activated. The calculation model of a single member with two joint zones is plotted in Fig.3, where the member is modelled by four elements for higher accuracy, and the joint zones are modelled by one element. The joint zone and the gusset plate were the same lengths, and the elastic modulus of the joint zone was set as 100 times that of the member to account for the strengthening effect of the gusset plate [20,21]. The non-linear spring element COMBIN39 was used to consider the non-linear out-of-plane bending stiffness of the aluminum alloy gusset joint [22]. The reliable tested four-fold model proposed by Guo et al. [23] was used to determine the real constants of the COMBIN39 element, as depicted in Fig.3 [22]. As shown in Fig.4, the four-fold model consists of four phases, with K_f, K_s, K_c, and 0 as the bending stiffness of each phase; M_f, M_s, and M_c as the bending moments corresponding to the three turning points, and φ_f, φ_s, and φ_c as the corresponding rotation. The DOFs for in-plane and torsional rotation, warping, and translation along the three directions were coupled between adjacent elements of the joint zone and the member.

To demonstrate the reliability of the FE model, a single-layer reticulated shell was tested in Ref. [22], and the test process was simulated at the same time using the model established above. A 5-ring K6 shell with an 8-meter span and a 0.5-m height was used as the test model. In the test process, the central joint of the reticulated shell was subjected to a unidirectional cyclic load. The test and FE load-displacement curves of the loading joint are illustrated in Fig.5 to verify the FE model. The FE load-displacement curve was found to be quite similar to the experimental curve. As a result, the FE model developed here is accurate for simulating the behavior of aluminum alloy single-layer reticulated shells with gusset joints.

2.2 Optimization problem

The non-linear buckling capacity typically controls the design of single-layer reticulated shells, where the material nonlinearity has a significant impact [24]. Typically, only the buckling capacity at the first limit/bifurcation point is considered in practical engineering, regardless of the post-buckling capacity [21,25]. Hence, this study selects the non-linear buckling capacity P_c as the objective function to be maximized. Note that P_c is the structure’s bearing capacity at the first limit/bifurcation point, where the geometric, material, and joint stiffness nonlinearities are considered.

To ensure the smoothness of the structural shape, a cubic spline was adopted to parameterize the shape of the reticulated shell, as depicted in Fig.6, and the surface of the reticulated shell could be obtained by rotating the spline about the Z-axis. In Fig.6, L represents the span of the shell. Because this structure is symmetric and pinned supports are located at the periphery, there are only three independent control points whose Z-coordinates are denoted as Z₁, Z₂, and Z₃. Hence, vector Z = (Z₁, Z₂, Z₃) can be regarded as the set of design variables.

Based on the design variables and objective function, the optimization problem can be described as

(1)\{\begin{array}{l}max.P_{\mathrm{c}}(\mathit{Z}),\\ \mathrm{s}.\mathrm{t}.\mathrm{\Delta }{Z}_{min}\le \mathrm{\Delta }{Z}_i\le \mathrm{\Delta }{Z}_{max},(i=1,2,3),\end{array}

where ΔZ = (ΔZ₁, ΔZ₂, ΔZ₃) is the set of variations in the design variables, which denotes the variance of the Z-coordinates of the control points. ΔZ_min and ΔZ_max are the minimum and maximum variation limits to avoid considerable shape differences.

Because the objective function is highly non-linear, mathematical programming and gradient-based methods may not be effective. Therefore, the GA was employed to solve the optimization problem in Eq. (1).

2.3 Optimization method

2.3.1 Optimization algorithm

Since implicit nonlinearities are taken into account in this work, i.e., the constraints and objective functions are not only non-linear but also implicit functions in the shape optimization process, we adopt GA as the optimization algorithm, and the GA toolbox built-in MATLAB software is used. The main elements of GA are described as follows [26].

1) Populations and chromosomes: The initial population is generated randomly, containing a number of individuals. Each individual is labeled by a chromosome, which is a string formed by encoding a parameter set. Then, each chromosome is allocated a fitness value based on the evaluation of the objective function;

2) Selection: Using the probabilistic computation weighted by the relative fitness values, pairs of chromosomes are selected as parents. The stochastic universal sampling is then applied to conduct crossover and mutation to generate the new generation;

3) Crossover: One pair of offspring, i.e., the individual in the new generation, is generated from the selected parent by crossover. Crossover occurs with a pre-set probability. The random selection of a crossover and the combination of the two parent’s genetic data are preceded. The technique used in this paper is position independent crossover where each chromosome has an equal chance of coming from either parent;

4) Mutation: Genetically, the mutation occurs with a pre-set probability of which the new and unexpected point would be brought into the GA optimizer’s search domain. It is an essential operation to improve the exploration ability of GA’s optimization;

5) Elitist strategy: The elitist strategy is applied to reserve the best gene in the previous generation and directly copy it into the next generation without modification, i.e., crossover or mutation. In this way, the best fitness values would not decrease during the optimization process;

6) New generation: After the reproduction process, which includes selection, crossover, mutation, and applying the elitist strategy, a new generation is generated.

The main procedures of GA are illustrated in Fig.7.

2.3.2 Evaluation of fitness value

In the process of GA optimization, the fitness function, i.e., the non-linear buckling capacity, is calculated by the ANSYS software. The calculation process is as follows.

1) Establish the geometric model of the initial spherical shell.

2) Adjust the geometric model by the design variables transmitted from the GA.

3) Assign the geometric model with the material properties, section dimensions, and the initial member cross-section eccentricity to consider the influence of member buckling. Then, mesh the geometric model to generate the FE model.

4) This paper considers a single-layer reticulated shell with a partially distributed load. Thereby, to accommodate both possibilities of emerging bifurcation points and snap-through buckling, a geometric initial imperfection mode Y considering both symmetric and antisymmetric modes [27–29] is introduced to the FE model, as presented in Eq. (2):

(2)\mathit{Y}=\frac{1}{2}({\mathit{Y}}_{\mathrm{a}}+{\mathit{Y}}_{\mathrm{s}}),

where Y_s is the lowest symmetric linear buckling mode and Y_a is the lowest antisymmetric linear buckling mode.

5) Use the arc-length method to obtain the non-linear buckling load of the shell, namely the fitness function.

3 Numerical example

3.1 Structural model parameters

The main design parameters of the aluminum alloy reticulated shell include geometric parameters, material properties, load parameters, support conditions, and imperfections. The parameters of the numerical example in this section are determined according to the code recommendations and practical engineering structures [30–32].

1) Geometrical parameters

Span: L = 40 m;

height-to-span ratio: f/L = 1/4;

number of rings: m = 12;

radius of the gusset plate: R = 300 mm;

member section: H400 × 200 × 10 × 16 (the numbers indicate the height, flange width, web thickness, and flange thickness, unit: mm);

thickness of the gusset plate: t = 16 mm.

2) Material properties

The 6061-T6 aluminum alloy is adopted. The yielding stress f_0.2 is 240 MPa, the ultimate tensile strength f_u is 260 MPa, the elastic modulus E is 70 GPa, and the Poisson’s ratio μ is 0.3 [20]. The Ramberg−Osgood model [33] and the SteinHardt suggestion [34] are used to describe the non-linear constitutive model of aluminum alloy. The Ramberg−Osgood model is described in Eq. (3), and the stress−strain relationship is shown in Fig.8.

(3)\epsilon =\frac{\sigma }{E}+0.002{\left(\frac{\sigma }{f_{0.2}}\right)}^n,

where n is the hardening constant and could be calculated through the Steinhardt suggestion, as shown in Eq. (4)

(4)n=f_{0.2}/10.

3) Load distribution

The load ratio (defined as the ratio of the full-span dead load g to the half-span live load q): γ = 1.

4) Boundary condition

Pinned supports at the periphery.

5) Imperfection

The imperfection was adopted as illustrated in Eq. (2). The initial eccentricity of each H-shaped beam member section is l/1000 along the weak axis, where l is the member length.

3.2 Optimization parameters

The boundaries ΔZ_min and ΔZ_max are selected as −0.3 and 0.3 m, respectively, so that adjusting the controlling points would not affect the macroscopic shape. The numbers of generations and individuals are 50 and 50, respectively. The crossover rate of GA is 80%. The Gaussian mutation is adopted with the inbuilt ‘MUTATION GAUSSIAN’ function, and the values for scale or shrinkage are specified as 1.0 and 1.0, respectively. To prevent the loss of the optimal individual of the current population in the next generation, failing GA to converge to the global optimal solution, the elitist strategy is adopted, which is to copy the best individual (elitist) of each generation directly to the next generation without crossover and mutation operations. Furthermore, due to the characteristics of GA and the adoption of the elitist strategy, repetition of individuals in the optimization process is likely to occur; the corresponding fitness function values can be directly obtained from the previous calculation results without repeating the non-linear buckling analysis, reducing the unnecessary calculation time.

3.3 Optimization results

To illustrate the validity of the proposed shape optimization method, the shell model described in Subsection 3.1 was optimized using GA.

The iteration history of the optimization process is illustrated in Fig.9, where n denotes the iteration step. Note that the individual with the highest objective function in each generation was selected to represent the entire generation. Due to the elitist strategy, the objective function P_c significantly increased in the first few steps and gradually converged to the maximum value. Specifically, P_c increased from 28.33 to 47.52 kN/m², by 67.74%. Fig.10 shows the difference in ΔZ between the optimal and initial shapes. The difference between the initial and optimal shapes is minor because ΔZ_i is restricted to the range of [−0.3,0.3] (unit: m). Therefore, it can be concluded that the proposed method can significantly improve the buckling capacity of aluminum alloy reticulated shells considering geometric, material, and joint stiffness nonlinearities without changing the macroscopic shape.

Because the minimum buckling capacity within the optimization process is P_c,min = 28.33 kN/m², Fig.11 shows the history of structural performance indicators during the optimization process under P_c = 28 kN/m². As depicted in Fig.11(a), the maximum joint displacement z_max (the positive sign corresponds to the downward direction of the Z-axis) significantly decreased in the first few iterations and then gradually converged to the minimum value with a slight fluctuation. After optimization, the maximum joint displacement was reduced from 93.7 to 51.0 mm, by 45.6%. This result indicates that the overall stiffness of the structure is improved after optimization. As shown in Fig.11(b), the maximum compressive force F_c,max decreased by 7.5%. Fig.11(c) and Fig.11(d) show that the strong axis’s positive and negative bending moments decrease significantly by 31.1% and 54.0% after optimization, respectively, while Fig.11(e) and 11(f) show that the positive and negative bending moments about the weak axis are slightly increased by 6.4% and 2.5%, respectively, as a result of optimization. These changes indicate that the internal forces tend to be more uniformly distributed after optimization, representing the mechanism of the increase in the non-linear buckling capacity.

3.4 Parametric analysis

The height-to-span ratio influences the initial shape of the spherical shell, and the load factor affects the load distribution. Therefore, the influence of these two parameters on the optimal shape needs to be investigated. In this section, three height-to-span ratios (1/4, 1/5, and 1/6) and four load ratios (0, 1/4, 1/2, and 1) were varied to explore the difference in the optimal solutions, while other parameters were fixed as described in Subsections 3.1 and 3.2. It is worth mentioning that both geometrical and material nonlinearities were considered in this section.

3.4.1 Influence of load ratios

Fig.12 compares the optimized ΔZ of the reticulated shell with different height-to-span ratios after optimization. The shapes were similar for \gamma = 0, 1/4, and 1/2, where the nodes between X = 0 and X = 12 m were moved upwards, and the other nodes were moved downwards. For \gamma = 1, most of the nodes were optimized downwards. The differences may be caused by the discontinuity of the non-linear buckling load with respect to the variation in \gamma.

3.4.2 Different height-to-span ratios

Fig.13 presents the changes in the optimized design variable Z-coordinates of the reticulated shell with different height-to-span ratios during optimization. The optimized Z-coordinates were significantly different, indicating that the variation trends of the variables, i.e., the Z-coordinates, under various initial shapes are different.

3.5 Practical design tables

To avoid the time-consuming non-linear analysis involved in the optimization process, practical design tables are provided in this section for engineers’ reference. To maintain the analytical formula of the spline curve, as depicted in Fig.6, which is fully consistent with the optimized coordinates, the spline curve of the reticulated shell is divided into three segments, and a cubic polynomial describes each part, as

(5)

(6)X_i=\{\begin{array}{ll}0,& i=1,\\ {\displaystyle \frac{1}{6}}L,& i=2,\\ {\displaystyle \frac{1}{3}}L,& i=3,\end{array}

where X and Z(X) are the X- and Z-coordinates, respectively, as shown in Fig.6. The parameters of the cubic polynomial and the corresponding increase rate of the objective function IR are listed in Tab.1–Tab.3.

4 Comparison of shape optimization with and without material nonlinearity

4.1 Increase rate of objective function

Because material nonlinearity is not considered in the study of Zhu et al. [35], it is necessary to compare the optimal solutions obtained in this study and Ref. [35] to investigate the impact of material nonlinearity on the optimal structural shape. To avoid redundant descriptions, the naming conventions of the analysis are listed in Tab.4.

To start with, the load−displacement curves of a typical aluminum alloy reticulated shell mentioned in Subsection 3.1 under elastoplastic and elastic analysis were compared, as shown in Fig.14. Note that the displacement was taken from the node with the maximum resultant displacement, and the analysis was terminated at the first limit point using the ‘ARCTRM,L’ command in ANSYS. It can be seen that the stability behavior of the reticulated shell varies when elasto-plasticity is taken into consideration. Moreover, existing research indicated that unexpected or antecedent buckling could occur when considering elasto-plasticity. In this case, the reliability of the optimal solution based on elastic analysis should be tested by comparing it with the optimal solution based on elasto-plastic analysis.

Tab.5 presents the optimal non-linear buckling capacity P_c for series MN-INI, MN-NZ, ML-INI, ML-LZ, MN-LZ, and ML-NZ, respectively, while the IR of the series is plotted in Fig.15. Note that the value of P_c varies with the model type, even if the design variables are the same, and all other parameters are the same as specified in Subsections 3.1 and 3.2.

It can be seen from Tab.5 that the non-linear buckling loads corresponding to the optimal cases are generally larger than those of the initial cases, regardless of the model type. Specifically, the non-linear buckling load was enhanced by 59.50% on average after optimization for the series MN-NZ, while the average improvement was 44.66% after optimization for the series MN-LZ. Meanwhile, the non-linear buckling load was enhanced by 49.25% and 51.20% for the ML-NZ and ML-LZ results, respectively. Although the average performance of the NZ results was superior to that of the LZ results, the difference was not significant. Because the MN models are closer to reality, it is better to conduct shape optimization considering both geometric and material nonlinearities.

4.2 Internal force and mechanism

This section compares the internal force distribution of the ML and MN models to gain a deeper insight into the mechanism underlying the increase in the non-linear buckling capacity. A reticulated shell with f/L = 1/4 and load ratio γ = 1 was used as an example. Series ML-INI, ML-LZ, MN-INI, and MN-NZ mentioned in Subsection 4.1, are compared under the following load levels.

① P = 28 kN/m². This value is slightly smaller than the non-linear buckling load of series MN-INI. Under this load level, none of the series enter the non-linear buckling phase. Therefore, the internal force conditions of all four series are compared in Fig.16.

② P = 48 kN/m². This value is slightly smaller than the non-linear buckling load of series MN-NZ. Under this load condition, because the load value is larger than the non-linear buckling load of series MN-INI and ML-INI, only series MN-NZ and ML-LZ were considered in the comparison, as shown in Fig.17.

③ P = 52 kN/m². This value is slightly smaller than the non-linear buckling load of the ML-LZ series. Under this load condition, because the load value is larger than the non-linear buckling load of series ML-INI, MN-INI, and MN-NZ, only the internal force distribution of the ML-LZ series is shown in Fig.18.

As shown in Fig.16, the pairwise comparisons between the ML and MN series demonstrate some consistent trends. First, the maximum compressive force and the positive and negative bending moments about the strong axis were reduced, particularly the positive and negative bending moments about the strong axis. The positive and negative bending moments about the strong axis decreased by 58.52% and 44.94% after optimization for series MN-NZ, while the decrease was 55.89% and 4.22% after optimization for series ML-LZ. Although there were some improvements in the bending moment about the weak axis, they were controlled with an average of 5% for both the MN and ML series. Second, the distribution of the internal forces appeared to have the same patterns before and after optimization. Among them, the change in the distribution of the maximum bending moment about the strong axis was the most prominent. In the initial stage, the maximum bending moments were concentrated near the top of the reticulated shell, which became more uniformly distributed after optimization. This change allowed the optimized structures to gain a greater bearing capacity. In contrast, the differences between the MN and ML series manifested in the values of the internal forces. Due to material nonlinearity, the MN series exhibited larger internal force values. Therefore, the MN model tends to be more conservative, which helps improve structural safety. Fig.17 shows that the internal force distributions of the MN-NZ and ML-LZ series are significantly similar, implying that the plastic development of the MN-NZ series is not in-depth under the non-linear buckling capacity. As illustrated in Fig.18, under load level ③, the members’ axial force and bending moment of the ML-LZ model continue to increase, and the entire structure is close to losing its stability.

Therefore, it can be concluded that aluminum alloy spherical reticulated shell structures can benefit from such shape optimization, resulting in a higher non-linear buckling load because the weak-axis bending moments do not present a significant increase. In contrast, the maximum values of the strong-axis bending moments and axial compressive forces are significantly reduced. Moreover, it is acceptable to directly use the optimal solutions obtained from the ML model when the computational time needs to be shortened. However, it is suggested that the actual material nonlinearity should be considered when a more optimal design is required because material nonlinearity is more reflective of the actual structure.

Indeed, this approach has its limitations. It is worth noting that the proposed method is suitable only for spherical reticulated shells, and a more generalized approach should be further developed for free-form reticulated shells. However, the spherical reticulated shell is one of the most widely-used structural types in spatial structures; hence, the findings of this study are of both scientific and practical significance. Besides, the results produced in this study, on the other hand, can provide some general guidance for users of the suggested algorithms in finding the optimal shape of aluminum alloy reticulated shell structures.

5 Conclusions

This study proposes a simple and easy-to-use method to improve the non-linear buckling load of reticulated structures without changing the macroscopic shape of the structure. The method uses the GA to enhance the buckling capacity of K6 aluminum alloy reticulated shells with initial spherical profiles, and the buckling analysis considers geometric, material, and joint stiffness nonlinearities. The main findings are as follows.

1) The proposed method, which selects the vertical coordinates of the control points as design variables, and the non-linear buckling capacity as the objective function to be maximized, effectively provides a favorable optimized shape. According to the numerical results, the reticulated shells with optimized shapes possess higher load-bearing capacities and improved structural performance.

2) During the optimization process, the internal forces tend to be more uniformly distributed, which represents the mechanism underlying the increase in the non-linear buckling capacity.

3) Practical design tables are proposed for obtaining optimized shapes of aluminum alloy spherical shells with different height-to-span and load ratios so that the high computational cost of the optimization process can be avoided.

4) The optimal solution obtained based on an objective function that considers material nonlinearity can better improve the non-linear buckling load of the actual structure than the objective function that considers material linearity with a higher computational cost.

5) Aluminum alloy spherical reticulated shell structures can benefit from the presented shape optimization and result in a higher non-linear buckling load because the weak-axis bending moments do not significantly rise. In contrast, the maximum values of the strong-axis bending moments and axial compressive forces are significantly reduced.

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