#### Abstract

The buckling loads of shell structures are sensitive to initial geometric imperfections. Conventional methods used to model geometric imperfections cannot determine the accuracy of buckling loads with high computational efficiency. A new computational approach based on particle swarm optimization (PSO) is proposed to obtain the lower bound of the buckling load of shell structures with geometric imperfections. The proposed approach assumes a nodal geometric position using uncertain parameters. The buckling loads of the shell structures are then optimized using the PSO-based approach. Both academic and practical numerical examples have been thoroughly investigated. Thus, the applicability and accuracy of the proposed method is critically validated.

#### 1. Introduction

Shell structures are widely used in large-span public construction owing to its novel shape, low dead weight, and excellent mechanical behavior. However, initial geometric imperfections, an inevitable phenomenon due to machining and construction errors, considerably influence the buckling of shell structures [1–4]. Therefore, shell structures are sensitive to initial geometric imperfections, and a few studies [5, 6] have shown that even a small amount of initial geometric imperfection may lead to a reduction of over 50% in the buckling load of the structure.

To incorporate the effects of initial geometric imperfections, several notable approaches have been developed in the past few decades. These diversely available approaches for buckling analysis of shell structures can be classified into four major categories according to the mode of imperfect shape. The first category is the consistent imperfection modal method (CIMM) [7], which assumes that the imperfect shape is similar to the elastic critical buckling mode of a structure. Another category is the consistent buckling mode method [8], which considers the relative nodal- displacement increment at the state of incipient collapse as initial geometric imperfections. In both methods, the limiting value of the initial maximum nodal geometric deviation, which represents the geometric imperfection of a structure, is equal to a prescribed value, i.e., *L*/300 in the current Chinese specification [9], where *L* represents the largest horizontal dimension of the structure. In addition, the Fourier decomposition method has been used to interpret the imperfections of cylindrical shells [10, 11], and a series of modal amplitudes and phase angles are used to model the imperfect surface of a cylindrical shell. Moreover, the stochastic imperfection modal method (SIMM) is widely used to assess the buckling of structures, which investigates conditions where uncertain system parameters are modeled by implementing the probabilistic theory. In [1, 2, 12–15], uncertain structural parameters are modeled as random variables with known probability density functions, and samples of imperfect structures can be generated using the Monte Carlo simulation. A stochastic finite element (FE) approach has been developed in [16, 17] to investigate the uncertain buckling behavior of structures involving random geometric properties.

In addition, considering the effect of imperfection, response of perforated glass/epoxy composite tubes subjected to axial compressive loading was investigated using numerical and experimental methods in [18], and the effect of two parameters of size and number of cutouts on the buckling load of cylindrical shell structures was investigated using linear buckling and modified linear buckling mode shape imperfection method in [19]. Meanwhile, the effect of a rectangular cutout on the buckling behavior of a thin composite cylinder was investigated using numerical and experimental methods in [20]. Reliability-based design optimization was investigated for stiffened shells in consideration of various uncertain factors [21, 22], such as variations in manufacturing tolerance, material properties, and environment aspects. A new method called the worst multiple perturbation load approach (WMPLA) was proposed by [23] to improve the accuracy and reliability of the prediction of knockdown factors for cylindrical shells. In the WMPLA, multiple dimple-shape imperfections are applied on the shell surface, and then, the most detrimental combination of these dimple-shape imperfections can be determined by optimization methods. To verify the accuracy of this method, the buckling test and imperfection sensitivity of one full-scale 4.5 m-diameter stiffened shell under axial compression were investigated with several types of assumed imperfections [24]. The results showed that the WMPLA is a realistic and effective approach to predict the knockdown factors in the design stage. Five unstiffened shells are tested in [25]. The measured imperfection approach, single-perturbation load approach (SPLA), WMPLA, and combined approach for measured imperfections and superimposed radial point load imperfections are compared with test results, and the results show that the knockdown factors by the WMPLA and the combined approach are similar to one another and very close to the test results.

Although such diverse approaches have successfully been proposed, some of which are even used in practical engineering applications, a key problem, i.e., the worst case imperfect shape of a nodal geometric position, has not yet been solved. This means that a method to obtain the lower bound of a buckling load, vital to the design, has not yet been discovered. The lower bound of a buckling load is commonly known to objectively exist when the boundary of a nodal geometric position is determined. The buckling analysis of shell structures that considers the parameters of geometric imperfection can be transformed into an optimal-value problem. However, because of the complexity of the nonlinear analysis, establishing direct theoretical computational approaches is difficult. To realize the purpose of the limit-state analysis of structural buckling, this paper presents a novel particle swarm optimization- (PSO-) based computational scheme to search the worst case imperfect shape of shell structures. PSO, which was initially developed by Kennedy and Eberhart [26], is a widely used multiagent metaheuristic algorithm. Its high capability of finding optimal solutions in a reasonable amount of time together with the relative ease of implementation and small number of parameters has continually encouraged researchers to use PSO for a diverse range of optimization problems in different disciplines. In structural engineering, PSO has been successfully applied to different types of optimization problems [27–29].

The rest of the paper is organized as follows: Section 2 presents the formulation of optimization problem for the shell structures with geometric imperfections. Subsequently, the proposed PSO-based approach is presented in Section 3, and the analysis framework of the approach is presented. Furthermore, two academic and practical examples that have been investigated using the proposed approach, namely, CIMM and SIMM, are presented in Section 4. Finally, some conclusions are provided in Section 5.

#### 2. Problem Statement

In a geometric-imperfection-constraint optimization problem, the objective is to minimize the buckling load of shell structures while satisfying the constraints on the boundary of the nodal geometric position. The optimization problem can be mathematically expressed as follows:

Here, is the vector of the nodal-coordinate variables. , , and , are the nodal coordinates of the structures. and , and , and and are the lower and upper bounds of variables , , and , respectively. is the buckling load of the shell structures. is the number of nodes.

#### 3. PSO-Based Approach for Buckling Analysis

Compared with the other population-based optimization methods, the advantages of PSO consist of its simple structure, immediate accessibility for practical applications, ease in implementation, speed for acquiring solutions, and robustness. Although PSO is a very well-known and common optimization algorithm, a basic form of the standard PSO is first summarized in brief. Then, a novel computational approach for searching the lower bound of the buckling load based on PSO is introduced.

##### 3.1. Basic Concept of the PSO

PSO is a type of a swarm-based optimization algorithm, which is a population-based metaheuristic algorithm, inspired by the social behavior of animals such as those of fish school, insect swarm, and bird flock. The population in PSO is called a swarm, whereas each individual in the PSO population is called a particle. A swarm consists of particles that move around an *N*-dimensional search space, where each particle represents a potential solution. The *j*th particle is characterized by its position vector and velocity vector . Each particle flies through the solution space searching for the global optimal solution. The velocity and position of the *j*th particle are updated according to the following equations during the flying process [30]:

Here, and are acceleration constants multiplied by two random numbers and , respectively, which are uniformly distributed in [0.0,1.0] and used to weigh the velocity toward the best previous position of the *j*th particle. represents the best ever position of the *j*th particle, and corresponds to the global best position in the swarm up to the *k*th iteration. Parameters and are both usually set to be equal to provide each component equal weight as the cognitive and social learning rates [31], respectively, and the most common setting is [32]. We should note that the standard PSO introduced by Kennedy [33] employs a set of different random numbers and instead of , and , for each component of the velocity vector. Inertia weight in interval [0.0,1.0] is used to control the influence of the previous velocity. A large inertia weight facilitates global exploration, which is particularly useful in the initial stages of optimization. In contrast, a small value allows for more localized searching, which is useful when the swarm moves in the neighborhood of the optimum value. An appropriate value for the inertia weight usually provides balance between exploration and exploitation and consequently results in a better optimum solution.

In the PSO algorithms, boundary conditions and convergence criteria are very vital, which are defined as follows:(i)*Boundary Conditions.* We need to note that appropriate boundary conditions improve the convergence rate and add a further random component to the search, thereby helping the algorithm avoid the local minima [34]. In the present study, if a new particle is not inside the domain region, a new position is reset at the boundary.(ii)*Convergence Criteria*. Convergence criteria are used to terminate the execution of PSO. Several convergence criteria have been proposed [35], including the maximum-iteration, minimum-error, exhaustion-based, improvement-based, and movement-based criteria, as well as other similar convergence criteria. Because the maximum-iteration criteria are widely used whose rule is simple, the maximum number of generations, namely, , is defined as the convergence criteria.

##### 3.2. PSO-Based Approach for Searching the Lower Bound of Buckling Load

In this study, a novel computational scheme is proposed to search the lower bound of the buckling load considering the effects of geometric imperfection. This novel computational approach offers a buckling-analysis framework that can handle the imperfection-shape problem. Similar to the conventional methods of buckling analysis, the nodal geometric imperfections are considered in this study, whereas the geometric imperfections of a member, such as the initial bending, are not considered. Moreover, we assume that the deviation between neighboring nodes is uncorrelated.

The main steps of the novel method are described as follows: *Step 1*. The parametric model of the shell structures is established. In the parametric model, the geometric position of the joints consists of the perfect geometric position and geometric-position error, which can be defined as Here, , , and are the perfect nodal geometric position in the *X*, *Y*, and *Z* directions, respectively. , , and are the nodal geometric-position errors in the *X*, *Y*, and *Z* directions, respectively. *Step 2*. Number of particles and maximum iteration are set, and particle position and velocity for each particle are analyzed. Initial particle position can be conveniently randomly chosen by employing the lower and upper bounds in the following manner: Here, indicates a random number between zero and one. In a similar manner, the initial velocity can be randomly determined as follows: Here, is a scale factor assigned by the user to limit the size of the initial velocity. *Step 3*. The individual particles are evaluated. By using a nonlinear analysis program such as ANSYS or Abaqus, every particle that represents a shell structure with a special imperfect shape is calculated to obtain the buckling load, which is used as the fitness function. *Step 4*. Best known position of each particle and best known swarm position are updated. By comparing the fitness function of the current iteration with that of the previous iteration, best known position of each particle can be obtained. The best known swarm position can then be determined by comparing the fitness of each particle. *Step 5*. The particle velocity is calculated using velocity equation (2). The particle position is updated using position equation (3), and the boundary conditions are verified. *Step 6*. The convergence criteria are evaluated. If the maximum iteration is attained, the execution of the algorithm is terminated; otherwise, the execution continues.

The flowchart of the PSO-based approach is shown in Figure 1.

#### 4. Numerical Investigations

In this study, two numerical examples are investigated to illustrate the applicability of the novel approach. The considered two numerical examples include both academic-sized and practically motivated examples. All presented numerical results are obtained using a desktop computer: CPU Intel® Xeon (R) E5-1607 v4 at 3.10 GHz with 32 GB memory and 1 TB hard drive.

##### 4.1. Case A

The first investigation concerns an academic-sized example to determine the buckling load of a shell structure subjected to externally applied concentrated load *P*. The structure shown in Figure 2 has been analyzed in the literature [1, 36]. In this case, the effect of geometric imperfection is considered. The three translational degrees for the supports in the circumference are fixed. The sectional area of the beam is 317 mm^{2}, elastic modulus is 3030 MPa, and shear modulus is equal to 1096 MPa [36]. Concentrated load *P* is located at the center node of the structures. In the present study, a parametric model is built using FE software ANSYS. To simulate the buckling of the structures, beams are modeled using Beam188 element. As a three-node linear finite strain beam element, the Beam188 element is in accordance with the Timoshenko beam theory. The arc-length method is used to trace the load-deflection behavior in the postbuckling range and subsequently capture the first maximum point that represents the buckling load.

**(a)**

**(b)**

To verify the accuracy of the FE model, the buckling analysis of a perfect structure is performed by considering the geometric nonlinear relationship between the load and the deflection. The load-deflection curve for Node 1 by the FE model and the results from the literature [36] are shown in Figure 3. From the curve, the critical buckling load of perfect structure from the literature [36] is equal to 612.3 N, and the critical buckling load of FE model is 609.4 N, which shows that the buckling loads are close to each other. Figure 3 also show that there are small deviations between the proposed method and the literature [36] at the range of postbuckling stage. The main reason that causes the deviations is that there are minor differences between finite element of the literature [36] and Beam188 based on the Timoshenko beam theory. On the whole, the solutions by our proposed method agree well with the results in the literature [36]. Therefore, the validity of the FE model is evidently verified, which means that the FE model can be confidently used to predict the buckling behavior of a dome.

To verify the availability of the novel method based on the PSO algorithm, comparison of the load-deflection curves between the novel method and those of the other three methods, which include the CIMM [7], SIMM [5], and stochastic imperfection modal superposition method [1], is carried out. In all four methods, the initial positional-deviation tolerance at each node along the three directional axes is 2 mm, and the total initial positional-deviation tolerance is 3.464 mm. For the CIMM, the first elastic critical buckling mode is considered as the geometric imperfect shape. The buckling load obtained by the CIMM is 437.1 N. For the SIMM, the initial nodal geometric imperfections are assumed to be truncated Gaussian-distributed random variables with the following statistical information: , , , and . We generate 10000 random samples, and the mean and standard deviation for the buckling-analysis results are as follows: , , minimum buckling load , and nominal minimum buckling load using the “” principle . For stochastic imperfection modal superposition method [1], the buckling load is 436.3 N. For the PSO-based approach, the parameters are set as follows: , , , and . The corresponding buckling load is 427.7 N. The load-deflection curves of the four methods are shown in Figure 4, which shows that the buckling load of the PSO-based approach is smaller than those of the other three methods.

The perfect shape and worst imperfect shapes are shown in Figure 5, and the resulting geometric-position deviations of the joints are listed in Table 1. The list in Table 1 clearly illustrates that all nodal geometric deviations in the PSO-based approach reach the limiting value. This result means that the PSO-based approach fully uses the nodal geometric deviation, which is the fundamental reason why the buckling load is the smallest load in all the methods.

The convergence curve of the PSO-based approach is shown in Figure 6. The convergence curve also shows that complete convergence is almost attained in two iterations; thereafter, the convergence curve becomes straight.

##### 4.2. Case B

To investigate the role of the initial geometric imperfections on the buckling load of a structure, a single-layer Kiewitt-6 spherical reticulated shell, which is modeled using FE software ANSYS, is used, as shown in Figure 7. In this case, the span and height of the shell are 40 and 5 m, respectively, and the shell consists of 90 steel tube members and 37 joints. The joints among the steel tubes are assumed to be rigid. The nodes on the perimeter ring are fixed against displacement and rotation. The diameter and thickness of the steel tube are 159 mm and 10 mm, respectively. For the member element, it is modeled by the Beam188 element with actual member section and material property.

The main load considered in the buckling analysis includes uniform gravity load and uniform live load, and their respective values are 0.3 and 0.5 kN/m^{2}. In the FE model, each member is divided into two beam elements which are modeled using Beam188, which is in accordance with the Timoshenko beam theory. The arc-length method is used to calculate the stiffness-matrix equation, and the two-norm condition for residual force *f* is used as the convergence criterion. The constitutive model of steel is perfectly elastoplastic with yield strength of 235 MPa and Young’s modulus of 206 GPa.

All the nodal geometric positions of the shell structures are considered as variables. According to the current Chinese specification [9], the initial positional-deviation tolerance of each node along the three directional axes is 0.0770 m, and the total initial positional-deviation tolerance is 0.133 m. For the PSO-based approach, the parameters are set as follows: , , , and . The buckling load factor of the shell structure is 3.13, which is derived from the ratio between the critical buckling load and the sum of gravity and uniform live load, thus the buckling load is equal to the times of 0.8 kN/m^{2}. Meanwhile, the buckling load factor of the shell with geometric imperfection is calculated using CIMM and SIMM, and the buckling load factor of the perfect shell is also calculated. All results are shown in Figure 8 and listed in Table 2.

The list in Table 2 clearly illustrates that the buckling load factor is the smallest in all the methods. The convergence curve of the PSO-based approach is shown in Figure 9, which shows that the buckling load factor is smaller than those in all the other methods even after only the first two iterations. The convergence curve also shows that complete convergence is almost attained in 67 iterations, and thereafter, the convergence curve becomes straight. Meanwhile, 10000 calculations are performed by the SIMM with a computational time of 8645 s, whereas the PSO-based approach performs 2000 simulations and consumes 1656 s under the same computational environment. Therefore, the PSO-based approach possesses the computational capability with much higher performance efficiency.

#### 5. Conclusion

The conclusions and outcomes of this work can be summarized as follows:(i)This paper has presented a novel computational analysis framework based on PSO to obtain the lower bound of the buckling load of shell structures with geometric imperfections. In the proposed approach, the nodal geometric positions are modeled using uncertain parameters.(ii)The proposed PSO-based approach and the analysis framework of the approach are presented. The buckling loads of the shell structures are optimized using the PSO-based approach, and the worst imperfect shape with boundaries is then obtained.(iii)Two numerical examples are thoroughly investigated, and the results demonstrate that the proposed approach has a good convergence speed and high accuracy.

It is believed that the PSO-based approach in this study can be effectively used to incorporate the buckling of shell structures with geometric imperfections. Subsequently, the lower bound of buckling load can be obtained by the proposed method, which can be regarded as safe defense-line, providing important reference to the design.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of China under grant no. 51808176 and by the National Science Foundation of Hainan Province of China under grant no. 518QN213.