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Grid-stiffened structures have excellent performance, such as high specific stiffness and strength, and are widely used as critical load-bearing structures in aerospace equipment, including launch vehicles and spacecraft. It is now a well-established fact that curvilinearly stiffened panels with curved stiffener paths and varying stiffness distributions significantly improve load-bearing efficiency. However, existing methods are usually designed based on two clusters of approximately orthogonal stiffeners, which may limit the design flexibility and load-carrying potential of such structures. In the proposed design method, various complex variations of curved stiffener layouts can be characterized with a simple set of variables by establishing a database of stiffener unit cells and feature dimension reduction, as well as using coordinate transformation and monotonicity preserving interpolation methods for geometric dimension reduction, respectively, which allows simultaneous selection of basic stiffener unit cells and adjustment of stiffener paths. Based on the structure characterization with high design flexibility, the hyperparameter dimension reduction of the Kriging model is introduced, thereby forming an optimization framework through multi-level dimension reduction, while ensuring the efficiency of the optimization process. Moreover, an accurate geometric modeling based on NURBS is proposed to meet the needs of designing stiffened structures on various types of complex surfaces. In two typical aerospace cases, the developed structural concept and design method can improve structural performance by over 80 % compared to initial design schemes with straight stiffeners and constant stiffness. The results demonstrate that both the developed structural concept and design methodology effectively enhance the upper performance design limit of stiffened structures. These findings are practically significant for the application of advanced lightweight thin-walled structures in aerospace equipment. [ABSTRACT FROM AUTHOR]

To resist external pressure, stiffened spherical domes have been widely used in underwater vehicles. In traditional design of these thin-walled structures, uniform stiffeners are usually utilized and overly conservative safety factor method is employed to envelop the multi-source uncertainties mainly caused by manufacturing. In this study, a concept of non-uniformly stiffened spherical dome (NUSPD) is developed to improve the structural buckling capacity. Furthermore, an efficient reliability-based design optimization (RBDO) framework is established for NUSPD, which includes a deterministic optimization loop to fix the stiffener layout, and subsequently followed by a RBDO loop to determine the detailed stiffener dimensions and decrease the structural weight under geometric and material property uncertainties. Since the RBDO of stiffened spherical domes is very complex, an improved enhanced chaos control method together with surrogate model is proposed to improve the convergence rate. Finally, the efficiency of the proposed RBDO framework is demonstrated by the NUSPD example. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) is particularly suitable for the prediction of buckling load and design optimization of variable-stiffness composite panels, since curvilinear fiber path can be described exactly to improve the analysis efficiency, moreover, analytical sensitivity can be derived to improve the optimization efficiency. In this study, an integrated framework of exact modeling, isogeometric analysis and optimization for variable-stiffness panels is developed for the global optimum. Due to the inherent feature of multiple local optima for this type of problems, a novel multi-start gradient-based strategy is developed to enhance the global optimization capacity, and multiple initial designs for gradient-based optimization are determined by space tailoring method, which can guarantee the convergence rate and efficiency. Once the constraint aggregation and parallel computing methods are employed, the computational efficiency will be further improved. For typical illustrative example, it can be demonstrated that the proposed method is able to provide a more efficient optimum design with significant less computational cost compared to other traditional methods, including FEA-based optimization, direction optimization using genetic algorithm, gradient-based optimization without K–S function, gradient-based optimization based on difference method. [ABSTRACT FROM AUTHOR]

Due to the non-uniform in-plane stress distribution, variable-stiffness panel with curvilinear fiber paths is a promising structural concept for cutout reinforcement of composite structures under axial compression, due to the more diverse tailorability opportunities than simply choosing the best straight stacking sequence. However, traditional representation methods of curvilinear fiber path are usually not flexible for cutout reinforcement. In this study, the flow field function containing a uniform field and several vortex fields is utilized to represent the fiber path due to its inherent non-intersect and orthotropic features, and a bi-level optimization framework of variable-stiffness panels considering manufacturing constraints is then proposed. A typical rectangular composite panel with multiple cutouts is established to demonstrate the advantage of proposed framework by comparison with other fiber path functions. Results indicate that the flow fiber path only needs few variables to finely represent the fiber path, which can provide satisfying and manufacturable fiber paths by combination use of curvature constraint. [ABSTRACT FROM AUTHOR]

Aiming at the optimization of aircraft panels, a modified version of harmony search (HS) algorithm is proposed based on the information of the harmony memory (a memory location where all the solution vectors are stored) for improvisation procedure, named as adaptive dynamic harmony search (ADHS) algorithm. In order to reduce the amount of calculation, response surface method is employed, and second-order polynomial with cross terms is used to construct the model. To demonstrate the advantage of the proposed algorithm, typical aircraft panels under buckling constraint are established, and several existing HS algorithms are compared. The effects of the number of improvisation ( NI ) and harmony memory size ( HMS ) are investigated and discussed in detail. Results indicate that the proposed ADHS can provide an optimum design in a robust manner, and local optimum solutions may be reduced based on the ADHS for optimization problems with multiple local minima. Finally, several useful information is obtained for the design of stiffened panels with cutouts. [ABSTRACT FROM AUTHOR]

Variable-stiffness panel with curvilinear fibers is a promising structural concept compared to constant-stiffness designs. However, for the traditional finite element analysis (FEA), there is no guarantee that the fiber angle is continuous and smooth due to element discretization. In this study, on the basis of Mindlin plate theory, the buckling behavior of composite variable-stiffness panels is investigated based on isogeometric analysis (IGA), whose main feature is that the continuity of fiber angle on the whole panel is guaranteed. In particular, since geometric stiffness matrix has a significant influence on the buckling behavior, it is obtained by performing a static analysis prior to the buckling analysis herein, which can further improve the prediction accuracy of current methods. Different fiber path functions, ply number, geometric parameter, as well as various boundary and loading conditions are adopted to verify the proposed buckling analysis method. Finally, the prediction accuracy, total degree-of-freedom and CPU time are compared with the traditional FEA, which indicates that the isogeometric buckling analysis method can provide an adequate accuracy in a more efficient manner. [ABSTRACT FROM AUTHOR]

A concept of hybrid-stiffness stiffened shell is proposed based on sub-panel elements to achieve a simultaneous buckling pattern, which can provide enhanced design flexibility to fully explore the load-carrying capacity of structures. Then, a novel hybrid model is established to improve the computational efficiency of post-buckling analysis for stiffened shells, where the Numerical Implementation of Asymptotic Homogenization Method is utilized to smear out the stiffeners. On this basis, an integrated optimization framework of sub-panel configurations and weld lands for stiffened shells is presented. Illustrative examples with single and multiple cutouts demonstrate the effectiveness of the proposed framework based on the concept of hybrid-stiffness stiffened shell. [ABSTRACT FROM AUTHOR]

Stiffened shells are affected by numerous uncertainty factors, such as the variations of manufacturing tolerance, material properties and environment aspects, etc. Due to the expensive experimental cost of stiffened shell, only a limited quantity of statistics about its uncertainty factors are available. In this case, an unjustified assumption of probabilistic model may result in misleading outcomes of reliability-based design optimization (RBDO), and the non-probabilistic convex method is a promising alternative. In this study, a hybrid non-probabilistic convex method based on single-ellipsoid convex model is proposed to minimize the weight of stiffened shells with uncertain-but-bounded variations, where the adaptive chaos control (ACC) method is applied to ensure the robustness of search process of single-ellipsoid convex model, and the particle swarm optimization (PSO) algorithm together with smeared stiffener model are utilized to guarantee the global optimum design. A 3 m-diameter benchmark example illustrates the advantage of the proposed method over RBDO and deterministic optimum methods for stiffened shell with uncertain-but-bounded variations. [ABSTRACT FROM AUTHOR]

The influence of imperfection distributions considering manufacturing characters on the buckling response of stiffened shells has not been satisfactorily understood. Stiffened shells with three types of weld lands were established, including axial weld lands, circumferential and sequential axial weld lands, as well as staggered axial weld lands. As a concept of equivalent imperfection, dimple-shape imperfections produced by perturbation loads were adopted to substitute the measured imperfections, in order to reduce experimental and computational costs. Firstly, the influence of imperfection positions on the collapse load was examined for single perturbation load. Then, the influence of imperfection distributions was investigated for multiple perturbation load based on Monte Carlo Simulation. Finally, detailed comparison of three types of weld lands was made from the point-of-view of load-carrying capacity and imperfection sensitivity. Results can provide general instructions about imperfection-critical areas for axially compressed stiffened shells, which are particularly crucial for the manufacturing process. [ABSTRACT FROM AUTHOR]

• An efficient bi-stage RBDO framework of composite structures is established. • Lamination parameters are used to obtain an approximated optimal layer number. • IGA is utilized for buckling analysis and derivation of analytical sensitivity. Composite variable-stiffness (VS) panels with curvilinear fiber paths are very promising for aerospace structures. Due to the inherent complexity of VS laminates, buckling analysis and design optimization are extremely time-consuming and challenging, especially when uncertainties are considered, i.e. reliability-based design optimization (RBDO). In this study, an efficient bi-stage RBDO framework via isogeometric analysis (IGA) is established to release the tremendous computational burden. In Stage I, the layer thickness and lamination parameters are used as design variables to obtain an approximated layer number, which greatly reduces the design variable size. In Stage II, intermediate density variables are introduced, and IGA is employed for the buckling analysis and derivation of the analytical sensitivity. Furthermore, the augmented step size adjustment (ASSA) algorithm is used to enhance the efficiency and robustness of the RBDO process. Numerical results of VS panels are used to validate the performance of the proposed RBDO framework. The optimal results indicate that the proposed framework can find the optimal lightweight design that satisfies the manufacturing constraints in an efficient and accurate manner. [ABSTRACT FROM AUTHOR]

With regard to future heavy-lift launch vehicles, the buckling analysis and optimization of large-scale stiffened shells by finite element method (FEM) suffer from unbearable computational cost. In spite of the high analysis efficiency, the traditional smeared stiffener method (SSM) is still not accurate enough owing to the assumptions of analytical derivations. In this study, an effective and efficient numerical-based smeared stiffener method (NSSM) is proposed for the buckling analysis of stiffened shells. Firstly, the representative unit cell of stiffened shell is divided, and then it is equivalent using a novel numerical implementation of asymptotic homogenization (NIAH) method. The equivalent stiffness coefficients can be obtained accurately. Then, the buckling load is calculated by means of Rayleigh–Ritz method. Comparing with the prediction results of SSM and FEM, the high prediction accuracy and efficiency of NSSM are observed. Then, the effectiveness of NSSM for different loading conditions and model scales are discussed. Finally, numerical examples illustrate the high prediction accuracy and widespread applicability of NSSM for various grid-patterns, and the advantage of the rotated triangle grid-pattern in load-carrying capacity among various grid-patterns is demonstrated. [ABSTRACT FROM AUTHOR]

As a potential aerospace structural concept, hierarchical stiffened plates under axial compression are characterized by multiple local features, which lead to the buckling analysis and optimization suffering from heavy computational costs, by means of exact Finite Element Method (FEM). Thus, an efficient and simple hybrid framework for the buckling analysis and optimization of hierarchical stiffened plates is presented in this study. Firstly, the skin and minor stiffeners are equivalent to an unstiffened anisotropic plate based on a novel numerical implementation of asymptotic homogenization method (NIAH), which can be easily realized using commercial software as a black box, indicating a strong applicability for rather complicated minor stiffener configurations. Then, the equivalent plate together with major stiffeners can be treated as a hybrid model to be calculated by FEM. Further, a surrogate-based optimization based on this hybrid model is performed. Finally, three illustrative examples are established to demonstrate the effectiveness and simplicity of the proposed framework. [ABSTRACT FROM AUTHOR]

For the design of thin-walled cylindrical shells under axial compression empirical knockdown factors are applied. These knockdown factors are based on experimental results from the beginning of the 20th century and have been shown to be very conservative for modern shell structures. In order to determine less conservative and physically based knockdown factors for the design of axially loaded shells, different analytical and numerical design approaches have been developed. In this paper common as well as new shell design approaches are presented in detail and evaluated regarding the lower-bound buckling load. Among these design approaches are the EN 1993 1–6, the reduced energy method, linear buckling eigenmode imperfections, perturbation approaches and the new threshold knockdown factors. Important analysis and modeling details of each design approach are described and test examples are given and validated. Advantages and disadvantages of each approach are listed and design recommendations are given. A comparison of deterministic design approaches with modern probabilistic design methods is shown and the range of application of both design philosophies is discussed. Orthogrid stiffened cylinders with weld lands from NASAs Shell Buckling Knockdown Factor Project (SBKF) are modeled, analyzed and lower-bound buckling load calculations for improved knockdown factors are shown. [ABSTRACT FROM AUTHOR]

Influenced by numerous local features, the post-buckling analysis and optimization for hierarchical stiffened shells suffer from heavy computational costs. By smearing the minor stiffeners based on a novel numerical implementation of asymptotic homogenization (NIAH) method, a novel hybrid model for hierarchical stiffened shells is presented. Then, the high prediction accuracy and efficiency of the hybrid model for post-buckling analysis and imperfection sensitivity analysis are validated. Furthermore, a grid-pattern optimization framework of novel hierarchical stiffened shells allowing for imperfection sensitivity is established. The illustrative examples indicate that, the hierarchical stiffened shell with candidate sub-structures is more competitive in load-carrying capacity than that with the fixed grid-pattern, and the presence of imperfections would greatly affect the results of grid-pattern optimizations. [ABSTRACT FROM AUTHOR]

Vibration correlation technique (VCT) is an effective non-destructive buckling experimental technique for shell structures. In this study, VCT is studied from the point-of-view of being a buckling load numerical prediction method by numerically simulating the experimental procedure of VCT. Firstly, the formulas of VCT are introduced for axially loaded cylindrical shells and conical shells under the clamped–clamped boundary condition. According to the VCT formulas, the numerical procedure of VCT is provided. In order to accelerate the repeated eigenvalue analysis of VCT, the proper orthogonal decomposition (POD) method is integrated into VCT, and the POD-VCT is developed. Extensive examples are presented to verify the effectiveness of the proposed method, including unstiffened cylindrical shell with real measured imperfection, unstiffened conical shell with single perturbation load imperfection, composite cylindrical shell with eigenmode imperfection, and hierarchical stiffened cylindrical shell with combined imperfection. In comparison to buckling test results, high-fidelity explicit dynamic method and VCT method, the high prediction accuracy and efficiency of the proposed POD-VCT are fully demonstrated. Additionally, example results indicate the strong applicability of the proposed POD-VCT for various types of structural configurations, materials and imperfections. Above all, the POD-VCT is verified to be a fast buckling load numerical prediction method for imperfect shells. [ABSTRACT FROM AUTHOR]