Stiffeners are secondary structural parts attached to main structure in order to make it stiffer, or more resistant to deformation, under bending loads. Increasing wall thickness is a well-known approach to improve the flexural stiffness and strength, but the same stiffness and strength can be obtained by using stiffeners with much less use of material. Stiffeners are commonly used to provide extra rigidity in such systems where weight plays a critical role such as rocket launch systems, aircraft engines, and aircrafts. Apart from stiffening the whole structure, sometimes stiffeners are only used for locally increasing rigidity especially around cutouts that may significantly decrease the buckling strength of the structure. For example, wind-turbine towers have a considerably large hole used as entrance and stiffeners need to be used to recover the loss in the load-carrying capacity of the structure due to the cutout. In weight critical applications, the topography and size of the stiffeners should be judiciously chosen to make the best use of material. For this purpose, a systematic methodology should be adopted to find the optimum stiffener configuration.
1.1. Literature Review
A number of researchers [1–24] studied the buckling behavior of stiffened cylindrical shells made of metals. Some of them [1–17] investigated the influence of stiffener geometry on the buckling strength of cylindrical shells made of steel or aluminum with no cutouts under axial compressive loading. Some optimized [7–17] the design of the stiffeners on the shell surface. Weller and Singer [1] experimentally studied the effect of stiffeners on the buckling behavior. Cylinders stiffened by stringers having different cross-sections and different spacing were tested. All stiffened shells provided larger buckling resistance compared to equal-weight uniform shells. Hotala and Skotny [2] experimentally and numerically investigated buckling strength of steel silos having stiffeners with different shapes and lengths. The results showed that use of short ribs interconnected with a circumferential ring highly decreased meridional stresses and increased load-carrying capacity of such structures. Hui-shen et al. [3] conducted a theoretical study on the buckling and post-buckling behavior of stiffened shells based on the boundary layer theory. Zhu et al. [5] presented a new design for a ring-type stiffened shell with a combination of large and small rings having T and L-shaped cross-sections. By optimizing the height of the stiffeners and the span of T-shaped stiffener rings, the buckling load-carrying capacity was maximized. Foryś [7] optimized the internal diameter and the number of rings of ring-stiffened shells to maximize the critical buckling load by using modified particle swarm optimization method. Constraints were imposed on the volume of the material and the stability of the post-buckling behavior. Bagheri et al. [8] also obtained optimum design of a ring-stiffened shell using genetic algorithm (GA). The design variables were shell thickness, number of stiffeners, stiffener dimensions, and spacing between the stiffeners. The results showed that the optimized design had lower structural weight, higher natural frequency, and higher buckling load carrying capacity. Sadeghifara et al. [9] conducted a multi-objective optimization of a cylindrical shell stiffened by orthogonal grids (stiffeners in the axial and circumferential directions) to maximize the buckling load and minimize the weight. Optimization was performed by varying thickness, width, and number of stiffeners, and spacing for rectangular, C, I, and hat-shaped stiffener sections. The I-section and the rectangular section proved to be the most and least effective geometry on the buckling strength, respectively. Wang et al. [11] maximized the load carrying capacity of cylindrical shells stiffened with different grid patterns by varying the shell thickness and the heights and number of axial and circumferential stiffeners. They considered four different types of hierarchical stiffener patterns, which included major and minor stiffeners having different dimensions. Zhao et al. [13] and Tian et al. [14] similarly, considered a hierarchical grid pattern having a primary grid and a secondary grid with a smaller cell size. The design variables were chosen as the width and height of the primary and secondary group of stiffeners and their numbers.
Cutouts significantly reduce the buckling strength of cylindrical shells [22, 23]. Reduction in strength significantly depends on the size of the cutout. Starnes [24] found that the buckling load of a cylinder with a circular hole depended on a parameter proportional to the hole radius divided by the square root of the product of the shell radius and thickness. Detrimental effects of cutouts can be offset by making use of stiffeners. Several researchers [18–21] investigated the effects of stiffeners on the buckling behavior of cylinders having cutouts and some of them [20, 21] optimized their geometry. Grazijahani et al. [18] studied the buckling behavior of cylinders with a cutout similar in form to wind turbine entrance doors. Steel tubes with rectangular, elliptical, oval- shaped cutouts of different sizes at different positions on the shell surface were tested to determine their effects on load carrying capacity. Then, two stringers with rectangular cross section were welded close to the rectangular cutout. It was found that the stiffened shell had buckling load–carrying capacity %33 higher than that of the unstiffened cylinder. Alsalah at al. [19] numerically investigated various stiffener configurations around cutouts and tried to recover buckling load-carrying capacity lost due to the opening. They considered cylindrical shells made of steel with rectangular cutouts and determined the influence of the shell thickness, cutout dimensions, and location on buckling load. The results showed that the cutout shape had minimal influence on the buckling load, whereas the size had much more influence. Besides, it was found that the frame ring configuration, where stiffeners were placed all around the cutout could fully recover the lost capacity. Hao et al. [20] minimized the weight of an aluminum cylindrical shell with rectangular and circular cutouts using a genetic algorithm. They subdivided the structure into panels and used a different stiffener pattern on the panels with a cutout. The design variables were the dimensions of the stiffeners, the shell thickness, and the number of stiffeners for each sub-panel with a constraint on the collapse load. In a similar study, Hao et al. [21] considered an aluminum cylindrical shell in a launch vehicle with multiple cutouts. They used curvilinear stiffeners around the cutouts and straight ones away from them. The objective of the optimization was to increase the collapse buckling load with a constraint on the mass. The design variables were the number of the axial and circumferential curvilinear stiffeners, shell thickness, stiffener dimensions, and the parameters defining the positions of the curvilinear stiffeners.
In early studies [7–14], researchers were mainly interested in the optimization of stiffener dimensions such as height and width and the stiffeners were applied uniformly over the entire shell surface. Besides, only few studies [18–21] considered geometrical imperfections like cutouts on shell surface and conducted optimization. In recent years, hybrid stiffener models with varying stiffener dimensions have gained importance [17, 20, 21], because more weight reduction is thus possible, but the same stiffener pattern was applied either on the entire surface or the sub-panels of the shell structures. In the present study, on the other hand, the stiffener pattern on the entire lateral surface and the stiffener dimensions are optimized.
It is worth pointing out that due to limitations of the convectional manufacturing techniques and weldability, it was not possible to produce such complex stiffener patterns on shell surfaces in the past. Now, with the help of the latest manufacturing technologies such as 3D printers, it is possible to produce complex metal structures; thus, any stiffener configuration is feasible for weight reduction and better structural performance especially for the aviation industry.
It should be noted that there were a great number of studies aimed to increase buckling resistance of stiffened cylindrical composite panels by optimizing stiffener cross-section, pattern, ply angle, material properties, panel thickness etc. However, because only metal cylindrical shell structures are considered in this study, the literature on the optimization of stiffened composite structures is not discussed.
1.2. Problem Statement and Objective
In this study, a thin-walled cylinder containing two holes is considered as shown in Fig. 1. The cylinder is subjected to an axial compressive force. For this reason, it is liable to buckling failure. It is well known that holes may cause significant decrease in the buckling strength of structures. Increasing thickness is not an effective way of improving the buckling strength in weight-critical applications compared to the use of stiffeners. For these reasons, the goal of this study is to find the optimum layout of the stiffeners that maximizes the buckling strength of the cylindrical shell structure with cutouts. The stiffeners are not modeled as separate structural parts added to the main cylindrical shell structure, but as integral parts of the main shell with locally increased shell thickness. In this way, the optimization problem is posed as finding optimum thickness distribution of a 2D structure, i.e. topography optimization. The optimization is achieved in two levels. First, topography of the stiffeners is optimized. In this way, the optimum stiffener layout is determined. In the second level, using the stiffener pattern determined in the first level, heights of the stiffeners and the shell thickness are optimized.