## 3 Stiffened Shell Structures
### Mechanics of Ring Stiffened Cylinder and Numerical Approaches
Additionally, the internal stiffeners favor the formation of checkerboard buckling, whereby positive eccentricity reduces the buckling load compared to the centrally stiffened cylinders. However, with outside stiffeners, eccentricity has almost no influence, as the cylindrical shell tends to fail with an axisymmetric buckling shape. Based on experimental observations, [PERSON] [28, 29, 30, 31] developed the \"smeared stiffener theory\", that is, the smeared consideration of the stiffeners in a calculation model (Figure 5). The basis of this simplification is the assumption that the stiffeners are so close together that no local buckling can occur, and thus, global failure becomes the design criteria. [PERSON] and [PERSON] [49] investigated the applicability of the \"smear stiffener theory\" for stiffened cylinders under internal pressure, constant temperature variation, and axial compression, as well as a proposed improvement to consider the shear transfer between the stiffener and the shell. They examined the interaction of shear and normal forces using representative shell sections with shells, rings, and longitudinal stiffeners. [PERSON] et al. [50] proposed a modified theory of applied stiffeners for the calculation of cross-stiffened shells by determining the bending stiffness and the coupling stiffness of the stiffening ribs and the shell to account for the interaction between the stiffeners and the shell. [PERSON] et al. [51] determined the global buckling loads of cross-stiffened shells using the \"smear stiffener theory\" and based on the energy method to determine the bifurcation loads for selected stiffener configurations and compared them to experiments on composite cylinders. [PERSON] et al. [52] attempted to improve the prediction of buckling loads using the \"smear stiffener theory\" based on their research results. Using a hybrid optimization algorithm, they showed the potential savings of the stiffened shells in terms of weight. [PERSON] et al. [53] proposed a numerically-based \"smear stiffener theory\". A representative section was considered, and a numerical implementation of asymptotic homogenization was employed. In this way, they were able to determine the equivalent stiffness coefficient very precisely and to use the Rayleigh-Ritz method to determine the critical load of the stiffened shell structures. However, due to the complexity of the stiffened cylindrical shell structure, the simplified analytical model of the stiffened cylindrical shell still needs to be improved in the future. With advances in computing and finite element theory, the numerical method can theoretically precisely analyze the complex nonlinear problems of shell stability by ignoring the geometrical complexity of the shell structures. On the basis of the recent efforts of researchers and engineers over a long period of time, the FEM program has become the most effective and popular tool for analyzing cylindrical shell buckling problems in recent years. A numerical method is an analytical tool aimed at researchers as well as engineers. They can use commercial finite element software, such as Abaqus(r) and Ansys(r) FE software, for structural design. However, some researchers are constantly striving for further improvements in numerical algorithms to solve specific problems. For analyzing the buckling of the stiffened cylindrical ring shell under axial compression, the time-dependent explicit dynamic method [54; 55] has high robustness and is suitable for buckling and post-buckling analysis at the expense of time and computing efficiency [56]. [PERSON] and [PERSON] et al. [43; 57] attempted to determine whether the shell reached the bifurcation load using an iterative algorithm based on the accompanying eigenvalue method [58] in Abaqus(r). This approach enables us to quickly obtain the ultimate load capacity of a stiffened cylindrical shell structure (Figure 6). Figure 5: Schematic representation of the smeared stiffener theory from [PERSON] [29].