A method for analysis of the stability and load-bearing capacity of imperfect smooth and ribbed shells is developed. This method is based on the finite-difference method and is implemented as an algorithm for fast calculation of critical forces, as opposed to the finite-element method. The theoretical results discussed include both early and recent results. The emphasis is on shells with local dents. The numerical results are successively corrected and compared with available experimental data for shells with a single dent and with other data. The method enables us to discover new features in the behavior of thin-walled structures under loading: development of precritical state, change in the dent shape, and exhaustion of load-bearing capacity. The lower local critical loads and upper stresses are determined. They correspond to general buckling and agree well with available experimental data.Introduction. Methods combining numerical and classical analytical approaches are of special interest in the theory of shells. This is pointed out in [50, etc.].The present paper briefly describes a numerical approach to the analysis of the stability and load-bearing capacity of imperfect shells. If theoretical solutions are unavailable, we will use experimental data to justify the reliability of the method. Solutions that could improve results obtained within the framework of shell theory can be derived from the three-dimensional theory of deformable bodies and its applications [28][29][30]48].Well-known theoretical studies into the stability of shells with a local dent did not addressed the issue of load-bearing capacity of such shells. Such studies are discussed in Sections 1-3.Background Information and Problem Statement. The new approach and the theoretical results compared with experimental data are described in Sections 4-6. The strength loss process of real thin-walled shells includes, at least, three stages: subcritical, critical, and postcritical. These stages are shown graphically in [40].Consider the following approach to the buckling analysis of shells: (i) analyze the behavior of imperfect shells with a local dent (generally dents) growing in amplitude;(ii) determine the critical (lower) load under which the originally nonloaded dent transforms into a new (loaded) dent; (iii) study the behavior (whether stable or not) of the shell with new dent, which subsequently turns into the postcritical state, using nonlinear shell theory and the generalized mixed equations describing the entire process at all the three stages.The theoretical results are supported by experimental data [33] on stability of shells with dents. Smooth cylindrical shells with a single local dent were the subject of detailed theoretical and experimental research. The experiments involved steel specimens. All the shells were made by the same method and tested under identical conditions. The theoretical analysis was based on the mesh method and nonlinear shell theory. The experimental and theoretical critical forces were compared for shells

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A new approximate approach is proposed to find upper-bound estimates for the critical loads of ribbed shells. Seventeen cases are considered, and the minimum critical load parameter is determined Keywords: new approximate approach, upper bound, ribbed shell, analytical solution Introduction. The following approaches are used to develop methods for stability analysis of reinforced thin-walled shells.1. Structurally orthotropic model of a momentless shell where the stiffness characteristics of the ribs are uniformly distributed over the casing [1, 13, 18, 20, etc.].2. Theory of perfect elastic thin-walled ribbed shells, with momentless subcritical state being homogeneous. The assumptions and hypotheses of this theory completely coincide with the assumptions of the classical theory of stability for smooth shells. The only difference is that the discreteness of ribs is taken into account. The basic results and methods are described in [2,4,5].3. Theory of imperfect thin-walled ribbed shells, with moment subcritical state being inhomogeneous [8,9,[14][15][16][17]. 4. The shell and ribs are regarded as solids, i.e., in a three-dimensional formulation. This is the most complex approach. The general theory and stability problems for beams and plates are addressed in [10][11][12]19].The first two approaches are convenient in that they allow us to develop analytic methods of shell design. The third and fourth approaches require the use of numerical methods such as the finite-element and finite-difference methods.Many studies employ the first and second approaches and then compare the results they produce. The first approach analyzes the general buckling mode, and the second approach introduces so-called special cases of buckling. From this we obtain various applicability conditions for the theory of structurally orthotropic shells, and this theory is discarded, as a rule. It is shown in [4] that with a small number of rigid ribs the polynomial approximation of displacements yields results that are substantially different from those obtained under the theory of structurally orthotropic shells.We will demonstrate here that comparing only the general case of buckling in the first approach with special cases in the second approach leads to significant differences in estimates of minimum critical loads. If in using the structurally orthotropic model and the monomial approximation of displacements (or unknown functions) we consider the special cases of buckling from the second approach, then the results produced by the theory of structurally orthotropic shells draw closer to those obtained under the theory of ribbed shells. The minimum critical loads obtained in the examples below differ by no more than 11% from those predicted by the second approach. However, the theory of ribbed shells is the only accuracy criterion for solutions to problems of specific classes. It is in particular used to validate the results from the polynomial approximation of displacements [3, 6, 7].1. Analytic Calculation Technique. Let us analyze the sta...