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...

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The paper proposes a new approach to estimate the lower bounds of critical loads for circular cylindrical shells. These bounds are compared with the ordinary lower bound of critical load under which a shell with initial deflections loses stability. The lower bound produced by the approach is higher than the ordinary bound and can be used in design Introduction. Available results on the stability of shells indicate that the minimum lower critical load predicted by the nonlinear theory of shells is rather small. This issue is addressed in many studies, which together with their results are presented in [4, p. 120 (Table 7.1)]. It follows from this book that p = N cr /N cl (N cr and N cl are the real and classical values of the critical load for a smooth shell) is equal to 0.3 for w/t = 8, to 0.156 for w/t = 23, and to 0.07 for w/t = 236 (where w/t is the ratio of deflection amplitude to shell thickness). Values of p corresponding to many varied parameters and large values of w/t should be rejected in order for the question of whether the Donnell equations are applicable not to arise. Moreover, these values appear unreal because they exceed experimental deflections by many factors of ten.Other authors obtained the following acceptable values of this parameter: p = 0.308 by Kirste (from geometrical considerations) [5], p = 0.296 by Pogorelov [7], and p = 0.352 by Sachenkov [8]. Alekseev [1] was the first to show, using the method of successive approximations, that p = 3.87 t r / , i.e., in contrast to the above-mentioned results, this minimum load parameter depends on the shell thickness, which is qualitatively consistent with experiment.Almroth [9] long ago found this parameter in the form of a function p = f (d/d cl ), where d is the compressive strain of the shell; and d cl = s cl /E, s cl is the critical stress in the perfect shell and E is the elastic modulus of the shell material. The radial displacement was expanded into a double Fourier series, and diamond-shaped buckling modes were considered. The paper [9] shows curves of p min plotted with three (curve A), four (curve B), six (curve C), and nine (curve D) terms in the series. The value of p min changes from 0.3 to 0.1. The minimum value of p corresponds to experimental values for shells [10] tested on a rigid testing machine.Other authors recognized [10] that the postcritical minimum lower bound of critical load is too conservative to be used in designing. The final conclusion was drawn by Grigolyuk and Kabanov in [4, p. 129]: the lower critical load (as the absolute minimum of the lower loads of nonadjacent states) cannot be used as an estimate of shell strength. It might be added here that these authors meant the use of the estimate in engineering. This reputedly completes the study of the lower critical load. However, Alekseev, Grigolyuk, Kabanov, and Almroth [1,4,5,10] acknowledged that not all issues had been studied exhaustively.It is necessary to study in more detail the behavior of shells with various geometrical imperfections considering that the ...

By using the equilibrium equations in displacements, we develop an analytic method for the evaluation of upper critical loads in elastic cylindrical shells with transverse reinforcement. As a result, we deduce analytic expressions for the evaluation of critical stresses in momentless shells and propose a procedure for the evaluation of the lower limit of the load-carrying capacity of shells based on the application of the method of reduced stiffness. The numerical results are compared with the available theoretical and experimental data.Keywords: stability of reinforced shells, load-carrying capacity, lower limit, analytic method.Introduction. The methods used for the analysis of stability of reinforced shells are, as a rule, based on the application of analytic or numerical methods. As a significant advantage of analytic methods, one can mention the possibility of evaluation of critical loads by using simple analytic expressions.In the present work, we propose a simple approach in which the subcritical state of reinforced shells is regarded as momentless and homogeneous. The well-known procedure based on the application of equations of mixed form, the analysis of eight cases of the loss of stability, the determination of the minimum upper critical load, and the comparison of theoretical results with experimental data are presented in [1].Unlike [1], in what follows, we use the equations in displacements and improve the values of critical loads corresponding to 17 different cases of the loss of stability. The values of the upper critical loads obtained in [1] are compared with the values obtained according to the proposed method. We also present an estimate of the lower limit of the load-carrying capacity of reinforced shells and compare this estimate with the experimental data. The same approach is used in [2,3].For the analyzed shells, the so-called panel form of the loss of stability (i.e., the loss of stability in the skin of the panels between stringers and ribs) proves to be the main form of the loss of stability.Procedure of Numerical Analysis. In the present work, we generalize the procedure proposed in [4] to the case of investigation of stability of a cylindrical shell reinforced by stiffening ribs in two directions (Fig. 1). The strains and curvatures of the median surface of the skin are expressed via the components of displacements as follows: