Numerical and Analytical Approaches to the Stability Analysis of Imperfect Shells

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 ...

show abstract

“…Note that similar conclusions were earlier drawn, both theoretically and experimentally, in [3,15] for shells with single local dents.…”

supporting

confidence: 87%

On lower-bound estimates of critical loads for cylindrical shells

2006

Self Cite

View full text Add to dashboard Cite

show abstract

“…To this end, they are linearized. The technique is detailed in [20,21,42]. The resulting linearized system of differential equations has right-hand sides that have the form of the original nonlinear equations for the unknown dimensionless functions ϕ (force functions) and w (displacement functions).…”

Section: Numerical Technique For Subcritical and Stability Analyses Omentioning

confidence: 99%

Stability and load-bearing capacity of smooth and ribbed shells with local dents

Gavrilenko

2004

Int Appl Mech

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Its assumptions and hypotheses fully comply with the so-called classical theory of smooth shells and additionally allow for the discrete arrangement of ribs. The discreteness of ribs is incorporated differently, depending on whether mixed equations [4] or displacement equations [2,22] are used.The approach that does not assume that the subcritical state is momentless and homogeneous in both shells and ribs is based on the theory of imperfect thin-walled ribbed shells with an inhomogeneous moment subcritical state [5,6,12,13,15].To determine the upper critical loads, use is made of either monomial approximation of displacements [2, 4], or polynomial [3], or a solution that includes a fixed number of longitudinal half-waves and an arbitrary circumferential function f(y) [17,18,22], or solutions that do not restrict the behavior of the circumferential and longitudinal functions.The solutions obtained have certain differences that reflect features of the buckling of ribbed shells. …”

mentioning

confidence: 99%

“…The approach that does not assume that the subcritical state is momentless and homogeneous in both shells and ribs is based on the theory of imperfect thin-walled ribbed shells with an inhomogeneous moment subcritical state [5,6,12,13,15].…”

mentioning

confidence: 99%

Some features of the buckling of stringer shells

Gavrilenko

Matsner

2006

Int Appl Mech

Self Cite

View full text Add to dashboard Cite

A technique for stability analysis of stringer shells is proposed. It is used to analyze the minimum critical stresses. The dependence of the dimensionless parameters σ cr /σ cl on the number of stringers is plotted. The linear and nonlinear theories of ribbed shells are used to examine the features of how stringer shells lose stability. It is shown that the minimum critical stresses determined using the theory of ribbed shells and a structurally orthotropic model are close within the range of stiffness parameters considered Keywords: stringer shell, numerical and analytical solutions, minimum critical stress, stability Introduction. The accuracy of determining the critical loads in perfect shells is known to depend on both the design model used and the error of specific boundary-value solutions.The basic design approaches for momentless shells are based on: (i) structurally orthotropic model [1,7,8,11,21] or model of equivalent orthotropic shells [9, 10, 19];(ii) theory of elastic thin-walled ribbed shells with a homogeneous subcritical state [2,4]. Its assumptions and hypotheses fully comply with the so-called classical theory of smooth shells and additionally allow for the discrete arrangement of ribs. The discreteness of ribs is incorporated differently, depending on whether mixed equations [4] or displacement equations [2,22] are used.The approach that does not assume that the subcritical state is momentless and homogeneous in both shells and ribs is based on the theory of imperfect thin-walled ribbed shells with an inhomogeneous moment subcritical state [5,6,12,13,15].To determine the upper critical loads, use is made of either monomial approximation of displacements [2, 4], or polynomial [3], or a solution that includes a fixed number of longitudinal half-waves and an arbitrary circumferential function f(y) [17,18,22], or solutions that do not restrict the behavior of the circumferential and longitudinal functions.The solutions obtained have certain differences that reflect features of the buckling of ribbed shells. Examples of Nonlinear and Linear Stability Analysis of Stringer Shells.Let us analyze the results of calculation of simply supported stringer shells described in [22]. For these shells, the following geometrical parameters and material constants are set: r = 100 mm; l = 300 mm; t = 0.25 mm; ν = 0.3; E 0 = 5.5⋅10 9 N/m 2 ; and E s = 2.9⋅10 9 N/m 2 . The stringers have a rectangular cross section: 2.5×6.25 mm (type 1); 3.75×6.25 mm (type 2); and 5.0×6.25 mm (type 3). The dimension 6.25 mm is measured in the circumferential direction. The stringers are attached to the inside surface of the shell. Their number was different (n s = 6, 9, 12, 18). Figure 1 shows the dependence of the dimensionless critical stress σ cr /σ cl , σ cl = 0.605Et/r, on the number of stringers. The dotted curves in Fig. 1a, b have been plotted based on the data from [22]. The numbers 1, 2, and 3 stand for the type of stringers. The solid curves have been plotted using the technique from [7,16]. It can be seen that the solid...

show abstract

Numerical and Analytical Approaches to the Stability Analysis of Imperfect Shells

Cited by 26 publications

References 16 publications

On lower-bound estimates of critical loads for cylindrical shells

On lower-bound estimates of critical loads for cylindrical shells

Stability and load-bearing capacity of smooth and ribbed shells with local dents

Some features of the buckling of stringer shells

Contact Info

Product

Resources

About