Analytic upper-bound estimates for the critical loads of perfect ribbed 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 ...

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“…Methods for analytic determination of upper bounds for critical loads are outlined in [16][17][18][19][20].…”

Section: Possible Shapementioning

confidence: 99%

On lower-bound estimates of critical loads for cylindrical shells

2006

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“…2) have been plotted based on the data from [3], and the dashed lines have been obtained using the technique [7,16]. Figure 2a, b represents shells with γ = 0.2 and γ = 0.6, respectively.…”

Section: Linear Analysis Of Different Cases Of Bucklingmentioning

confidence: 99%

“…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 and dotted curves (Fig.…”

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Some features of the buckling of stringer shells

Gavrilenko

Matsner

2006

Int Appl Mech

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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...

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“…The branching of solutions to equations of the nonlinear theory is numerically analyzed in [20]. Using these approaches would refine the lower bounds for critical loads.The objective of the present study is to show that the potentialities of structurally orthotropic theory have not yet been exhausted.An analytic approach to estimate the minimum upper bounds of critical loads is outlined in [15]. In [16], a procedure to determine the upper and lower bounds of critical loads is set forth and formulas to calculate the upper bounds for critical stresses and to evaluate the load-bearing capacity of shells are derived,…”

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Influence of axisymmetric dents in ribbed shells on minimum critical loads

Gavrilenko

Matsner

2007

Int Appl Mech

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The effect of axisymmetric dents in ribbed shells on the minimum critical loads is studied analytically. The upper and lower bounds for the critical stresses in imperfect cylindrical shells reinforced with stringers, rings, and both are estimated. The upper bounds are compared with those obtained from the known solutions for perfect ribbed momentless shells and with experimental data. The effect of the amplitude of initial dents and their number on the upper and lower bounds of critical stresses is examined. The procedure used is the most efficient to determine the load-bearing capacity of ring-reinforced and ribbed shells Keywords: ribbed shells, dents, minimum critical stresses, upper and lower bounds Introduction. Methods for analytic and numerical stability analyses of smooth and ribbed cylindrical shells are outlined in [4-18, etc.]. Early studies on stability of reinforced shells employed a structurally orthotropic scheme [1, 8-10, 21-23, etc.], examined only the general case of buckling, and predicted only one value of critical load. Comparing these results even with the prediction by the linear momentless theory of perfect ribbed shells turns out not to support them [3]. Solutions of nonlinear problems are described in [18,19]. The branching of solutions to equations of the nonlinear theory is numerically analyzed in [20]. Using these approaches would refine the lower bounds for critical loads.The objective of the present study is to show that the potentialities of structurally orthotropic theory have not yet been exhausted.An analytic approach to estimate the minimum upper bounds of critical loads is outlined in [15]. In [16], a procedure to determine the upper and lower bounds of critical loads is set forth and formulas to calculate the upper bounds for critical stresses and to evaluate the load-bearing capacity of shells are derived,

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Analytic upper-bound estimates for the critical loads of perfect ribbed shells

Cited by 6 publications

References 6 publications

On lower-bound estimates of critical loads for cylindrical shells

On lower-bound estimates of critical loads for cylindrical shells

Some features of the buckling of stringer shells

Influence of axisymmetric dents in ribbed shells on minimum critical loads

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