Some features of the buckling of stringer shells

Abstract

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...

On lower-bound estimates of critical loads for cylindrical shells

“…The second value is an upper bound of p min . Figure 5 shows curves 8-11 of the third family with the following shapes: n = 2, 4, 6, 8, 12, 18, 24, 10, 20, 30, 40 (curve 8); n = 2, 4, 6, 8, 12, 18, 24, 10, 20, 30, 40, 14, 18, 42, 56 (curve 9); n = 2, 4, 6,8,12,18,24,10,20,30,40,14,28,42,56,36, 54, 72 (curve 10); and n = 2, 4, 6, 8, 12, 18, 24, 10, 20, 30, 40, 14, 28, 42, 56, 36, 54, 72, 22, 44, 66, 88 (curve 11).…”

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

Stability and Approximate Analytical Periodic Solution of a Structurally Orthotropic Stringer Shell

“…For the case of a cutout in the shell, Palazotto [1] performed a bifurcation and collapse analysis of the stiffened cylindrical shell. In recent years, when the stringer buckles, Gavrilenko and Matsner [2] used the linear and nonlinear theory of the ribbed shell to examine how the stringer shells lose their stability characteristics. Schilling and Mittelstedt [3] derived a clear analytical formula for critical buckling loads by using the Ritz-like principle of minimum potential energy.…”

Critical loads of shells with high-modulus reinforcement

“…Ribs made of a high-modulus material can enhance the stability of shells by a factor of tens Keywords: upper and lower critical loads, analytical method, high-modulus material, imperfections Introduction. Methods for analytic and numerical stability analysis of smooth and reinforced shells are outlined in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Early publications on stability of reinforced shells employed structurally orthotropic approaches, examined only one general case of buckling, and determined only one critical load [1,9,10,[23][24][25][26][27].…”