Stability of Cylindrical Composite Shells under Torsion

Abstract

The stability of fiber-reinforced cylindrical shells under torsion is analyzed in the case where the principal directions of elasticity in the layers do not coincide with the coordinate directions. The solution to the linearized equations of the technical theory of anisotropic shells is obtained in the form of trigonometric series. It is shown that for some reinforcement configurations the critical loads may depend on the direction of the torsional moment. It is also established that the minimum (in absolute value) eigenvalue does not always correspond to the critical load. This fact should be taken into account not only in the case of torsion but also in more complicated cases of loading Keywords: stability, torsion, cylindrical shell, composite with one plane of symmetry, critical loads Introduction. Problems of stability of cylindrical shells under torsion differ from those under loading of other types in both formulation and solution methods. It is obvious that the direction of the torsional moment is of no importance for isotropic and orthotropic shells. The mathematical solution should account for the fact that the absolute value of the critical load is also independent of the torsion direction, but its sign may be either plus or minus. The buckling of a shell is only possible under compressive loads of one sign such as axial load or pressure. This fact is also incorporated into the formulation of the eigenvalue problem. It is for these two types of loading that the most accurate solutions of stability problems have been obtained [2,4,8]. Less attention was given in the literature to the problem of stability under torsion [4,8,9]. Among the available analytic solutions, the most accurate seems to be Batdorf's solution to the stability problem for an isotropic cylindrical shell under torsion [8,9]. Its characteristic equation has the critical load parameter raised to the second power. If we determined its minimum value, it would become possible to calculate two critical values of the torsional moment differed only by sign. There are approximate analytic solutions for orthotropic shells [4]. For materials of lower order of symmetry, no satisfactory analytic solution has been found yet.In this paper, we solve the stability problem for cylindrical shells filament-wound in directions different from the coordinate axes. As in [9], the unknown functions are expanded into trigonometric series satisfying the hinged-support conditions. It should be noted that the numerical method proposed in [12,13] for shells with one plane of symmetry also applies to the case of loading under consideration. The importance of allowing for low orders of symmetry can be judged from [6, 10], wherein it was established that not only quantitative but also qualitative effect of this factor on the natural vibrations of anisotropic bodies is significant.1. Consider shells with an odd number of homogeneous anisotropic layers that have at each point only one plane of elastic symmetry parallel to the mid-surface. From the formal viewpoint, the...

Stability of axially compressed cylindrical shells made of reinforced materials with specific fiber orientation within each layer

“…However, they are rather difficult to solve by the technique in question, which hinders parametric analysis. The analytic solution, found in [7], to the stability problem for anisotropic cylindrical shells under torsion is free from this disadvantage, but is applicable only to one class of shells and boundary conditions. We will use such a solution below to analyze the typical features of buckling of anisotropic laminated cylindrical shells made of materials with one plane of symmetry and subjected to axial compression.…”

“…For composites with one plane of symmetry, we can derive from (1.20) one matrix equation that additionally includes λ. This means that the critical loads of shells made of such materials depend on the direction of twisting moment [7].…”

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).…”

Solving the torsion problem for an anisotropic prism by the advanced Kantorovich–Vlasov method

“…In the latter case, all the stresses are nonzero yet constant throughout the height, the prism axis bends, and the cross-sectional warping linearly varies throughout the height. This case of torsion is observed in materials with general anisotropy or anisotropic materials with one plane of elastic symmetry nonorthogonal to the prism axis.The classical torsion problem was generalized in different ways such as inclusion of large strains or stress concentration around cracks, analysis of stability and vibrations under various torsional loads, complication of the body shape, material properties (viscosity), and loading and boundary conditions [10][11][12][13][14][15]17].The present paper addresses the torsion problem for a rectangular prism made of anisotropic materials with low order of symmetry and the stress-strain state varying throughout the height. Use will be made of the general problem statement in linear elasticity.…”

“…The classical torsion problem was generalized in different ways such as inclusion of large strains or stress concentration around cracks, analysis of stability and vibrations under various torsional loads, complication of the body shape, material properties (viscosity), and loading and boundary conditions [10][11][12][13][14][15]17].…”